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is therefore automatically H-equivariant, which completes the proof. 0 EXAMPLE 1.5.13. Consider a Riemannian manifold N of dimension n. By example (iii) of 1.5.1, N carries a canonical Cartan geometry of type (Euc(n) , O(n)). Now take the subgroup O(n-l) C O(n) and consider the associated correspondence space eN. We can realize O(n - 1) as the stabilizer of a unit vector in ]R.n and, as in 1.1.1, identify the homogeneous space O(n)jO(n - 1) with the unit sphere sn-l C ]R.n. Now the associated bundle 'P xO(n) ]R.n is the tangent bundle TN. Hence, eN = 'PXO(n)sn-l can be identified with the unit sphere bundle SN c TN of all tangent vectors of length one. By the proposition, we obtain a natural Cartan geometry 'P ~ SN of type (Euc(n), O(n - 1)) on the unit sphere bundle. Realizing Euc(n) as a matrix group as in 1.1.2 and O(n - 1) C O(n) as the stabilizer of the first standard basis vector in ]R.n, we get Euc(n) =
{G
~): v E ]R.n,A E o(n)} ,
and the subgroup O(n - 1) corresponds to the matrices in which v = 0 and A is of the form (~ g). Looking at the Lie algebras, we see that, as an O( n - 1)-module, we have euc(n) = ]R. EEl]R.n-l EEl]R.n-l EEl o(n - 1), with o(n) corresponding to the last two summands. The first three summands provide us with an O(n - I)-invariant complement n to o(n - 1) c euc(n). Using the standard inner products on these three summands, we obtain an O(n-l)-invariant inner product on n, which induces a canonical Riemannian metric on SN. By construction, TSN decomposes into the orthogonal direct sum of three sub bundles. By part (3) of the proposition, the last summand is the vertical subbundle of SN ~ N. The other two summands constitute the horizontal subbundle for the Levi-Civita connection (with the lifted metric). This decomposes further into the line subbundle formed by multiples of the foot point and its orthogonal complement. 1.5.14. Characterization of correspondence spaces. We continue working in the setting of a Lie group G with closed subgroups K c H c G. In 1.5.13 we have shown how to associate to a Cartan geometry of type (G, H) a Cartan geometry of type (G, K) on the correspondence space. Now we want to characterize Cartan geometries of type (G, K) which are locally isomorphic to correspondence spaces. Let (p : 'P ~ M, w) be a Cartan geometry of type (G, K). As in part (3) of Proposition 1.5.13, the K-invariant subspace ~jt c gjt determines a smooth subbundle V M c TM. In the case of a correspondence space eN, this bundle becomes the vertical subbundle of the projection eN -7 N, so, in particular, it must be involutive. If M is locally isomorphic to a correspondence space, then this isomorphism is compatible with the subbundles, so V M must be involutive, too. We can characterize involutivity in terms of the torsion of the Cartan connection w. Recall from 1.5.7 that the torsion T E S)2(M, TM) of w is obtained by applying the projection II: AM ~ TM to the values of the Cartan curvature /'i, E S)2(M,AM). LEMMA 1.5.14. Let (p: 'P -7 M,w) be a Gartan geometry of type (G,K) with torsion T E S)2(M,TM), and let VM c TM be the sub bundle corresponding to ~jt c gjt. The VM is integrable if an only ifT(VM, VM) c VM.
102
1.
CARTAN GEOMETRIES
PROOF. Let ~ and TI be local sections of V MeT M, and choose local lifts ~,ij E X(P). Then [~, ij] is a local lift of the Lie bracket [~, TI]. Thus, we have to check whether Tp . [~, ij] lies in V MeT M. Since the identification of T M with P XK (gft) is obtained from (u, X + t) f-> Tup' w(u)-l(X), this is the case if and only if w([~, ij]) has values in ~ C g. The assumptions that ~ and TI are sections of V M likewise is equivalent to the fact that w(~) and w(ij) have values in ~ C g. In this case, also ~. w(ij) and ij. w(~) have values in~. Hence, we see that [~, TI] E r(V M) is equivalent to dw(~, ij) having values in ~. Since ~ is a Lie subalgebra, also [w(~), w(ij)] automatically has values in ~, so we can equivalently replace dw(~, ij) by K(~, ij). But this having values in ~ is equivalent to T( ~, TI) having values in the sub bundle of T M corresponding to ~~. 0 Suppose that the geometry (p : P -) M, w) of type (G, K) satisfies this necessary condition for being locally isomorphic to a correspondence space. Then we can actually construct a candidate for a space N such that M may be locally isomorphic to CN. Namely, in the case of a correspondence space, N is simply the (global) space of leaves of the foliation corresponding to the subbundle VCN. Returning to M, we have to consider spaces which locally parametrize the leaves of the foliation defined by V MeT M. Due to the origins of this whole circle of ideas in twistor theory, such spaces are called local twistor spaces for M. DEFINITION 1.5.14. Let (p: P -) M,w) be a Cartan geometry of type (G,K) such that the subbundle V MeT M is integrable. Then a (local) twistor space for M is a local leaf space for the foliation defined by V M, i.e. a smooth manifold N together with an open subset U C M and a surjective submersion '¢ : U -) N such that ker(Tx'¢) = VxM for all x E U. Existence of local twistor spaces follows immediately from the local version of the Frobenius theorem (see [KMS, Theorem 3.22]) by projecting onto one factor of an adapted chart. Note that for two local twistor spaces '¢i : Ui -) Ni there is a unique diffeomorphism ¢ : '1PI (U1 n U2 ) -) '¢2(U1 n U2 ) such that ¢ 0 '¢l = '¢2' We know already that correspondence spaces satisfy a much stronger curvature condition than the one from the lemma, since by part (3) of Proposition 1.5.13 we must have ie/'i, = 0 for any section ~ of VM c TM. Surprisingly, this curvature condition is actually equivalent to local isomorphism to a correspondence space: THEOREM 1.5.14. Let (p: P -) M,w) be a Cartan geometry of type (G,K) with curvature /'i,. Suppose that i~/'i, = 0 for all ~ E r(V M). Then for any sufficiently small local twistor space '¢ : U -) N of M, one obtains a Cartan geometry of type (G,B) on N such that (p-l(U),wl p -l(U)) is isomorphic to an open subspace in the correspondence space CN. If B f K is connected, then this Cartan geometry is uniquely determined. PROOF. The composition '¢ 0 p : p-l(U) -) N is a surjective submersion, so it admits local smooth sections. Choosing U sufficiently small, we may therefore assume that there is a global smooth section (7 : N -) p-l (U) of ,¢op. In terms of the curvature form K E n 2 (p,g), the condition on /'i, implies 0 = K(w-1(A),w-1(B)) for all A,B E ~ c g. This can be written as 0 = -w([w-1(A),w-1(B)]) + [A,B], which means that A f-> w-1(A) defines a Lie algebra homomorphism ~ -) X(P), i.e. an action of ~ on P. By Lie's second fundamental theorem (Lemma 1.5.11), this
1.5. CARTAN CONNECTIONS
103
Lie algebra action integrates to a local group action. There is an open neighborhood -+ P such that • F(u,e) = u and !tlt=oF(u,exp(tA)) = w-I(A)(u) for all u E P and all A E I) . • F(F(u,g), h) = F(u,gh) provided that (u,g), (u,gh) and (F(u,g), h) all lie in W. Possibly shrinking the leaf space further, we find an open neighborhood V of e in H such that (a( x), g) E Wand (F( a( x), g), e) E W for all x E N and all 9 E V. Then we define 4> : N x V -+ P by 4>(x,g) := F(a(x),g). For x E N the tangent map T(x,e)4>: TxN x I) -+ Ta(x)Q is evidently given by (~, A) ~ Txa· ~ + w-I(A)(a(x)), so it is a linear isomorphism. Possibly shrinking U and V, we may assume that 4> is a diffeomorphism onto an open subset U c P, and we arrive at the following picture: W of P x {e} in P x H and a smooth map F : W
N x V ~p-I(U)~p
lp~!p Nt
(j(u))(Tuj· ~ + (A (j(u))) :=
a(w(u)(~))
+A
for A E e, and verifies that it is well defined using property (i) of a. By definition, we see that We> (j (u)) reproduces the generators of fundamental vector fields. Suppose that We> (j(u))(Tuj . ~ + (A (j(u))) = O. Projecting to lie, we see that
o=
a(w(u)(~))
+ e=
~(w(u)(~)
+ ~).
By condition (iii) on a, this implies that w(u)(~) =: X E ~, i.e. ~ = (x(u). But this means that our original vector was vertical, and by construction w'" (j (u)) is injective on vertical vectors. Thus, We> (j (u)) is a linear isomorphism. As in the proof of Theorem 1.5.6 we then define
w(j(u) . k)("7) = Ad(k- 1 )(w(j(u))(Tr k - 1 • "7)) and verify that this is well defined using property (ii) of a. Having defined We> E 0 1 (P Xi K, I), we see that by construction the value in each point is a linear isomorphism, j*we> = aow and w'" is uniquely determined by this property. Equivariancy of We> immediately follows from the construction. Finally, the fact that w'" reproduces generators of fundamental vector fields follows by equivariancy from the fact that We> (j (u)) has this property. 0 Fixing the data (i, a), which are equivalent to a G-invariant Cartan geometry of type (L, K) on G I H, we can now associate to each Cartan geometry of type (G, H) a Cartan geometry of type (L, K). In fact, this construction is functorial: THEOREM 1.5.15. Let G and L be Lie groups with Lie algebras I} and I, and let H c G and K c L be closed subgroups. Fix a homomorphism i : H --> K and a linear map a : I} --> I with properties (i)-(iii) from the proposition. Then mapping (P --> M,w) to (PxiK --> M,we» defines a functor from Cartan geometries of type (G, H) to Cartan geometries of type (L, K). Passing from (i, a) to (i, &) as described in the proposition, one obtains a naturally isomorphic functor. PROOF. As in the proof of part (2) of Theorem 1.5.6, a morphism q> : (P --> M,w) --> (1' --> it,w) induces a principal bundle map F(q» : P Xi K --> l' Xi K. Since q> is a local diffeomorphism, the same is true for F(q» and hence F(q»*we> is a Cartan connection on g Xi K. Then the fact that F(q»*we> = W'" follows as in Theorem 1.5.6, and functoriality is obvious. Suppose that i(h) = koi(h)ko1 and & = Ad(ko) 0 a. For a fixed geometry (P --> M,w) of type (G, H), the map P X K --> P X K defined by (u, k) 1-+ (u, kok) induces an isomorphism P Xi K. Denoting by j and 3 the two inclusions, this satisfies rko 03 = j(c(t)) . g(t) for some smooth function 9 : 1-- G. From the proof of Theorem 1.5.17 we see that deve(t) = [u, g(t)-1 ·0] for all tEl. In particular, the curve t I--> g(t)-1 ·0 belongs to C. Now take tl E I, and consider the curve t I--> j(c(t + td) . g(t + tdg(tl)-1 in P. This is obtained by the principal right action of a fixed element of G on a horizontal curve, so it is horizontal, too. Its value in t = 0 is j(C(t1)) E j(P) and it lifts the curve Ctl. By Theorem 1.5.17 the development of Ctl is, locally around zero, represented by t I--> [c(td, g(tl)g(t +td- 1 ·0]. But since g(t)-l ·0 lies in C, by admissibility g(tdg(t + td- 1 ·0 also lies in C. Since tl E I is arbitrary, the result follows. 0 PROOF.
c : I -- P
Together with Theorem 1.5.17 we see that the structure of the local canonical curves of type C through any point x in any Cartan geometry looks exactly as the structure of local curves through 0 in G / H which are in C. This means that many questions about canonical curves (e.g. how many derivatives in the point x are needed to uniquely specify a canonical curve of type C) can be reduced to looking at 0 E G / H. Questions of this type in the realm of parabolic geometries will be studied in Section 5.3. Notice further, that the proof of Theorem 1.5.17 also shows how canonical curves of type C can be constructed as projections of solutions of appropriated ODEs. As a simple example let us look at the case of exponential curves. In 1.4.11 we have started from a subspace neg which is complementary to ~ and we have considered the curves t I--> gexp((t - to)X)H for X E n and 9 E G. If this maps 0 to 0, then gexp(-toX) =: h E H, and our curve is given by t I--> hexp(tX)· 0 = exp(tAd(h)(X))· o. More generally, for X E g, consider cX(t) := exp(tX)· o. If g-I· 0 = exp(toX) ·0, then as above we see that g.c x (t+to) = CAd(h)(X) (t). Hence,
1. CARTAN GEOMETRIES
112
if we suppose that A egis any subset such that Ad(h)(A) C A for all h E H, then the family CA := {c X : X E A} of curves through 0 is admissible. Hence, we have the notion of canonical curves of type CA on arbitrary Cartan geometries of type (G, H). In this case, we can describe the canonical curves explicitly: COROLLARY 1.5.18. Let A C 9 be a subset which is invariant under Ad(h) for all h E H. Let (P -+ M,w) be any Cartan geometry. Then a curve c : I -+ M is canonical of type CA if an only if locally it coincides up to a constant shift in parameter with the projection of a flow line of a vector field w- 1 (X) E X(P) for some X E A.
-leX)
Put c(t) = Fl~ (u) for some u E P and some X E A. From the definitions and using that Ad(exp(tX)) (X) = X, one immediately verifies that the curve j(c(t)) . exp( -tX) is horizontal in P. By Theorem 1.5.17, the development of the projection to M is represented by [c(O), exp(tX) . oD, so by the proposition it is canonical of type CA. Therefore, any curve which locally coincides with such flow lines (up to a constant shift in parameter) is canonical, too. Conversely, if c is canonical, then around each point, it develops to an element of CA. Shifting the parameter, the development is given by [u, exp(tX) ·oD. Now we can locally reconstruct c as in the proof of Theorem 1.5.17. But the resulting data for the ODE is S(p,q) be the obvious smooth projection. The tangent space of C at v E C is v.l, and this contains v since v is null. The tangent map to 1f induces a linear isomorphism v.l /lRv --> Trr(v)S(p,q). Now the inner product on lR m +2 induces an inner product v.l /lRv, which is nondegenerate of signature (p, q) and can be carried over to Trr(v)S(p,q). Replacing v by av for some a E lR, this inner product gets multiplied by a 2 , so we get a conformal structure on S(p,q). From this description it is evident that G acts by conformal isometries. Transitivity of the action follows from elementary linear algebra: Let us denote the inner product by ( , ). For a null vector v, one can find a vector w such that (v, w) = 1. By adding an appropriate multiple of v to w, we can further achieve (w, w) = O. Then on the subspace generated by v and w, ( , ) is nondegenerate of signature (1, 1). Hence, we can complete {v, w} to a basis by choosing an orthonormal basis of the orthocomplement {v, w}.l. Starting with another null vector il one obtains a basis of the same form. Hence, the linear map which maps the first basis to the second is orthogonal (since the inner product has the same form in both bases). This orthogonal map sends v to il, and transitivity follows. From this, it is clear that S(p,q) ~ G / P. To obtain the explicit description of S(p,q) , we identify lR m +2 with lRp+l x lRq+l and endow the first vector with the standard inner product and the second factor with the negative of the standard inner product. Then a point (x, y) # (0,0) lies in C if and only if Ixl = Iyl. In particular, we get SP x sq C C. Evidently, any line in C meets SP x sq in exactly two points, namely (x, y) and (-x, -V). This gives the two-fold covering as claimed. Since the inclusion SP x sq .. E JR, then this gives (2m - 2)>"gij. For P ij symmetric and tracefree, we obtain (m - 2)P ij and for P ij skew symmetric we get mP ij . Since m ~ 3, these three factors are all nonzero. Now any tensor Pij E JRm* ® JRm* may be uniquely decomposed as P ij
= !ngabpab9ij
+ ~(Pij + P ji -
~gabPabgij)
+ ~(Pij -
P ji )
into trace part, symmetric tracefree part and skew part. Applying the composition of the contraction and 8, each of these parts is multiplied by a nonzero factor. Hence the map Pij I-t (8P)klj is bijective. Given", we therefore find a unique P such that "'kikj = -(8P)klj' which completes the proof of the first part. Obtaining the explicit formula is now easy. We have to decompose -"'kikj(U) into trace-part, tracefree symmetric part and skew symmetric part. Then we have to multiply these parts by 2';-2' m~2' and !n, and add them up. This immediately leads to the claimed formula. 0 1.6.7. The conformal Cartan connection. Having the necessary algebraic background at hand, we formulate the normalization condition and establish the existence of a canonical Cartan connection. DEFINITION 1.6.7. A Cartan connection w of type (G, P) is called normal if and only if it is torsion free, and the go-{;omponent "'0 of its curvature function has the property that ("'o)kl j = O. THEOREM 1.6.7. Let (Po : go ---t M, (J) be a conformal structure on a smooth manifold M of dimension m ~ 3. Let p : g ---t M be the P-principal bundle constructed in 1.6.4. Then the canonical form (J-l EB(Jo on g from Proposition 1.6.4 uniquely extends to a normal Cartan connection w on g. This construction induces an equivalence of categories between first order Gstructures with structure group CO(p, q) and normal Cartan geometries of type (G,P). PROOF. Using Lemma 1.6.6, we see that for each point u E g, there is a unique linear map WI (u) : Tug ---t gl such that w( u) := (J-l (u) EB (Jo( u) EB Wl (u) : Tug ---t g is a linear isomorphism which reproduces the generators of fundamental vector fields and has the property that the associated map "'w(u) lies in the kernel of the Ricci type contraction. Looking at the point u . 9 for 9 E P, we may consider ¢:= Ad(g-l) 0 w(u) 0 Tr g- 1 : Tuogg ---t g. Equivariancy of (J-l EB (Jo as proved in part (2) of Proposition 1.6.4 shows that the components of this map in g-l EB go coincide with (J_l(U· g) EB (Jo(u· g). Using this and the definition of "'.p, we conclude that writing 9 E P as go exp(Z) according to Proposition 1.6.3, we obtain ",.p(Ad(go) (X), Ad(go) (Y)) = Ad(go)
Ad(go)-I. m But Ad(go) is just the standard action of CO(p, q) on JR , so "'.p lies in the kernel of the (by construction CO(p, q)--equivariant) Ricci type contraction. Uniqueness in the lemma shows that ¢ = w(u· g). To prove existence of the canonical normal Cartan connection, it only remains to show that the w(u) fit together to define a smooth one-form w E n1 (g,g). 0
"'w(u) (X, Y) 0
128
1.
CARTAN GEOMETRIES
Applying the contraction, we obtain
(8P)k/j = (m -
l)Pij -
P ji
+ gabpabgij.
In particular, if Pij = Agij for some). E ~, then this gives (2m - 2).gij. For Pij symmetric and tracefree, we obtain (m - 2)Pij and for Pij skew symmetric we get mP ij . Since m ;:::: 3, these three factors are all nonzero. Now any tensor Pij E ~m* ® ~m* may be uniquely decomposed as
Pij = fngabpabgij
+ !(Pij + Pji -
~gabpabgij)
+ !(Pij -
Pji)
into trace part, symmetric tracefree part and skew part. Applying the composition of the contraction and 8, each of these parts is multiplied by a nonzero factor. Hence the map Pij ~ (8P)ki k j is bijective. Given", we therefore find a unique P such that "'kikj = -(8P)ki\' which completes the proof of the first part. Obtaining the explicit formula is now easy. We have to decompose -"'k/j(U) into trace-part, tracefree symmetric part and skew symmetric part. Then we have to multiply these parts by 2';-2' m~2' and fn, and add them up. This immediately 0 leads to the claimed formula.
1.6.7. The conformal Cartan connection. Having the necessary algebraic background at hand, we formulate the normalization condition and establish the existence of a canonical Cartan connection. DEFINITION 1.6.7. A Cartan connection w of type (G,P) is called normal if and only if it is torsion free, and the go-component "'0 of its curvature function has the property that ("'O)kikj = O. THEOREM 1.6.7. Let (Po: go --t M, B) be a conformal structure on a smooth manifold M of dimension m ;:::: 3. Let p : 9 --t M be the P-principal bundle constructed in 1.6.4. Then the canonical form B-1 EI1Bo on 9 from Proposition 1.6.4 uniquely extends to a normal Cartan connection w on g. This construction induces an equivalence of categories between first order Gstructures with structure group CO(p, q) and normal Cartan geometries of type (G,P). PROOF. Using Lemma 1.6.6, we see that for each point u E g, there is a unique linear map WI (u) : Tug --t gl such that w(u) := B-1 (u) EI1 Bo(u) EI1Wl (u) : Tug --t g is a linear isomorphism which reproduces the generators of fundamental vector fields and has the property that the associated map "'w(u) lies in the kernel of the Ricci type contraction. Looking at the point u . 9 for g E P, we may consider ¢ := Ad(g-l) 0 w( u) 0 Tr g- 1 : Tu.gg --t g. Equivariancy of B-1 EI1 Bo as proved in part (2) of Proposition 1.6.4 shows that the components of this map in g-1 EI1 go coincide with B_ l (u· g) EI1 Bo(u· g). Using this and the definition of "'"" we conclude that writing 9 E P as go exp(Z) according to Proposition 1.6.3, we obtain
",,,,(Ad(go)(X), Ad(go)(Y)) = Ad(go) 0 "'w(u) (X, Y)
0
Ad(go)-I.
But Ad(go) is just the standard action of CO(p, q) on ~m, so "'''' lies in the kernel of the (by construction CO(p, q)--equivariant) Ricci type contraction. Uniqueness in the lemma shows that ¢ = w(u· g). To prove existence of the canonical normal Cartan connection, it only remains to show that the w(u) fit together to define a smooth one-form w E 01(g, g).
1.6.
CONFORMAL RIEMANNIAN STRUCTURES
129
Remarkably, this follows from the existence of some Cartan connection extending B-1 EB Bo. To get such an extension, we use Weyl connections. By part (3) of Proposition 1.6.4, there are Weyl connections on Po : go -+ M and they are in bijective correspondence with Go-equivariant sections a : go -+ g. From the construction of the tautological form it follows immediately that the affine connection corresponding to a is given by a*(B-1 + Bo). For a point Uo E go and u = a(uo), the tangent space Tug splits as im(Tuoa) EB ker(Tu7r). Hence, in this point we can uniquely extend B-1 EBBo to a form W U by requiring that the 91-component vanishes on the first summand and reproduces generators of fundamental vector fields. As in the proof of Lemma 1.5.15, one verifies that this extends to a smooth Cartan connection W U on g which extends B-1 EB Bo. Now in each point u E g, we can write
By the uniqueness part of the lemma, we can obtain P(u) by inserting KU(U) into equation (1.31). Here we have to view the 90-component K U of the curvature function of W U as acting on lR. m via B_ 1 • Anyway, the result evidently depends smoothly on u, so smoothness of W U implies smoothness of w. We just sketch the proof of the remaining claims leaving out some straightforward verifications since we will prove a much more general version of this theorem later. Let us assume that (p : Q -+ M, w) is a Cart an geometry of type (G, P). Via the identification TM = Q Xp (9/1'), the P-invariant conformal class of inner products on 9/1' ~ 9-1 induces a conformal structure on M. Denoting by go the conformal frame bundle as before, we obtain a homomorphism 0 : Q -+ go over the projection P -+ Go which covers the identity on M and such that W-1 = *B-1. Next, for U E Q the component (W-1 + wo)(u) induces a linear isomorphism TuQ/P+ -+ 9-1 EB 90· On the other hand, Tuo induces a linear isomorphism TuQ/P+ -+ T4>o(u)go, so together these two maps induce a linear isomorphism T4>o(u)go -+ 9-1 EB 90· By construction, the 9_1-component of this isomorphism is B-1 (0 (u)) and it reproduces the generators of fundamental vector fields. Assuming that w is torsion free, this defines an element in g lying over o(u) E go. Hence, we obtain a smooth map : Q-+ g lifting 0. One easily checks that by construction this map is P-equivariant and thus an isomorphism of principal bundles and from the definition of the tautological form we conclude that *(B-1 + Bo) = W-1 + woo Thus, (-1)*w is an extension of B-1 + Bo to a Cart an connection on g. The curvature function of this Cartan connection is given by K, 0 -1. Hence, if we assume that w is normal, then also (-1)*W is normal, and hence equals w. The first part of this argument shows that any Cartan geometry of type (G, P) has an underlying first order Go-structure, and clearly this defines a functor from Cartan geometries to Go-structures. The last part of the argument shows that any morphism of first order G-structures uniquely lifts to a morphism of Cartan geometries, which implies that the functors constructed above establish an equivalence of categories. 0 REMARK 1.6.7. The relation to extension functors for Cartan geometries as developed in 1.5.15 that was briefly hinted at in the proof, can be exploited much further. Using this theory, we could have obtained a quicker way towards existence and uniqueness of normal Cartan connections. This would, however, be less
130
1. CARTAN GEOMETRIES
transparent from a geometric point of view, so we preferred to use the more complicated traditional approach. The general construction of normal Cartan connections for parabolic geometries in Section 3.1 is closer to the arguments using extension functors, so for comparison we sketch this line of argument here. Consider the affine extension B of CO(p, q). In view of part (2) of Proposition 1.6.3, we can do this starting from Go acting on g-l via the adjoint action. Hence, we can view elements of B as pairs (go, X) and the multiplication given by (gO, X)(gb, X') := (gogb, Ad(gb)(X) + X'). Using that g-l is abelian, one immediately verifies that (go, X) f--+ go exp(X) defines a homomorphism ¢ : B --+ G. Following Example 1.5.16, we consider i = ¢Icu : Go --+ P, which is just the standard inclusion, and a := ¢' : b --+ g, which is the inclusion of the subalgebra g-1 Ef)go into g. In particular, a induces a linear isomorphism b/go --+ g/p. A Cartan geometry of type (B, Go) on a manifold M is then a conformal structure (p : go --+ M,(Ld as considered in the last few subsections together with a principal connection 'Yon go. Now we can form the extended bundle g = go Xi P, which is a principal P-bundle. By Lemma 1.5.15, there is a unique Cartan connection won g such that j*w = a 0 (ILl Ef) 'Y), where j : go --+ g is the natural map. Since a is a homomorphism of Lie algebras, Proposition 1.5.16 shows that pulling back the curvature of wone obtains the curvature of ILl Ef) 'Y. In particular, if we use a Weyl connection for 'Y, then wis torsion free. Next, we can modify wby adding a horizontal, P-equivariant one-form P with values in gl. Using Lemma 1.6.6 one easily shows that there is a unique choice for P such that w+ P is normal. In particular, the result is independent of the Weyl connection we started with. The fOrIa P can even be computed explicitly from the curvature of wand hence from the curvature of (L l Ef)'Y using formula (1.31). Hence, we have extended a given conformal structure to a normal Cartan geometry of type (G,P). If f
: M --+ M is a conformal isometry, then it lifts to a morphism 11>0 of Go-structures. Starting with a Weyl connection on go --+ M and considering the pullback of this connection on go, the map 11>0 becomes a morphism of Cartan geometries of type (B, Go). The functoriality result in Theorem 1.5.15 shows that 11>0 extends to a morphism of Cartan geometries of type (G, P) . Normalizing on one side and pulling back the result to the other side, the pullback is normal and hence has to coincide with the normalized Cartan connection on the other side. Hence, f lifts to a morphism between the associated normal Cartan geometries, and the equivalence of categories follows. 1.6.8. The curvature of the conformal Cartan connection. In the proof of Theorem 1.6.7, we have constructed the conformal Cartan connection w from the Cartan connections w17 associated to a Weyl-structure (j. We want to use this correspondence to determine the curvature of w. By construction, (j*w17 has values in g-1 Ef) go and coincides with the (affine) Weyl connection associated to (j. Moreover, the curvature function K17 has vanishing g_l-component, and from this and the definitions of the curvatures, we conclude that (j* K17 is exactly the curvature of the Weyl connection. In particular, the component Kg, interpreted as a section of the associated bundle A2T* M ® L(T M, T M), is exactly the classical curvature Ri/ t of the induced linear connection ';;r on the tangent bundle T M. Thus, the trace Rkik j is exactly the Ricci curvature ~j, and from above we see that the deformation Pij to the conformal Cartan connection, viewed as a section
1.6. CONFORMAL RIEMANNIAN STRUCTURES
131
of T* M ® T* M is given by Pij =
;-!2 (Rij + !n.(Rji -
Rij) -
2(~_1)gabRabgij).
In particular, taking the Levi-Civita connection of a metric gij in the conformal class, the Bianchi identity implies symmetry of the Ricci curvature Rij, and R := gab Rab is exactly the scalar curvature. In this case we obtain P ij
=
;-!2(Rij
+ 2(~_1)Rgij),
which (up to a sign) is the usual Rho tensor associated to a metric in the conformal class. Next, we know that the curvature component 1\;0 of the conformal Cartan connection is given by 1\;0' + 8(P), where P is the Rho tensor as above. Of course, it suffices to compute this for a Levi-Civita connection. In that case, Pij is symmetric, and using formula (1.33) from 1.6.6 for 8P, we conclude that I\;i/t = ~/t
+ Pj tt5f -
Pu t5% -lapjagil + laPiagjl .
By construction, we know that I\;kl j = 0, so let us look at the other possible trace, I\;i/ k. The corresponding trace for the curvature of the Levi-Civita connection vanishes, since this has values in o(TM), while for the remaining terms one obtains 2(Pij - Pjd, which vanishes by symmetry of the P-tensor of a LeviCivita connection. So we see that 1\;0 is totally tracefree. On the other hand, 1\;0 is obtained from the Riemann curvature by adding trace terms, which shows that we may alternatively characterize 1\;0 as the totally tracefree part of the Riemann curvature. Hence, 1\;0 is exactly the classical Weyl curvature, which is well known to be conformally invariant. Since also in the case of a general Weyl connection we obtain 1\;0 from R by adding trace terms, we conclude that the Weyl curvature coincides with the totally tracefree part of the curvature of any Weyl connection. What remains to be done is interpreting the 91-component of the curvature of the conformal Cartan connection. This is a bit more subtle, since this component does not define an equivariant function and thus a geometric object unless the 90component of I\; vanishes. A good way to circumvent this problem is to choose a Weyl connection and consider the pullback of this component along the corresponding section 0' : go ---t g. Equivariancy of I\; implies Go-equivariancy of the individual components 1\;0 and 1\;1 of the curvature function, so 1\;1 0 0' : go ---t A2(9jp)* ® 91 is a Go-equivariant function and hence corresponds to a T* M -valued two-form yO' on M, called the Cotton-York tensor associated to the given Weyl connection. Of course, equivariancy of I\; implies that 1\;1 is determined by its restriction to the image of 0'. Consider a vector field e E X(M) and let e hor E X(go) be its horizontal lift with respect to the chosen Weyl-connection. Then by construction 0'*W1 (e hor ) = P(O-1 (TO' . e hor )), where P is viewed as a function with values in L(9-1, 91). But the right-hand side of this may as well be viewed as the Go-equivariant function go ---t 91 representing the section p(e) ofT* M obtained by interpreting P as a T* Mvalued one-form. Since wo(e hor ) vanishes along the image of 0' we conclude that, along this image, the 91-component of [w(TO'·e hor ), w(TO"1]hor)] vanishes identically. From the definition of the curvature we thus conclude that YO'(e, 1]) E r(T* M) is represented by the function go ---t 91 given by dO'*W1 (e hor , 11hor ) = e hor ,p(1])_1]hor ·p(e)-p([e, 11]) = V'~ (P(1]))- V'~ (p(e) )-p([e, 11]).
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By definition, this is the covariant exterior derivative with respect to \lU of P E O,l(M, ToO M). Alternatively, using that any Weyl connection is torsion free, we may express the Lie bracket [e,17] as \le 17 - \l~e (see 1.3.5) and conclude that YU(e,17) = (\lup)(e, 17) - (\luP)(17, e), or in index notation Yijk = \lfP jk - \l'JPik . Collecting the information, we obtain the first two parts of the following result, which completely describes the relation between the curvature of the conformal Cartan connection and the curvature of any Weyl connection. COROLLARY 1.6.8. Let (Po: go -+ M,O) be a conformal structure on a smooth manifold of dimension m 2:: 3, p : g -+ M the prolongation, W E 1 (g, g) the conformal Cartan connection and", its curvature. Consider a Weyl connection on go and let (J' : go -+ g be the corresponding Go -equivariant section. Let P = Pij be the Rho tensor of the Weyl connection, R = ~/ e its curvature, and yu = Yijk its Cotton- York tensor. (1) The totally trace free part W = Wile of R represents the go-component of the curvature of wand thus is independent of the choice of the Weyl connection. It is explicitly given by
n
will = Ri/l + Pjlof -
Pieo; -lapjagu
+ lapiagje - (Pij - Pji)o},
for any metric gij in the conformal class with inverse gi j . (2) The Cotton- York tensor yu corresponds to the pullback along (J' of the g1 component of the curvature of w. It is the covariant exterior derivative of P E 1 (M,ToOM), or equivalently the alternation of\lP, i.e.
n
Yijk = \lfP jk
-
\ljPik.
(3) In dimensions m > 3, the conformal structure is fiat if and only if W vanishes. For m = 3, one always has W = 0, the Cotton- York tensor yu is independent of the choice of the Weyl connection, and the geometry is fiat if and only if Y vanishes. PROOF. It remains to prove part (3). First we observe that W vanishes if and only if "'0 vanishes, and since "'-1 is always identically zero, vanishing of "'0 implies that "'1 : g -+ A2(gjp)oO ® g1 is P--equivariant. But on the right-hand side, the subalgebra g1 C P acts trivially, so the subgroup P+ c P acts trivially. Thus, "'1 descends to a Go--equivariant function on go, which clearly coincides with the pullback of"'1 along any Go--equivariant section (J'. In conclusion, we see that vanishing of the Weyl tensor implies that the Cotton-York tensor is independent of the choice of the Weyl connection and thus a conformal invariant. In this case vanishing of the Cotton-York tensor Y is equivalent to vanishing of"" which by Proposition 1.5.2 is equivalent to the given Cartan geometry being locally isomorphic to the homogeneous model (G -+ G j P, wG). In view of the equivalence of the category of normal Cartan geometries with the category of conformal structures, this is equivalent to M being locally conformally flat. Let us first consider the case m = 3. The curvatures of Levi-Civita connections have values in A2ToO M ® o(TM), which, for m = 3, is a space of dimension nine. The trace map used for the normalization condition has values in ToO M ® ToO M, which also has dimension nine. For the standard basis {ei} of :lR3 with dual basis {ei } one may obtain the element ei ® ej E :lR3 oO ® :lR3 oO by applying the trace to ei /\e k ® (ek ®e j - ej ®e k ) E A2 :IR3 * ®o(3). Thus, the trace is surjective and hence a linear isomorphism by dimensional reasons. In particular, the totally tracefree part
1.6.
CONFORMAL RIEMANNIAN STRUCTURES
133
of the curvature of any Levi-Civita connection in three dimensions automatically vanishes, so the Weyl curvature is always trivial. From above, we conclude that the Cotton-York tensor Y is a well-defined conformal invariant, which is a complete obstruction to local conformal flatness. To complete the proof, it remains to show that in dimensions m > 3 vanishing of the Weyl tensor implies vanishing of the Cotton-York tensor. This is an application of the Bianchi identity: Let us consider the Bianchi identity in the form (1.26) from the proof of Proposition 1.5.9. We know that ~-1 = 0 and by assumption W = 0 and thus ~o = O. Since g-1 is abelian and ~-1 = 0, vanishing of the go-component of the Bianchi identity is equivalent to vanishing of L:cycI [~1 (X, Y), Z]. Let us again pass to an index notation, using the convention that ~1(X, Y)k = Xiyj~ijk. Formula (1.32) for the bracket from the proof of Lemma 1.6.6 gives [~1(X, Y), Z]~ = _Zi~ktjXkyt
+ gia~ktaXkytgjbZb -
za~ktaXkyt6}.
The first two summands form the tracefree part of this expression, while the last summand is pure trace, and of course the two parts have to vanish individually after forming the cyclic sum over the arguments. Renaming the indices of the entries to a, b, c, we see that the vanishing of the tracefree part of the cyclic sum is equivalent to vanishing of the cyclic sum over a, b, c of -6~~abj + gik ~abkgjc. Forming this cyclic sum and contracting over i and c, we obtain the equation
o= and using the
~
-m~abj - ~baj - ~baj
+ ~abj + 9ik ~iakgjb + 9ik ~bikgja,
is skew symmetric in the last two indices, this reduces to
0= (3 - m)~abj + 9 ik (~bikgja - ~aikgjb). Contracting with gjb, we obtain 0 = (4 - 2m)gik~aik' Since m > 3, this together with the above equation implies 0 = (3 - m)~abj, and thus vanishing of ~1' 0 REMARK 1.6.8. The approach to the construction of the canonical Cartan connection via extension functors discussed in Remark 1.6.7 above, has another interesting aspect. It may happen, that for some choice of Weyl connection, the extension functor directly produces the normal Cartan connection. This naturally defines a subclass of conformal structures which should be particularly well behaved. By equivariancy, the extended Cartan connection is normal if an only if it is normal along the image of j. Hence, starting with a Weyl connection 'V with curvature R i / £, the extension functor directly produces a normal Cartan connection if and only if Rkik j = O. Hence, the subclass in question is exactly conformal structures admitting a Ricci flat Weyl connection. Using the Bianchi identity, this implies that the curvature has values in o(n) c co(n). In the language introduced in 1.6.5, the given Weyl structure is closed and hence locally exact. Hence, we see that we obtain exactly the subclass of conformal structures, which locally contain Ricci flat representative metrics. 1.6.9. The Liouville theorem. There are various versions of what is called the Liouville theorem. A version for general Cartan geometries can be found in 1.5.2. This says that the automorphisms of the homogeneous model are given by left translations and local automorphisms between connected open subsets of the homogeneous model uniquely globalize. We can apply this to conformal structures using the equivalence to the category of normal Cartan geometries. Hence, the group of conformal isometries of the Mobius space S(p,q) is exactly the group PO(p+
134
1.
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1, q + 1) which acts on S(p,q) as described in 1.6.2. There we have also used the natural chart ~m 3:! 9-1 -> S(p,q) defined by X f---t exp(X) . o. We have noted in 1.6.2 that this chart is a conformal diffeomorphism of ~m onto a dense subset of S(p,q).
In particular, we conclude that conformal isometry between connected open subsets of ~m comes, via the natural chart, from the action of a unique element g E G = PO(p + 1, q + 1). Using this and some facts we have already proved, we can now easily derive a complete description of the pseudo-group of local conformal isometries for the standard metric of signature (p, q) on ~m. This result is also referred to as the Liouville theorem in the literature. The classical proof is rather involved. One has to completely describe the solutions of the overdetermined system of PDE's which characterizes a conformal isometry. There are several obvious examples of conformal isometries of ~m, in particular, translations, dilatations X f---t AX for A E ~ \ {O}, and orthogonal linear maps A E O(p, q). Further, we have already noted in 1.6.2 that the inversion ¢ in the unit circle, defined by ¢(X) = (X~X) is a conformal isometry from the set of all points X such that (X, X) i= 0 onto its image. Using this we now formulate COROLLARY 1.6.9 (Liouville theorem). For m = p + q ;:::: 3 consider ~m with the conformal structure of signature (p, q) defined by the standard inner product of that signature. Then the pseudo-group of all local conformal isometries of ~m is generated by translations, dilatations, orthogonal linear maps and the inversion in the unit circle. PROOF. Since any local conformal isometry is induced by an element of G = PO(p + 1, q + 1) it suffices to show that G is generated by elements which induce the maps listed in the theorem on ~m 3:! 9-1 via the natural chart. Since 9-1 is a commutative subalgebra of 9, we see that for X, Y E 9-1 we have exp(Y) exp(X) = exp(X + Y) in G. Hence, under the natural chart, translations are given by the action of exp(Y) for Y E 9-1' In 1.6.3 we have seen that elements of Go C P exactly correspond to conformal linear maps on ~m, which are generated by orthogonal maps and dilatations. It is also not hard to guess the element of G that induces the inversion in the unit circle. Consider the matrix
o
-2)
o . o o
id
It is immediately seen to be orthogonal with respect to the inner product S used in 1.6.2, so we may consider its class in G = O(p + 1, q + 1)/ {±id}. For X E ~m, this matrix maps (l,X,-~(X,X)) to ((X,X),X,-1/2) and if (X,X) i= 0, the latter point lies on the line through (1, (X~X)' 2(X\))' Thus, the class of Q implements the inversion. Note also that Q has determinant -1, so in case that G has two connected components, Q does not lie in the component of the identity. Further, one immediately verifies that the adjoint action by Q maps 9-1 to 91 by a linear isomorphism. Thus, for Z E 91, the element exp(Z) E G lies in the subgroup generated by the class of Q and all elements of the form exp(Y) for Y E 9-1. Putting things together, we see that the subgroup in question contains all elements of the form exp(Y)goexp(Z) for Y E 9-1, go E Go and Z E 91. By part (3) of Proposition 1.6.3, any element of P can be written as go exp(Z), so it
1.6. CONFORMAL RIEMANNIAN STRUCTURES
135
contains all elements of the form exp(Y)g for Y E 9-1 and 9 E P. But the images of elements of the form exp(Y) contain an open neighborhood of 0 in GIP, so we see that the elements of the form exp(Y)g as above contain an open neighborhood of e in G. Since these elements also meet each connected component of G, they generate G. 0 1.6.10. Invariant operators and invariants of conformal structures. Finding invariants of conformal structures and describing conformally invariant differential operators are two related problems, whose discussion in a more general setting will be among the main topics of volume two. Here we just sketch some basics to give the reader a rough impression of what is going on. We start by describing the problems in classical terms, i.e. without reference to Cartan geometries. This classical formulation is phrased in Riemannian terms. For simplicity, let us avoid spinors and consider two tensor bundles E and F over a Riemannian manifold (M, g). Then we can use the metric g, its inverse, the Levi-Civita connection and its curvature, and the volume form corresponding to 9 to write down a differential operator r(E) -+ r(F). If we can replace all the ingredients determined by 9 by the corresponding data for a conformally equivalent metric without changing the resulting differential operator, then the operator is called conformally invariant. Similarly, the question of (local) invariants consists in using the above ingredients to write down a function which then should be independent of the choice of the metric from the conformal class. To obtain conformally invariant operators, one has to use density bundles, which rarely are used in Riemannian geometry otherwise. Recall that for a smooth manifold M of dimension m and a number a E IR the bundle of a-densities is the associated bundle to the full frame bundle pI M with respect to the onedimensional representation A ~ Idet(A) I-a of GL(m, 1R). The sections of the bundle of I-densities are the right objects to integrate on nonorientable manifolds, while in the oriented case the bundle of I-densities is canonically isomorphic to AmT* M. By construction, any density bundle is trivial, but there is no canonical trivialization. Any Riemannian metric 9 trivializes the bundle of I-densities via the volume density, which is locally given by JI det(gij)I. Since this induces a trivialization of all density bundles, they usually are not useful in Riemannian geometry. Changing from 9 to a conformally related metric, however, also changes trivialization of density bundles. The usual convention in conformal geometry is to define e[w] to be the bundle of (-~)-densities, and to add the expression [w] to any bundle in order to indicate the tensor product with e[w]. Such tensor products are referred to as weighted bundles. Any choice of a metric 9 from the conformal class gives an identification of sections of e[w] with smooth functions on M but changing from 9 to 9 = n2 g, these functions transform as j = nw f. Otherwise put, the bundle e[w] may be interpreted as the associated bundle to the bundle C of scales of the conformal structure with respect to the representation t ~ t- W of 1R+. From 1.6.5 we know that rescaling 9 = n2 g, the corresponding one-form I which describes the change of Weyl connections is given by I = dlog(n), i.e. Ia = n-1V'an, and we have derived a formula for the change of the principal connection on C there. This immediately implies that for the covariant derivative of a section f of e[w], the change under a conformal rescaling as above reads as Vaf = V' af + WI af.
1. CARTAN GEOMETRIES
136
The role of the density bundles is easy to understand in the picture of conformal structures as first order G-structures with structure group CO{p, q). Representations of CO{p, q) can be restricted to the semisimple part O(p, q) and the center, which is isomorphic to 1R+. The point is now that the representation of the center may be varied, while the action of O{p, q) remains fixed, and this exactly corresponds to tensoring with different density bundles. The conformal structure itself may be viewed as a canonical section gij of S2T* M [2] with inverse gij, which is a section of S2T M [- 2]. Hence, similarly as in the case of Riemannian structures, one may raise and lower indices in conformal geometry, but at the expense of a weight. Having the possibility to form conformally invariant contractions, we immediately get examples of conformal invariants. Namely, the Weyl tensor Wilt is by construction conformally invariant, so taking any tensor power of W and forming a complete contraction, one obtains a conformally invariant density. Getting more general conformal invariants is a surprisingly hard problem, and only a few other examples are known classically. A trivial example of a conformally invariant differential operator is given by the exterior derivative of differential forms, which is just the alternation of the covariant derivative. To obtain a slightly less trivial example, let us analyze first order operators on weighted vector fields. The formulae for the change of the Levi-Civita connection under a conformal rescaling from 1.6.4 and the formula for densities above immediately imply that rescaling fJ = 0 2 g implies that for a section ~ = ~a of T M[w] we obtain Va~b = Va~b
+ {w + I)Ya~b -
gac~cgbdy d + Y c~c6:
as a section of T* M ® T M[w]. Tracing over the two free indices, we obtain Va~a = V a~a + (m + w) Y a~a, so putting w = -m, we obtain a conformally invariant first order differential operator mapping sections ofTM[-m] to sections of £[-m]. This is the conformally invariant divergence. Another possibility is to lower the free upper index in the above equation to obtain gcb Va~c = gcb V a~c + (w + I)Y agbc~C - Y bgac~C + Y c~Cgab as a section of T* M ® T* M[w + 2]. Obviously, if we put w = -2, the deformation term is symmetric, so alternating defines a conformally invariant operator from sections of TM[-2] to A2T* M, which exactly corresponds to the exterior derivative under the identification TM[-2] 9:! T*M induced by gab. Finally, for w = 0, the symmetric tracefree part of the deformation vanishes, so projecting to the symmetric tracefree part gives a conformally invariant differential operator from X( M) to S5T* M[2]. This is exactly the conformal Killing operator, whose solutions are the infinitesimal conformal isometries. The general theory of conformally invariant operators of first order looks very similar, as long as one restricts to bundles corresponding to irreducible representations of CO(p, q). It is an exercise in representation theory to show that for any irreducible representation V of O(p, q) the tensor product IRm * ® V decomposes into a direct sum of pairwise non-isomorphic irreducible representations. Further, one proves that for any of these components there is a unique choice of a weight w such that one obtains a first order conformally invariant operator from sections of E[w] to sections of F[w], where E and F are the bundles corresponding to V and the
1.6. CONFORMAL RIEMANNIAN STRUCTURES
137
irreducible representation in question. This result is due to [Feg79]. We will prove the vast generalization of this result to all parabolic geometries due to [SlSo04] in volume two. For higher order operators the situation becomes much more complicated, since in many cases one has to include curvature correction terms in order to get conformal invariance. We just mention the simplest and best known example of such an operator, namely the conformal Laplacian or Yamabe operator. This operator maps sections of £[2-;m] to sections of £[-2;-m] and is defined in terms of a chosen metric as ~ - 2-;mp, where ~ = gab'Va'V b and P = gabp ab is the contraction of the Rho tensor. Proving invariance of this operator directly is a nice exercise that will convince the reader that this direct approach gets out of hand for higher order operators very quickly. The general calculus for Cart an connections introduced in Section 1.5 offers a completely different approach to this problem. Iterated fundamental derivatives provide an invariant way to capture arbitrarily high jets of sections of any natural bundle as sections of tensor powers of the adjoint tractor bundle and the given natural bundle. Passing to quotients can be used to construct invariant operators with values in simpler bundles. Doing this systematically in a much more general situation leads to the concept of BGG-sequences, which were originally introduced in [CSS01] and will be one of the main objectives of volume two. Similarly, iterated fundamental derivatives of the Cartan curvature provide examples of conformally invariant sections of certain bundles, and constructing natural bundle maps (possibly nonlinear) from these to density bundles leads to conformally invariant densities. 1.6.11. Remarks on further developments. Let us conclude this section with a few remarks on generalizations. We have not explicitly used representation theory in our treatment of conformal structures. The main algebraic ingredients can, however, be proved very efficiently using tools from representation theory. In particular, the analysis of the maps in 1.6.4 and 1.6.6 which were both denoted by can be obtained as a corollary of the description of some Lie algebra cohomology groups. These groups are described by Kostant's version of the Bott-Borel-Weil theorem. Having switched to this language, one can deal with gradings of the form g = g-l EB go EB gl of any semisimple Lie algebra g in a similar way: Given a group G with Lie algebra g, one defines subgroups Go c PeG corresponding to the subalgebras go C go EB gl C g. The homomorphism Go -4 GL(g-d defined by the adjoint action is always infinitesimally injective, so one has the notion of a first order G-structure with structure group Go on manifolds of dimension dim(g_l). Under a cohomological condition which is satisfied in almost all cases, one then obtains an equivalence of categories between such structures and normal Cartan geometries of type (G, P). These structures are called almost Hermitian symmetric structures or AHS structures in the literature, and a treatment of the canonical Cartan connections for such structures along the lines sketched above can be found in [CSS97b]. Of course, one may try to prolong more general first order G-structures. While in our presentation we used a priori the Lie algebra g with the grading, this may be bypassed. Namely, one may start with I) C g[(V) and consider V EB I) EB 1)(1), where 1)(1) denotes the first prolongation as introduced in 1.6.1. We may view V and 1)(1) as abelian Lie algebras, and since both spaces are I) modules by construction, we
a
138
1. CARTAN GEOMETRIES
get bracketsl)®V _ V and 1)®1)(1) _1)(1). Finally, by definition, 1)(1) is a subspace of L(V, I)) which can be used as the definition of a bracket 1)(1) ® V - I). While these brackets do not make V EB I) EB 1)(1) into a graded Lie algebra in general, one can build a Lie group HI corresponding to the Lie algebra I) EB 1)(1) as a semidirect product. Next, one chooses an appropriate complement to the image of 8, and it is possible to develop a general version of the prolongation of first order G-structures in that style; see for example [864]. The first prolongation can be viewed as a first order G-structure with structure group HI on the total space of the principal bundle one started with. Hence, one may prolong once more, and hope that this stops at some step and gives rise to a Cartan connection. Unfortunately, there are general results which show that in many interesting cases, this cannot work. A result of S. Kobayashi and T. Nagano (see [KN64]) treats the case of a subalgebra I) C gl(V) which acts irreducibly on V. This means that there is no nontrivial I)-invariant subspace in V. In this situation, there are exactly three possibilities: Either the first prolongation 1)(1) is trivial, so we are in the situation dealt with in 1.6.1. Secondly, we may have V = g-1 and I) = go for a grading 9 = g-l EB go EB g1 of a simple Lie algebra g. Then we are in the situation of AHS-structures discussed above. In all other cases, process of iterated prolongations as outlined above does not stop after finitely many steps, so there is no hope to get a Cartan connection on a finite-dimensional bundle in this way. In spite of this result, there still is a number of interesting geometric structures that may equivalently be described as normal Cart an geometries. One way to view these structures is as subclasses of first order G-structures. Consider, for example, manifolds of dimension 2n + 1 endowed with a complex subbundle E of rank n in the tangent bundle T M. Such structures may be equivalently viewed as first order G-structures with structure group a certain subgroup H of GL(2n + 1, JR.). More precisely, one has to view JR.2 n +l as en EB JR., and then define H to be the subgroup of those real linear automorphisms which preserve the subspace en and restrict to complex linear maps on this subspace. It is then obvious how to obtain a reduction of the frame bundle to the structure group H from the complex subbundle E C T M and vice versa. Following the theory of first order G-structures, the first step to understand this structure is to consider the struct.ure function, which in this special case can be described as follows: Consider the quotient bundle T M / E and t.he canonical projection q : TM - TM/E. For two sections f""., E r(E) consider q([f"".,]) E r(T M / E). This is visibly bilinear over smooth functions, and thus defines a tensor e : E x E - T M / E, which is equivalent to the structure function. One choice that leads to a Cartan geometry is then to require that for each x E M the value ex is nondegenerate and compatible with the almost complex structure, which forces ex to be the imaginary part of a nondegenerate Hermitian form. This leads to the definition of a nondegenerate partially integrable almost CR-structure, for which a canonical Cartan connection turns out to exist. In his pioneering work culminating in [Tan79], N. Tanaka showed that such an approach applies in to a large class of examples, arriving at an approach to parabolic geometries; see Appendix A for a survey. Conceptually, it is preferable to take the different point of view of filtered manifolds. By definition, a filtered manifold is a smooth manifold M endowed with a decreasing filtration TM = T-kM :J T- k+1 M :J ... :J T- 1 M of the tangent
1.6. CONFORMAL RIEMANNIAN STRUCTURES
139
bundle by smooth subbundles, which is compatible with the Lie bracket in the sense that for ~ E f(Ti M) and 'TJ E f(Tj M) we have [~, 'TJ] E f(Ti+j M). Here we agree that TR. M = T M for all e ::; - k. Then one considers the associated graded bundle gr(TM) = gr_k TM EB ... EB gr_ 1(TM), where gri(TM) = Ti M/Ti+1 M. Similarly, as above, the Lie bracket of vector fields induces tensorial maps gri (T M) x grj(TM) ---+ gri+j(TM) (where grt(TM) = 0 for e < -k and e 2: 0) which, for each x E M, make gr(TxM) into a nilpotent graded Lie algebra. If one requires that all these Lie algebras are isomorphic to a fixed nilpotent graded Lie algebra n, then one naturally gets a frame bundle for gr(T M) with structure group the group Aut(n) of automorphisms of the Lie algebra n. Now one can look at reductions of structure group of this frame bundle as a filtered analog of first order G-structures. This is the point of view of [Mo93], which constructs canonical normal Cartan connections for a wide variety of structures in this sense. In the example of almost CR-structures above, one would start by looking at manifolds of dimension 2n + 1 with real rank 2n subbundles in the tangent bundle. Requiring that the induced algebraic bracket in each point is nondegenerate means exactly looking at contact structures. The associated graded gr(TxM) in each point is then a Heisenberg algebra, and a compatible complex structure on the subbundle can clearly be interpreted as a reduction of structure group of gr(TM). In this picture, parabolic geometries can be characterized as those structures in which the nilpotent graded Lie algebra n is the nilradical of a parabolic subalgebra of a semisimple Lie algebra 9, while the reduction of structure group is to a group with Lie algebra the Levi part of the parabolic. Equivalently, 9 has to admit a grading of the form 9 = 9-k EB ... EB 9k such that n = 9-k EB ... EB 9-1 and we need a reduction to a structure group Go with Lie algebra 90. Such a reduction of structure group of the associated graded gr(T M) is called a regular infinitesimal flag structure. We will prove in Section 3.1 that in the situations coming from parabolic subalgebras there is a categorical equivalence between regular infinitesimal flag structures and Cartan geometries which satisfy a normalization condition on their curvature. The approach presented there shows that the algebraic properties of the normalization condition are the crucial ingredient for this equivalence result, while the constuction of the Cartan bundle plays only very little role. More traditional prolongation procedures which put more effort into the construction of the Cartan bundle and then obtain the Cartan connection from tautological forms are sketched in Appendix A.
CHAPTER 2
Semisimple Lie algebras and Lie groups Going through 1.4 one notices that understanding the geometry of homogeneous spaces in most cases boils down to understanding (finite-dimensional) representations of Lie groups and equivariant mappings between such representations. Moreover, from 1.5 we know that this also gives the basis to understanding Cartan geometries. The Lie group / Lie algebra correspondence (see 1.2.3) implies that there is a close relation between representations of a Lie group G and its Lie algebra g, as well as for equivariant mappings between such representations. While representations of general Lie algebras may be very complicated, there is a satisfactory representation theory for semisimple Lie algebras, which can also be used to analyze more general situations. Moreover, the analysis of the adjoint representation leads to the classification of semisimple Lie algebras. We will start this chapter by briefly discussing elementary properties of Lie algebras and the basic structure theory. Next, we will discuss complex simple Lie algebras and their representations. Then we study real forms, arriving at the description of real semisimple Lie algebras via Satake diagrams. We will briefly sketch the classification of simple real Lie algebras and study real representations. Throughout this chapter, we restrict our attention to real and complex Lie algebras and do not consider other ground fields.
2.1. Basic structure theory of Lie algebras 2.1.1. Abelian, nilpotent and solvable Lie algebras. By definition, a Lie algebra 9 over ][{ = IR or C is a vector space together with a ][{-bilinear mapping [, j : gxg - g, called the Lie bracket, which is skew symmetric, i.e. [Y, Xj = -[X, Yj and satisfies the Jacobi identity, i.e. [X, [Y, Z)) = [[X, Yj, Zj + [Y, [X, Z)) for all X, Y, Z E g. If 9 and ~ are Lie algebras, then a homomorphism ¢ : 9 - ~ of Lie algebras is a ][{-linear mapping which is compatible with the brackets, i.e. such that [¢(X), ¢(Y)j = ¢([X, Y)) for all X, Y E g. The simplest choice for the bracket is the zero map, and in this way we get an abelian Lie algebra, which is just a vector space. If (g, [ , )) is a Lie algebra and A, Beg are nonempty subsets, then we denote by [A, Bj the linear subspace spanned by all elements of the form [a, bj with a E A and b E B. Then there is the obvious notion of a Lie subalgebra ~ in a Lie algebra g, namely a linear subspace which is closed under the bracket, i.e. such that [~, ~j c ~. We write ~ ::; 9 if ~ is a subalgebra of g. Clearly, the intersection of an arbitrary family of subalgebras of 9 is again a subalgebra. Thus, for any subset A c g, there is a smallest subalgebra of 9 which contains A, called the subalgebra generated by
A. To form quotients of Lie algebras, one needs a strengthening of the notion of a subalgebra. We say that a subalgebra ~ ::; 9 is an ideal in 9 and write ~ Cartan's criteria for solvability and semisimplicity: THEOREM 2.1. 5. (1) Let V be a vector space and 9 C gl(V) a Lie subalgebra. If Bv is zero on g, then 9 is solvable. (2) A Lie algebra 9 is solvable if and only if its Killing form has the property that B(g, [g, g]) = O. (3) A Lie algebra 9 is semisimple if and only if its Killing form is nondegenerate. PROOF. (1) By complexifying V, we can view 9 as a subalgebra of the Lie algebra of complex endomorphisms of Ve, and then the complexification ge of 9 is a complex subalgebra in there. Since solvability of ge implies solvability of 9 (see 2.1.4) we may, without 10:>:> of generality, assume that V is a complex vector space and 9 is a complex Lie subalgebra of gl(V). Since g/[g, gj is abelian, it suffices to show that. [g, gj is nilpotent. For an element X E [g, g]let Xs be the semisimple part in the Jordan decomposition of X : V -> V, and we define Xs to be the linear map which has the same eigenspaces as X s , but complex conjugate eigenvalues. Using a basis of V such that X is in Jordan normal form, we see that tr(Xs 0 X) = L: IAjI2, where the Aj are the eigenvalues of X. If we show that this trace vanishes, then it follows that X is nilpotent, which by Engel's theorem implies that [g,g] is nilpotent; see 2.1.1. If Xs would lie in g, then this would follow from the vanishing of Bv. Since this is not the case in general, we need an additional argument: By the lemma, ad(Xs) is the semisimple part of the endomorphism ad(X) of gl(V), so this can be written as a polynomial in ad(X), and therefore ad(Xs)(g) c g.
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On the other hand, from the proof of the lemma we see that ad(Xs) has the same eigenspaces as ad(Xs ), but complex conjugate eigenvalues. Since the projections onto the eigenspaces of ad(Xs) can be written as polynomials in ad(Xs ), we see that ad(Xs) is a polynomial in ad(Xs), so ad(Xs)(g) c g. Now since X E [g,g], we can write it as a finite sum, X = ~]Yi, Zi]. But then
(2) We have seen the necessity of the condition already. Conversely, we show that even B([g, g], [g, g]) = a implies solvability of g. Indeed, by (1) this implies that the image of [g, g] under the adjoint representation is solvable. Since the kernel of the adjoint representation of [g, g] is the center of [g, g], which is an abelian (and hence solvable) ideal, we conclude that [g, g] is solvable. Since the quotient of 9 by the solvable ideal [g, g] is abelian, we conclude that 9 is solvable. (3) For semisimple 9 consider the null space ~ := {X E 9 : B(X, Y) = a VY E g} of the Killing form. By invariance of the Killing form, this is an ideal in g, and by (1) the image ad(~) c gl(g) is solvable. Since ker(ad Ill) = J(g) n ~ is abelian, we conclude that ~ is solvable, and thus ~ = {a}. Conversely, let us assume that B is nondegenerate and that ~ egis an abelian ideal. For X E ~ and Y E g, we see that ad(Y) 0 ad(X) maps 9 to ~ and ~ to zero, so this map is nilpotent and thus tracefree. Hence, X lies in the null space of B, so X = a. This shows that 9 has no nontrivial abelian ideal, which by 2.1.2 implies that 9 is semisimple. D Next we note several important consequences of this result. In particular, as promised in 2.1.4 we show that semisimplicity is well behaved with respect to complexification and we prove that the study of sernisimple Lie algebras reduces to the study of simple Lie algebras. COROLLARY 2.1.5. (1) If 9 is a semisimple Lie algebra, then there are simple ideals g1, ... , gk in 9 such that 9 = g1 61· . '61gk as a Lie algebra. Moreover, 9 = [g, g] and any ideal in 9 as well as any homomorphic image of 9 is semisimple. (2) A real Lie algebra 9 is semisimple if and only if its complexification gc is semisimple. (3) If 9 is a complex simple Lie algebra and IP : 9 x 9 -> C is a g-invariant complex bilinear form, then IP is a multiple of the Killing form. In particular, if IP is nonzero, then it is automatically symmetric and nondegenerate. PROOF. (1) If ~ egis an ideal, then the annihilator ~.l with respect to the Killing form is an ideal by invariance of the Killing form. The Killing form restricts to zero on the ideal ~n~.l, so this ideal is solvable, By semisimplicity, ~n~.l = {a}, and hence 9 = ~ 61~.l and [~, ~.l] = a. The last fact implies that the Killing form of ~ is the restriction of the Killing form of 9 and, in particular, nondegenerate. Thus, ~ is semisimple and cannot be abelian, so the decomposition of 9 into a sum of simple ideals follows by induction. The fact that 9 = [g, g] then follows immediately, since gi = [gi, gil for each of the simple ideals. Finally, let a be any Lie algebra and let 4> : 9 -> a be a homomorphism. Then 4>(g) ~ g/ker(4)), and from above we see that this is isomorphic to the ideal ker(¢).l c g, which is sernisimple.
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(2) From 2.1.4 we know that 9 is semisimple if glC is semisimple. But the converse now immediately follows from part (3) of the theorem and the fact that the Killing form of glC is the complex bilinear extension of the Killing form of g. (3) The adjoint representation of 9 is a complex representation which is irreducible, since a g-invariant subspace in 9 by definition is an ideal in g. Now a complex bilinear form : 9 X 9 - 0, we conclude that SOi (a) E A +. We can continue this process until we reach a simple root. On the one hand, this shows that all the coefficients aj are integers. On the other hand, we conclude that any positive reduced root a may be written as w(aj) for some wE W'. Since -aj = SOj(aj) we conclude that any reduced root may be written in this form. Finally, if a = w(aj), then So = w 0 SOj 0 w- 1 and since S20 = So this implies W' = W. (2) It is almost obvious that for w E W, also W(AD) = {w(al),"" w(a n )} is a simple subsystem. (To obtain a representation of a in terms of the w(aj) use a representation of w-l(a) E A in terms of the aj.) Hence, we obtain a well-defined map from W to the family of simple subsystems. Suppose that w E W has the property that w(ai) E AD for all i. Writing temporarily Si for SOi we know from above that we may write w = Sit 0 ... 0 Si , for some choice of indices i j . Now by assumption w(ait) E AD C A+ while si,(ai,} = -ait E -A+. Thus, there is a minimal r ~ 2 such that for w' = Sir_l 0'" 0 Si, we have w'(ai,} E -A+, but Sir (w' (ail )) E A +. We have seen above that for a E A + with a =J air we have Sir(a) E A+, so we must have w'(ai,} = -ai,.. Since Sir = S-Oi r ' we obtain Sir = SW'(Oil) = w , 0 Si,O (W')-1 . Hence, Sir 0 W" = W 0 Si, = Si"_l 0 ... 0 Si2' W h'ICh implies that w can be written as the product of r - 2 simple reflections. Iterating this, we see that w = id if r is even and w is a simple reflection if r is odd. But the latter case cannot occur, since si(ai) = -ai ~ AD. Hence, we conclude that the mapping w f--+ w(AD) is injective. On the other hand, suppose that A C A is any simple subsystem and D+ C A is the corresponding system of positive roots. Then A C A + implies (and thus is equivalent to) D+ C A + and hence to D+ = A +. If this is the case, then AD C D+, so any element of AD may be written as a linear combination of elements of A with nonnegative integral coefficients and vice versa, which easily implies A = AD. Hence, if A =J AD we can find a root a E An-A+. Since a E A, we see that so(D+) is obtained from D+ by replacing a by -a, so so(D+) n A + is strictly larger than D+nA +. Clearly, so(D+) is the positive system associated to the simple subsystem PROOF.
j
= 1, ... ,n. Suppose that a = L: ajaj
lk~ ,Q~
1
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2. SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
sa(A). Inductively, this implies that we find an element w E W such that the positive system associated to w(A) equals a+ and thus w(A) = 6.0. 0
This immediately implies that the Cart an matrix and hence the Dynkin diagram is independent of the choice of the order (or equivalently of the simple subsystem). We have just seen that any two simple subsystems are related by an orthogonal isomorphism and since the Cartan matrix depends on inner products only, it does not change. We can also conclude that a reduced root system is completely determined by any simple subsystem, since the roots are exactly given by the images of the simple roots under the group generated by the simple root reflections. Finally, let us discuss the concept of dominant elements and Weyl chambers which will be important in the sequel. Let a c V be an abstract root system with a distinguished simple subsystem a ° (or equivalently a distinguished positive subsystem a +). Then an element v E V is called dominant (or a °-dominant) if and only if (v, a) ~ 0 for all a E a ° (or equivalently for all a E a +). The closed dominant Weyl chamber is defined to be the set of all dominant elements in V. In general, one defines an open Weyl chamber to be one of the connected components of the complement of all hyperplanes orthogonal to the roots. Hence, an element of v E V lies in some open Weyl chamber if and only if (v, a) i= 0 for all a E a. Two such elements v and v' lie in the same open Weyl chamber if (v, a) and (v', a) have the same sign for all a. Obviously, the closed dominant Weyl chamber is the closure of the open Weyl chamber for which all the inner products are positive. By construction, any root reflection permutes the open Weyl chambers, so the same is true for any element of the Weyl group. On the other hand, choosing an open Weyl chamber C, one obtains a positive subsystem a + as those roots whose inner product with all elements in the chamber C are positive. This positive subsystem in turn determines a simple subsystem a o. From the construction it is clear that for wE W the simple system associated to w(C) is w(a O), so from above we conclude that the Weyl group acts simply transitively on the set of all open Weyl chambers. In particular, given any element v E V there is an element w E W such that w(v) is dominant. We shall analyze the Weyl group in much more detail in Section 3.2. Here we just note what the Weyl groups and dominant Weyl chambers look like in the classical examples from 2.2.6. For the root system A n - 1 of s[(n, q from 2.2.6(1), the dual space ~o is the quotient of the space of all E ajej by the line generated by el + ... + en, so we may view it as the space of all E ajej such that E aj = O. One easily verifies that the root reflection Sej -ej : ~o ~ ~o is induced by the map which exchanges ei and ej and leaves the ek for k i= i,j untouched. Hence, the Weyl group of A n - 1 is the permutation group 6 n of n elements. Elements of the dominant Weyl chamber by definition are represented by expressions E ajej such that a1 ~ a2 ~ ... ~ an. For the root system Bn of so(2n + 1, q from 2.2.6(3), the reflections in ei - ej again exchanges ei and ej, while the reflection in ej changes the sign of ej and leaves the other ek untouched. Thus, we may view the Weyl group W as the subgroup of all permutations 0' of the 2n elements ±ej such that 0'( -ej) = -O'( ej) for all j = 1, ... , n. Otherwise put, W is a semidirect product of 6 n and (Z2)n, so in
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particular, W has n!2n elements. The dominant Weyl chamber by construction consists of all element E ajej such that al ~ ... ~ an ~ O. Since the reflection corresponding to 2ej coincides with the reflection for ej, for the root system Cn from 2.2.6(4) we get the same Weyl group and the same positive Weyl chamber as for Bn. Finally, for the even orthogonal root system Dn from 2.2.6(2), the reflections in the roots ei - ej again generate permutations of the ej, while the reflection in ei + ej maps ei to -ej and ej to -ei while all other ek remain untouched. Consequently, W can be viewed as the subgroup of those permutations 7r of the elements ±ej which satisfy 7r( -ej) = -7r( ej) and have the property that the number of j such that 7r(ej) = -ek for some k is even. In particular, the number of elements in W equals n!2n- 1 • The positive Weyl chamber consists of all E ajej such that al ~ a2 ... ~ an and an- l ~ -an. 2.2.8. The classification of Dynkin diagrams. To classify reduced abstract root systems, it suffices by our observations in 2.2.7 to classify abstract Cartan matrices or equivalently Dynkin diagrams. Clearly, we may restrict ourselves to irreducible reduced root systems and thus to connected Dynkin diagrams. Apart from the classical examples An, B n , C n and Dn that we have met in 2.2.6, there are five exceptional irreducible root systems which are called G 2 , F 4 , E 6 , E7 and Es. The index refers to the dimension of the Euclidean space in which these systems sit. The Dynkin diagram of G 2 is 0;$0=0, for F4 one obtains 0 ~ o. For the E o
series, the Dynkin diagram of Es is , and the diagrams for E7 and E6 are obtained from this by removing the leftmost, respectively, the two leftmost vertices and edges. Now the classification of abstract root systems reads as: THEOREM 2.2.8. Any irreducible reduced root system is isomorphic to exactly one of the systems An (n ~ I), Bn (n ~ 2), Cn (n ~ 3), Dn (n ~ 4), E 6 , E7, E s , F4 , or G 2. Any non-reduced irreducible root system is isomorphic to (BC)n := Bn U C n for some n ~ 2. SKETCH OF PROOF. For the reduced case, there is a very elegant proof based on extended Dynkin diagrams which we have taken from [CSM95j: We have seen in 2.2.5 that for the Cartan matrix A of an abstract root system there is a diagonal matrix D with positive entries, such that the matrix S := DAD- 1 is symmetric and positive definite. One immediately sees that the entries Sij of S are given by Sii = 2 and Sij = - vn;; for i f. j, where nij is the number of edges in the Dynkin diagram joining the ith and the jth vertex. The proof is done by showing that the Dynkin diagrams listed in the theorem are the only connected diagrams such that the associated matrix S is positive definite. Since S is obviously independent of the arrows in the diagram, we may forget about them for the purpose of the classification. It is elementary to see that the quadratic form associated to S remains positive definite if some of the nij are decreased. On the other hand, if we take a connected subset of a diagram corresponding to a positive definite S, then the associated matrix just describes the restriction of the inner product to some subspace, and hence
17S
2. SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
is positive definite, too. For the rest of this proof, we use the term "subdiagram" for any diagram which can be obtained from a given one by these two operations. The upshot of this is that if we find a diagram for which the corresponding matrix S has zero determinant, then this cannot occur as a sub diagram in the Dynkin diagram of any irreducible abstract root system. A smart way to construct such diagrams is to add to the Dynkin diagram of an irreducible abstract root system one more vertex corresponding to the largest root of the system (in the given ordering) and to add the edges corresponding to the inner products with the simple roots. This leads to the following diagrams, which are called the extended Dynkin diagrams. (Any diagram with index k in this list has k + 1 vertices.)
-
Es:
?
0-0-0---0--0--0,
-
F4 :
0--0-0=0---0,
-
G2 :
0
o.
The matrix associated to any of these diagrams by construction has linearly dependent lines and thus zero determinant. This may also be verified directly for each of the extended Dynkin diagrams. With the extended Dynkin diagrams at hand, the classification proceeds very quickly: The diagram Ak shows that there are no cycles in the Dynkin diagram of an abstract root system. We already know that there may be at most triple edges. The diagrams ih, 62 , D4 and th show that there are at most three edges connected to one point. In particular, if there is a triple edge, then G2 is the only possibility. Let us call a vertex in which three single edges meet a branch point. Suppose that a Dynkin diagram contains a double edge. Then in view of Bk and 6k there may only be one double edge and no branch points. The diagram F4 shows that F4 is the only possible Dynkin diagram in which further edges are attached to both vertices joined by a double edge. The remaining possibilities in case of a double edge are only Bn and en. Thus, we are left with the case that the Dynkin diagram contains only single edges. If there is no branch point, the absence of loops implies that the diagram is of type An. Moreover, in view of Dk , there may be at most one branch point in each Dynkin diagram. From the diagram E6 we see that at least one of the three chains meeting at a branch point must consist of a single vertex, and E7 shows that one of the two remaining chains consists of at most two vertices. But then Es shows that D n , E 6 , E7 and Es are the only possible diagrams. In the non-reduced case, one shows that the subset of all roots O! such that 20 is not a root forms an irreducible reduced abstract root system. Using this, the result easily follows from the reduced case; see [Kn96, 11.8]. 0 2.2.9. The classification of complex simple Lie algebras. We have seen how to pass from a Lie algebra to a root system and further to a Dynkin diagram. We have also noted in 2.2.7 that this Dynkin diagram does not depend on the choices
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of a Cartan subalgebra and a set of positive roots. In particular, this implies that isomorphic Lie algebras lead to the same Dynkin diagram. Hence, there are two remaining questions. On the one hand, we do not know whether there exist complex simple Lie algebras corresponding to the exceptional root systems of type E, F and G from 2.2.8. On the other hand, we do not know whether two Lie algebras having the same Dynkin diagram must already be isomorphic. Both questions can be simultaneously answered (positively) by giving a universal construction for a simple Lie algebra with a given Dynkin diagram, using the so-called Serre relations. These will also be important in the study of real semisimple Lie algebras later on. Let us start from a complex simple Lie algebra 9 with a chosen Cartan subalgebra I), the corresponding set ~ of roots and a chosen simple subsystem ~o = { a1, ... , an}. For any j = 1, ... , n choose elements E j and Fj in the root spaces ga j , respectively, g-a such that B(EJ·, FJ·) = 2( .)' Recall from 2.2.4 that this means ~~. that H j := [Ej, Fj ] satisfies aj(Hj ) = 2, so {Ej, Fj , H j } is a standard basis for the subalgebra Saj ~ 5[(2, C). Moreover, the elements H j for j = 1, ... ,n span the Cartan subalgebra I). By part (3) of Proposition 2.2.4 we have ga+13 = [ga, g13] for all a, f3 E ~. Together with the above, this easily implies that {Ej, Fj , H j : 1 ~ j ~ n} is a set of generators for the Lie algebra g. Such a set of generators is called a set of standard generators for g. Next, there are some obvious relations. Since all H j lie in I), we have [Hi, H j ] = o for all i, j. By definition and the fact that the difference of two positive roots is not a root, we further have [Ei' Fj ] = 8ijHi. Next, by definition of the Cartan matrix A = (aij) of g, we have [Hi, E j ] = aijEj and [Hi, Fj ] = -aijFj . Finally, the formula for the length of the ai-string through aj from 2.2.4 implies that ad(Ei )-a ij +1(Ej ) = 0 and ad(Fi )-aii +1(Fj ) = 0 for i =f:. j. These six families of relations are called the Serre relations for g. The essential point for our questions now is that this is a complete set of relations. To be more precise one proves
.
THEOREM 2.2.9. Let A = (aij) be an abstract nxn Cartan matrix. Let'J be the free complex Lie algebra generated by 3n elements E j , Fj and H j for j = 1, ... ,n, and let 9l be the ideal generated by the Serre relations. Then 9 := 'J/9l is a finitedimensional simple Lie algebra. The elements H j span a Cartan subalgebra of g, and the functionals aj E 1)* defined by aj (Hi) = aij form a simple subsystem of the corresponding root system. In particular the Cartan matrix of 9 is exactly A. PROOF. The proof is rather involved; see [Kn96, 11.9-11.11].
0
COROLLARY 2.2.9. (1) Any reduced irreducible abstract root system is isomorphic to the root system of some finite-dimensional complex simple Lie algebra. (2) Two complex simple Lie algebras are isomorphic if and only if their root systems are isomorphic, i. e. if and only if they have the same Dynkin diagram. PROOF. (1) is obvious from the theorem in view of the bijective correspondence between Cartan matrices and reduced irreducible abstract root systems described in 2.2.7. (2) Let 9 be any complex simple Lie algebra and let A be its Cartan matrix. Let 'J/9l be the Lie algebra constructed from A in the theorem. By the universal property of a free Lie algebra, choosing a set of standard generators for 9 gives a surjective homomorphism 'J ---t g, which factors to 'J/9l since the Serre relations hold in g. But from the theorem we know that 'J/9l is simple, which implies that this homomorphism must be injective, and thus an isomorphism. 0
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REMARK 2.2.9. (1) The description of a complex simple Lie algebra by generators and relations corresponding to a Cartan matrix is the basis for various stages of generalizations of these Lie algebras in the direction of Kac-Moody algebras. Weakening the conditions on an abstract Cartan matrix from 2.2.5 slightly and then defining a Lie algebra by generators and relations as in the theorem, one obtains infinite-dimensional Lie algebras, which, however, behave similarly to complex simple Lie algebras in several respects; see for example [Kac90j. (2) While the theorem asserts the existence of the exceptional complex simple Lie algebras, i.e. Lie algebras corresponding to the exceptional root systems, it does not offer a good description of these. There are various ways to give more explicit descriptions of the exceptional Lie algebras. One way is to describe them in the form W EEl go EEl W*, where go is a certain complex semisimple Lie algebra, W is a representation of go with dual representation W*. The bracket is described by the bracket of go, the action of go on W and W*, and certain pairings on the representation spaces. This description of the exceptional algebras is outlined in [FH91, §22.4j. Other conceptual descriptions of the exceptional algebras are related to the octonions (or Cayley numbers) 0, the unique 8-dimensional (non-associative) normed real division algebra, and the exceptional Jordan algebra .JJ of dimension 27 formed by Hermitian 3 x 3-matrices with entries from O. The simplest case is the Lie algebra G 2 , which is the complexification of the Lie algebra of derivations of O. Similarly, the complexification of the Lie algebra of derivations of.JJ is a Lie algebra of type F 4 • For a discussion of the relation between octonions and exceptional Lie algebras see [Ba02j. Finally, one should also note that the Lie algebra of type G 2 can also be viewed as the algebra of endomorphisms of a seven-dimensional vector space V, which preserves a generic element of the third exterior power A3V. This point of view is important for the study of G 2-structures on seven-dimensional manifolds.
2.2.10. Finite-d.imensional representations. Let us fix a complex simple Lie algebra g, a Cartan subalgebra ~ S; g and let .6. be the corresponding set of roots. Fix an order on ~*, and let .6.+ and .6.0 be the corresponding sets of positive respectively simple roots. By ~o c ~* we denote the real span of the roots. Let ( , ) be the complex bilinear form on ~* induced by the Killing form. By part (1) of Theorem 2.2.4, the restriction of ( , ) to ~o is positive definite. From 2.2.2 we know that any finite-dimensional representation of g admits a weight decomposition into a direct sum of joint eigenspaces for the actions of the elements of~. These are the weight spaces and the eigenvalues, called weights, are linear functionals on ~. We will also use the term weights for general linear functionals on ~. To describe properties of the weights of finite-dimensional representations, we have to introduce a few notions. A weight A E ~* is called real if it lies in the subspace ~o, or equivalently, if (A, a) E lR. for all roots a E .6.. A (real) weight A is called algebraically integral if 2«A,"'}} E Z for all a E .6.. Ct,Ct Recall from 2.2.7 that the simple roots form a complex basis for ~* and a real basis for ~O. Writing.6.° = {a1, ... ,an }, we define elements W1, ... ,Wn E Ct ~o by requiring that 2(.. Then we have: (1) Any weight A of V is algebraically integral and at least one weight of V is dominant. (2) For any weight A E wt(V) and any w E W, also W(A) E wt(V) and the two weights have the same multiplicity in V. (3) Suppose that for A E wt(V) and a E A the integer k := 2((>.,,,,» is positive. "','" Then for each £ E {1, ... , k} we have A - £a E wt(V) with at least the same multiplicity as A. PROOF. Consider a fixed positive root a E A +, let 5", = 9-",61[9"" 9-0]619-", ~ 51(2, q be the corresponding subalgebra; see Proposition 2.2.4. For A E wt(V) consider V' := EBnEZ V>'+n C V. This subspace is invariant under the action of 5"" so we can apply Proposition 2.2.3. In 2.2.4 we have constructed a standard basis of 5", which contained the element (",~",)H", E [9""9-,,,]. By Proposition 2.2.3, this element has integral eigenvalues on any finite-dimensional representation, which shows that 2((>.,,,,» E Z. Further, the eigenvalues of 2() H", form an unbroken a,o Q,Q string of the form a, a - 2, a - 4, ... ,-a. Now, 2(A - £a, a) _ 2(A, a) _ 2£ (a, a) - (a, a) ,
so we see that, starting from A, we must also have the weights A - £a for £ 1, ... , 2(~:/ which proves (3). Moreover, for the maximal value of £, by definition, we obtain the weight S",(A). We also see that V>. and Vs ,,(>.) must have the same dimension. Since the Weyl group is generated by the reflections s"', this implies (2). Since any weight can be mapped to the dominant Weyl chamber by an element of W, we obtain the last claim in (1). 0
'
2.2.11. Highest weight vectors. For the next step, we will briefly leave the realm of finite-dimensional representations, and also allow infinite-dimensional ones. However, we stay in a purely algebraic context, so we do not need topologies on the representation spaces. We continue to fix 9 and IJ and use the notation of the last subsection. Assume that V is a representation of 9 such that the action of any element of the Cart an subalgebra IJ is diagonalizable. This means that V admits a (possibly infinite) decomposition into weight spaces. A highest weight vector in V is a weight vector v E V such that X . v = 0 for any element X lying in a root space 90 with a E A + . Let us fix a set {Ei,Fi,Hi : i = 1, ... ,n} of standard generators for 9; see 2.2.9. Then a weight vector v E V evidently is a highest weight vector if and only if Ei . v = 0 for all i = 1, ... ,n. THEOREM 2.2.11. Let 9 be a finite-dimensional complex semisimple Lie algebra. Let IJ :s 9 be a fixed Cartan subalgebra and let A be the corresponding set of roots. Fix an or-der- on the real span 1J 0 of A, let A + and A 0 be the sets of positive and simple roots with respect to this order, and let {Ei' Pi, Hi : i = 1, ... , n} be a standard set of generators. Let V be a representation of 9 such that the elements of IJ act simultaneously diagonalizable.
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(1) For any highest weight vector v E V the elements of the form Fi! ... Fit· v span an indecomposable subrepresentation V' of V. Denoting by A the weight of v, all weights occurring in V' have the form A - I: niai for ai E A O and nonnegative integers ni, and the weight space V~ is one-dimensional. (2) Suppose further that V is finite-dimensional. Then the weight of any highest weight vector is dominant and algebraically integral, and there exists at least one highest weight vector. In this case, the submodule V' C V from (1) is irreducible.
PROOF. (1) The subspace V' spanned by the elements of the claimed form is by construction invariant under the action of all Fj . Now for H E ~ and w E V we compute
H· Fi . W = [H, Fi ]· w + Fi . H . w = -O!i(H)Fi . W + Fi . H . w. Inductively, this shows that V' is ~-invariant and that Fi! ... Fit·v is a weight vector of weight A - ail - ... - ait. Likewise, E j • (Fi! ... Fit . v) is a linear combination of Fi! . E j . Fi2 ... Fie· v and Hi . Fi2 ... Fie· v. Again, by induction, this implies that Viis invariant under the action of each E i . Since the elements Ei , Fj , and H j generate 9 as a Lie algebra, we see that V' C V is a g-invariant subspace. Moreover, the weight spaces of V' have the claimed form and dim(VD = 1. To prove that V' C V is indecomposable, suppose that V' = V{ ffi V2 as a representation of g. By assumption, V is a direct sum of weight spaces, so any element in V can be written as a finite sum of weight vectors of different weights. Exchanging the two summands if necessary, we may assume that V{ contains an element of the form v + VI + ... + VN, where the Vi are weight vectors of pairwise different weights, which are all different from A. Fix an element H E ~. Then each Vj is an eigenvector for the action of H and only finitely many eigenvalues aI, ... , ar occur. Now by g-invariance, the element
+ ... + VN) By construction, this is a nonzero multiple of v + ViI + ... + Vi., where the (p(H) - al)
0 ••• 0
(p(H) - ar)(v +Vl
lies in V{. Vi. are those Vj on which the action of H has the same eigenvalue as on v. Doing this construction step by step for the elements of a basis of~, we obtain a nonzero multiple of v plus the sum of those Vi on which all elements of ~ have the same eigenvalue as on v. But since the weight space V~ is one-dimensional, this means that v E V{ and hence V{ = V'. (2) By Proposition 2.2.10, all weights of V lie in ~o and the choice of positive roots leads to an ordering on this space. Since V has only finitely many weights, there is a weight AO E wt(V) such that A ::; AO for all A E wt(V). If v is any nonzero element of V>'o' then v must be a highest weight vector, since Ei . v has weight AO + ai, which is strictly larger than AO. If (AO, O!i) < 0 for some simple root aj, then SCij (AO) = AO - 2(~o::/» O!j is strictly larger than AO. By Proposition 2.2.10, this is a weight of V which again contradicts maximality of AO. Finally, by Theorem 2.1.6, finite-dimensional indecomposable representations are irreducible. 0 For a finite-dimensional representation V, the maximal weight used in the proof of part (2) is often called the highest weight of V, in particular if V is irreducible.
2.2.12. Existence of finite--dimensional irreducible representations. Let V be a finite-dimensional irreducible representation of a semisimple Lie algebra g. Then by Theorem 2.2.11 there is exactly one weight space V>. which contains
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a highest weight vector, and the weight A is dominant and algebraically integral. Moreover, if V and V' are irreducible representations with the same highest weight A, then we choose highest weight vectors v E V and Vi E V'. Then (v, Vi) is a highest weight vector in the finite-dimensional representation V E9 V' of g, so it generates an irreducible subrepresentation if C V E9 V'. The restrictions of the two projections to if define homomorphisms if -+ V and if -+ V'. The homomorphism if -+ V is nonzero, since v lies in the image. By irreducibility it must be an isomorphism. Similarly, the other homomorphism if -+ Viis an isomorphism, so V S:! V'. Conversely, isomorphic irreducible representations obviously have the same highest weight. Thus, to get a complete hand on the finite-dimensional irreducible representations (and thus by complete reducibility 2.1.6 on all finite-dimensional representations), the remaining question is for which dominant algebraically integral weights A there exists a finite-dimensional irreducible representation with highest weight
A. THEOREM 2.2.12 (Theorem of the highest weight). If 9 is a finite-dimensional complex semisimple Lie algebra, then for any dominant algebraically integral weight A E ~o there is a (up to isomorphism) unique finite-dimensional irreducible representation with highest weight A. We will next sketch two different general proofs for the existence part of this theorem, namely via Verma modules and using the Borel-Weil theorem. A more pedestrian approach to the proof on a case by case basis will be discussed in 2.2.13 below. Verma modules. The first approach uses the universal enveloping algebra U(g) of the Lie algebra 9 and induced modules as discussed in 2.1.10. Fix a Cartan subalgebra ~ ~ 9 with roots ~ and an ordering on ~o with corresponding sets ~ + and ~0 = {a1, . .. , an} of positive and simple roots. Let n± C 9 be the sum of all positive (respectively negative) root spaces. Clearly, these are nilpotent subalgebras of g. Now the standard Borel subalgebra b ~ 9 corresponding to these choices is defined as b = ~E9n+. Clearly, this is a subalgebra of 9 and the commutator algebra [b, b] is the nilpotent subalgebra n+, so b is solvable. It is easy to see that, in fact, b is a maximal solvable subalgebra of g. The fact that any two Cart an subalgebras as well as the choice of the order can be absorbed in an inner automorphism of 9 (see 2.2.2 and 2.2.7) can be rephrased as the fact that any two maximal solvable subalgebras of 9 are conjugate under an inner automorphism. It is easy to describe all irreducible representations of b. In 2.1.3 we have observed that such representations are given by linear functionals on b/[b, b] S:! ~. Choosing A E ~* we thus obtain an irreducible representation C A of b. Since beg we may view U(b) as a subalgebra of U(g) and we define the Verma module Mb(A) with highest weight A as the U(g)-module induced by C A, i.e. Mb(A) = U(g) 0U(b) C A ; see 2.1.10. The action of U(g) (and thus also the action of g) comes from multiplication from the left in U(g). Since the subalgebra n_ egis complementary to b, we see from 2.1.10 that as an n_-module the Verma module Mb(A) is isomorphic to U(n_) 0 CA' Let us number the positive roots as ~ + = {t3b ... , t3d and choose an element F{3i E g-{3i for each i = 1, ... , e. Then from 2.1.10 we know that the elements F~~ ... F~! 0 1 form a linear basis of Mb(A). To obtain the action of an element H E ~ in
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this picture, one has to use the commutation relation H Ff3i - Ff3i H = [H, Ff3J = -!3i(H)Ff3i to commute H through the F's, and finally bring H across the tensor product, which adds a factor A(H). In particular, this implies that F~~ ... F~! 01 is a weight vector of weight A - (i 1 !31 + ... + it!3t). Hence, Mb(A) is a direct sum of weight spaces and each weight space is obviously finite-dimensional. By construction, 101 is a highest weight vector in Mb(A) which generates the whole module. In particular, Mb(A) is indecomposable by part (1) of Theorem 2.2.11. Note also that Frobenius reciprocity becomes particularly simple for Verma modules. By Proposition 2.1.10, restriction to 10C.>. induces an isomorphism Homg(Mb(A), V) ~ Homb(C,>" V) for any g-module V. Of course, a linear map C.>. -+ V is determined by the image v of 1 E C.>., and v gives rise to a homomorphism if and only if n+ acts trivially on v, while each H E ~ acts by multiplication by A(H). Hence, we conclude that g-homomorphisms Mb(A) -+ V are in bijective correspondence with highest weight vectors of weight A in V. From the proof of Theorem 2.2.11 we see that for a g-submodule N C Mb(A) an element x E N can be written as a finite sum of weight vectors of different weights and each of the components of this sum again lies in N. This immediately implies that for a proper submodule N there may never be a nonzero component of weight A. This in turn shows that the subspace spanned by an arbitrary family of proper submodules of Mb(A) is again a proper submodule, and hence Mb(A) contains a unique maximal proper submodule. Let L(A) be the quotient of Mb(A) by this maximal proper submodule. Then by construction L(A) is irreducible and has highest weight A. Hence, we see that for any A E ~* there is an irreducible representation of highest weight A, so to prove the theorem of the highest weight it remains to show that L(A) is finite-dimensional if A is dominant and algebraically integral. To do this, the main step (see [Kn96, Theorem 5.16]) is to show that under this assumption the set of weights of L(A) (including multiplicities) is stable under the action of the Weyl group. Having shown this, it is easy to see from the explicit description of Mb(A) above that there are only finitely many dominant weights, which implies that L(A) is finite-dimensional. The description above shows that any irreducible finite-dimensional representation of 9 is a quotient of a Verma module. In fact, a much more precise description is possible. It turns out that any finite-dimensional irreducible representation of 9 admits a finite resolution by Verma modules, the so-called Bernstein-GelfandGelfand resolution, and the Verma modules showing up in this resolution can be described precisely in terms of the Weyl group. We will study this resolution and give a geometric construction in volume two. The Borel-Weil theorem. The Borel-Weil theorem offers a geometric construction of finite-dimensional irreducible representations. We only briefly outline the statement here, since we will discuss the extension of this theorem due to Bott in more detail in Section 3.3. Starting with a complex semisimple Lie algebra g, let G be a connected and simply connected complex Lie group with Lie algebra g. Let b :$ 9 be a Borel subalgebra (which gives a choice of a Cartan subalgebra and a notion of positivity), and let BeG be the normalizer of b, i.e. B = {g E G : Ad(g)(b) c b}. This is obviously a closed subgroup and thus a Lie subgroup of G, and one easily shows that the Lie algebra of this subgroup is beg. The homogeneous space G / B turns out to be a compact Kiihler manifold
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(see 3.2.6 and 3.2.8), which is called the full flag variety of G. Next, one shows that for a dominant integral weight A E ~o the one-dimensional representation C_>. of b integrates to a representation of B, so we can form the associated line bundle G x B C_>. -+ G / B. From 1.4.4 we know that the space of smooth sections of this bundle naturally is a G-representation. Since G acts holomorphically on G / B, the (global) holomorphic sections form a subrepresentation. The Borel-Weil theorem then states that this representation is irreducible and the corresponding irreducible representation of g has highest weight A. 2.2.13. Fundamental representations - examples. The case-by--case approach to proving existence of irreducible finite-dimensional representations is based on the following observation: Suppose that V and Ware finite-dimensional irreducible representations of g with highest weights A and f.L, respectively. If v E V and w E W are highest weight vectors, then v ® w is a highest weight vector in the tensor product representation V ® W with weight A + f.L. By Theorem 2.2.10, this vector generates an irreducible subrepresentation of V ® W with highest weight A + f.L. Moreover, if V = EB V>" and W = EB WIL' are the weight decompositions, then the weight spaces in V ® W have the form EB>"+IL'=V V>" ® WIL" In particular, any weight 1/ of V ® W is ~ A + f.L. The subrepresentation generated by the highest weight vector above is usually called the Cartan product of V and W and is denoted by V@W. Recall from 2.2.10 that the dominant algebraically integral weights of g are exactly the linear combinations of the fundamental weights WI,'" ,Wn , with nonnegative integral coefficients. The irreducible finite-dimensional representation Vi with highest weight Wi is called the ith fundamental representation. Suppose that we have constructed the fundamental representations Vb"" Vn . Given a dominant integral weight A = alwl +.. ·+anwn , consider the representation V1181a1 ® .. ·®Vnl8l"n. From above we see that this contains a unique (up to scale) highest weight vector of weight A. We can also use the symmetric powers sa; Vi of the fundamental representations rather than the tensor powers. In any case, we obtain a unique irreducible subrepresentation of highest weight A. In simple cases we can even obtain an explicit description of this highest weight representation, since we have to find all homomorphisms to irreducible representations of lower highest weight and consider the intersection of their kernels. Hence, the essential step that remains is to construct the fundamental representations, which, at least for the classical algebras, is fairly simple. Let us discuss this for the classical examples using the notation of 2.2.6. Representations of s[(n, q. As in 2.2.6(1), the Cartan subalgebra ~ consists of all tracefree diagonal matrices, and the functionals ei for i = 1, ... ,n are given by extracting the ith diagonal entry. The simple roots al, ... , a n - l are given by ai = ei - ei+l, and the Killing form satisfies (ai, ej) = 2n(di,j - di+l,j)' By definition of a fundamental weight, we must have ~(W.i,OI.i» = dij, from which one OtJ,Ck J easily concludes that Wi = el + ... + ei for i = 1, ... ,n - 1. Let V = cn be the standard representation of s[(n,q. If {VI, ... ,Vn} is the standard basis, then obviously each Vi is a weight vector of weight ei. Consequently, the highest weight of V is el, so V contains the fundamental representation VI as an irreducible subrepresentation. Next, consider the exterior powers AjV of the standard representation for j = 2, ... , n - 1. The elements Vi 1 1\ ... 1\ Vij for
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1 ~ il < ... < ij ~ n form a basis for Aiv, which implies that the weights of AjV are given by all expressions of the form eil + ... + eij for 1 ~ i1 < ... < ij ~ n. In particular, W j = e 1 + ... + ej is a weight (and in fact the highest weight) of Ai V, so this contains the fundamental representation Vj as an irreducible subrepresentation. By Proposition 2.2.lO, the set of weights of a finite-dimensional representation must be invariant under the action of the Weyl group. From 2.2.7 we know that the Weyl group of s[{n, q is the permutation group 6 n which permutes the ei. But this shows that for each of the representations Aj V all weights are obtained from the highest weight by the action of the Weyl group. In particular, the fundamental representation cannot be strictly smaller, so AjV = Vj for j = 1, ... , n - 1, and we have found all fundamental representations. Note that An V is the trivial representation, since the action of g[{ n, q on Ane is given by multiplication by the trace. This implies that the wedge product induces a duality between the s({n, q representations AjV and An-jV, which shows that An-jv ~ AjV*. From the description of the fundamental representations we see that any finitedimensional irreducible representation shows up as a subrepresentation of a sufficiently high tensor power of the standard representation V = en. There is a general method how to obtain these subrepresentations, which is based on the description of dominant algebraically integral weights in terms of partitions of integers. These in turn may be described by Young diagrams, which define Young symmetrizers in an appropriate permutation group. This symmetrizer then acts on a tensor power of V by permutations of the factors, and the image is exactly the irreducible representation corresponding to the partition. This construction goes under the name of Schur functors, an account can be found in [FH91, §15.3].
Representations of so{2n, q. Following 2.2.6(2), the simple roots 01, ... , On are given by OJ = ej - ej+1 for j < n and On = en -1 + en. Using that the Killing form is a multiple of the standard form, one easily verifies that Wj = e1 + ... + ej for j < n - 1, Wn-1 = He1 + ... + en-1 - en) and Wn = ~(el + ... + en). For the standard representation V = e 2n , one immediately sees that the weights are ±ei for i = 1, ... , n. Since by 2.2.7 the Weyl group acts by permuting the ei and changing the sign of an even number of e's, we immediately see that the orbit of the highest weight el under the Weyl group is exactly the set of all ±ej, so we see that e2n is the first fundamental representation Vl. As in the case of s({ n, q above, we next conclude that for j = 2, ... , n - 2 the exterior power AjV has to contain the fundamental representation Vj. It turns out that AjV is irreducible for j = 1, ... , n - 1 and splits into two irreducible components for j = n. This can be either verified directly (see [FH91, Theorem 19.2]) or deduced from the Weyl-dimension formula; see 2.2.18 below. In particular, for j = 1, ... , n - 2 the representation AiV is the jth fundamental representation. The remaining two fundamental representations are the two spin representations. They cannot show up in any tensor power of the standard representation, since a tensor power contains only weights which are integral linear combinations of the ej, while half integers cannot occur. A construction of the spin representations is described in detail in [FH91, Lecture 20]. For irreducible representations whose highest weight is an integral linear combination of the ej, there is again a version of Schur functors; see [FH91, §19.5].
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Representations of so(2n + 1, C). From 2.2.6(3) we know that the simple roots 01, ... , On are given by OJ = ej - ej+1 for j < n and OJ = en, and the Killing form is a multiple of the standard form. Using this, one easily verifies that Wj = e1 + ... +ej for j = 1, ... , n-1, while Wn = ~(e1 + .. ·+en ). So here only the last fundamental representation is of different nature than the others. Similarly, as for the even orthogonal algebras, one shows that for the standard representation V = C2n+ 1 the exterior power Aiv is the jth fundamental representation for j = 1, ... ,n -1. In fact, even An V is irreducible, but not a fundamental representation. It should be noted, however, that for the odd orthogonal algebras the weights of the standard representation are not in one orbit of the Weyl group any more, since apart from ±ej also 0 is a weight. The last fundamental representation is the spin representation; see [FH91, Lecture 20]. A version of Schur functors is available for representations of so(2n + 1, C) whose highest weight is an integral linear combination of el. ... ,en. Representations ofsp(2n, C). Following 2.2.6(4), the simple roots 01,···, On are given by OJ = ej - ej+1 for j < n and On = 2e n . Since the Killing form is again a multiple of the standard form, one concludes that Wj = e1 + ... + ej for all j = 1, ... ,n, so the situation here is simpler than for the orthogonal algebras. The weights of the standard representation V = C2n are given by ±ej for j = 1, ... , n, so they lie on one orbit of the Weyl group, which acts by permutations and sign changes of the ej; see 2.2.7. Thus, V is irreducible and hence coincides with the fundamental representation V1 . Obviously, the fundamental weight Wj shows up as the weight of a highest weight vector in Aj V, but in contrast to the earlier cases, the exterior powers are not irreducible any more. The point here is that the symplectic form on V is an invariant element of A2V*. Contracting with this form defines a sp(2n,C)-homomorphism AjV -+ Aj- 2V. Since this map is clearly nonzero, its kernel is a nontrivial subrepresentation Vi, which turns out to be the jth fundamental representation. This can be proved directly (see [FH91, Theorem 17.5]) or using the Weyl dimension formula. Again, the formalism of Schur functors can be adapted to the symplectic case. 2.2.14. The isotypical decomposition. We next discuss some tools which can be used to decompose representations into components. From Theorem 2.1.6 we know that any finite dimensional representation of a complex semisimple Lie algebra 9 splits into a direct sum of irreducible representations, but the proof of this result does not tell us how to construct such a splitting. For most purposes, it is better to stick to a slightly coarser but more natural splitting. The point here is that for a direct sum of copies of a single representation, there is no canonical choice of the summands. To avoid this problem, it is better to only split into components corresponding to different irreducible representations, which leads to the isotypical decomposition. From part (2) of Theorem 2.2.11 we know that any finite-dimensional representation V of 9 contains at least one highest weight vector. For a given weight >., any linear combination of highest weight vectors of weight >. again is a highest weight vector of that weight, so these form a subspace V~ C V. The dimension of this subspace is called the multiplicity mA(V) of the irreducible representation fA with highest weight>. in V. By Theorem 2.2.11, any element v E V~ generates an irreducible subrepresentation of V which is isomorphic to fA, so we see that fixing
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a highest weight vector in r A leads to an isomorphism Homg(rA' V) 2'! V~. Thus, the multiplicity of r A in V can also be interpreted as the dimension of the space of all homomorphisms r A -+ V. Next, we define the isotypical component VA C V of highest weight A to be the g-subrepresentation of V generated by V~. Choosing a basis of V~ gives rise to an isomorphism VA 2'! (rA)m-\(V). Alternatively, one may also view the isotypical component V A as the image of the evaluation homomorphism r A12) Hom g (r A, V) -+ V defined by x 12) ¢ t-'> ¢(x). THEOREM 2.2.14. Let V and W be finite-dimensional representations of a complex semisimple Lie algebra g. Then we have: (1) V = EB VA 2'! EB(rA)m-\(V), where the sum goes over all weights A such that mA(V) > 0. (2) Restriction to the subspaces V~ induces an isomorphism A:m",(V»O
PROOF. (1) The sum of all isotypical components clearly is an invariant subspace of V. By complete reducibility (Theorem 2.1.6) this admits an invariant complement, which by construction cannot contain any nonzero highest weight vector. Thus, it must be zero by part (2) of Theorem 2.2.11, so the isotypical components span V. On the other hand, the highest weight vectors in VA are exactly the elements of Vr Starting with the maximal weight A for which mA(V) > 0, we see from Theorem 2.2.11 that all weights in the sum of the other isotypical components are strictly smaller than A. Hence, the intersection of VA with this sum cannot contain any highest weight vectors and must be zero. This shows that V A splits off as a direct summand and inductively we conclude that V is the direct sum of the isotypical components. (2) If ¢ : V -+ W is a g-homomorphism and v E V~, then ¢(v) must be a highest weight vector of weight A, and thus ¢(V~) C wf. Moreover, the restriction of ¢ to V A is determined by the restriction to V~, so by part (1) we obtain a well-defined injective linear map Homg(V, W) -+ EE1AL(V~, Wf)· Conversely, assume that for each A such that mA(V) > 0, we have given a linear map ¢A : V~ -+ Wf. Then for each such A and each v E V~, the element (v, ¢A(V)) is a highest weight vector of weight A in V EE1 W. Let if C V EE1 W be the g-submodule generated by all of these elements. Then one immediately verifies that the restriction of the first projection to if is an isomorphism if -+ V. The composition of the restriction of the second projection with the inverse of this isomorphism clearly defines a g-homomorphism ¢ : V -+ W, which restricts to ¢A on each V~. 0 Note that the first part of this theorem implies that V is determined up to isomorphism by the multiplicities mA(V). On the other hand, the second part evidently is a generalization of Schur's lemma from 2.1.3. 2.2.15. The Casimir element. Recall from 2.1.10 that representations of a Lie algebra 9 are the same thing as representations of the universal enveloping algebra U(g). Now if u is any element in the center of U(g), then the action of u on any representation commutes with the action of any element of g. In particular, by
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Schur's lemma, u has to act by a scalar multiple of the identity on any irreducible representation of g. It even turns out (see [Kn96, Proposition 5.19]) that there is an analog of Schur's lemma for arbitrary irreducible U(g)-modules, so also on these modules elements of the center of U(g) have to act by a scalar. At first sight it is not clear how large the center of U(g) is, but we can easily construct one nontrivial element. Observe first that there is a canonical action ad of 9 on U(g), defined by ad(X)(u):= Xu - uX. Since XY - YX = [X,Y] holds in U(g), this indeed defines an action. Since U(g) is generated by g, it follows that u lies in the center Z if and only if ad(X)(u) = 0 for all X E g. Recall that U(g) can be realized as the quotient of the tensor algebra T(g) by the ideal generated by all elements of the form X ® Y - Y ® X - [X, Y] for X, Y E g. Now consider an element YI ® ... ® Yn E ®ng C T(g). The natural action of 9 on this space is given by X . (Y1 ® ... ® Yn ) = LY1 ® ... ® [X, Yi] ® ... ® Yn · i
Under the projection to U(g), this element goes to Ei Y1 .•• (XYi - YiX) ... Yn , and this telescopic sum reduces to XY1 ••• Yn - Y1 ••• YnX. Hence, the action ad on U(g) really comes from natural action on T(g). In particular, we may look at 9 ® g, which via the Killing form is isomorphic to g* ® 9 = L(g, g) as a g-representation . The identity map is a g-invariant element in L(g, g), which gives an invariant element of 9 ® g. Projecting to U(g), we obtain a g-invariant element 0 E U(g), called the Casimir element of g. To describe the properties of the Casimir element, we need one more ingredient: The lowest form 8 of a complex semisimple Lie algebra 9 is defined to be half the sum of all positive roots. This lowest form has another important description: In 2.2.7 we have seen that for a simple root O!i the reflection Set; maps O!i to -O!i and permutes the other positive roots. Using this, we compute
seti(8) = ~
L etE~+
Seti(O!) =
-O!i
+~
L
O!
= 8-
O!i'
etE~+
By definition of Set,-, this implies that 2(Q"o:t. (a,et;» = 1 for each simple root O!i, so we see that 8 is the sum of all fundamental weights. In particular, 8 is the smallest algebraically integral element lying in the interior of the dominant Weyl chamber and if A is any dominant weight, then A + 8 lies in the interior of the dominant Weyl chamber. PROPOSITION 2.2.15. Let 9 be a complex semisimple Lie algebra with Casimir element n E U(g), and let V be a finite-dimensional representation of g. Then o acts on the isotypical component V>' of highest weight A by multiplication with IAI2 + 2(A, 8), where 8 is the lowest form and the inner product is induced by the Killing form B. PROOF. Let {Xi} be any basis of 9 and let {Xd be the dual basis with respect to the Killing form, i.e. B(Xi' X j } = 8ij . Then we may write the element of 9 ® 9 corresponding to the identity map as E j Xj ® Xj, so = E j XjXj . Specializing the basis, let {Ht. ... ,Hn} be an orthonormal basis for ~ (recall that the restriction of B to ~o is positive definite). Further, for any root O! E .6. choose vectors E et E get and E_ et E g-et such that B(Eet , E_ et ) = 1. Since ~ is perpendicular to all root spaces, we get iIj = H j for all j, and since get and g/3 are perpendicular unless
n
2.2. {3
COMPLEX SEMI SIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
191
= -a, we see that Eo. = E_o.. Thus, we obtain
=
L HJ + L j
([Eo., E-o.l
+ 2E-o.Eo.).
o.EA+
From 2.2.4 we know that [Eo., E-o.l = Ho., the element characterized by a(H) = B(Ho., H). Summing over all 0: E ~+, these elements add up to 2H6 by definition of the lowest form 8. Hence, we finally arrive at the expression j
o.EA+
Now suppose that v E V is a highest weight vector of weight A. Then v is killed by all Eo. with a E ~ +, so the last sum acts trivially. The first sum acts on v by multiplication by ~j A( Hj)2, and since the Hj form an orthonormal basis, this equals IAI2 = (A, A). The middle term simply gives 2(A, 8). Thus, we conclude that n acts on V~ by multiplication with IAI2 + 2(A, 8). Since the action of n by construction is a g-homomorphism, it acts in the same way on the whole isotypical 0 component V A. Suppose that for a given representation V of 9 we know the weights Ai for which m Ai (V) > O. Then we can compute the eigenvalue ai of the Casimir element on the isotypical component VAi. If these are all different, then we may write the projection on the ai-eigenspace of n as I1#i (ai~aj) (n-ajid), and this gives us the projection onto the isotypical component V Ai. Hence, in this case we get an explicit realization of the isotypical decomposition. Let us add another useful observation: For a dominant weight A we have (A, 8) ~ 0, which implies that the eigenvalue of n is positive unless A = O. Hence, we see that the kernel of n always equals the trivial isotypical component in V with highest weight O. 2.2.16. Tensor products of irreducible representations. A problem that one often meets in applications of representation theory is to decompose the tensor product of two representations. As before, let us denote by r A the irreducible representation of highest weight A. For any finite-dimensional representation V we have the multiplicities mA(V) of r A in V as introduced in 2.2.14 and V ~ E9 A:m>. (V»O (V). For another representation W we therefore get
r:;>.
mv{V ® W) =
L mA(V)ml'(W)mV(r A® r ",). A,I'
Hence, it suffices to deal with the case of the tensor product of two irreducible representations. Recall from 2.2.10 that the multiplicity nA(V) of the weight A in the representation V is the dimension of the weight space VA' PROPOSITION 2.2.16. Let 9 be a complex semisimple Lie algebra, and for each dominant algebraically integral weight A let r A be the irreducible representation with highest weight A. Then we have: If mV(r A ® r "') > 0, then II = A+ JL' for some weight JL' of r w If this is the case, then, in addition, mV(r A ® r "') :5 n",' (r1').
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2. SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
Any weight vector in fA Q9 f I-' of weight 1/ can be written in the form V1 Q9 W1 + ... + Vk Q9 Wk, where the Wi are linearly independent weight vectors in f I-' and the Vi are weight vectors in fA' Denoting by Ai the weight of Vi we order the sum in such a way that A1 2 ... 2 Ak. Now assume that E Vi Q9 Wi is a highest weight vector and let Ea be an element in a positive root space ga' Then by definition Ea annihilates Vl @ Wl + ... + Vk @ Wk, whence we may write (Ea' vt) @ W1 as a linear combination of elements of the form Vj Q9 (Ea' Wj) for j 2 1 and (Ea' Vj) @ Wj for j 2 2. Applying the tensor product of the identity on f A with a linear functional, which is one on W 1 and vanishes on all the other W j , we conclude that we may write Ea . Vl as a linear combination of the Vj for j 2 1. But if Ea' Vl is nonzero, then it would be a weight vector of weight Al + 0::, which is strictly larger than all Aj, so we get a contradiction. This implies that Vl is a highest weight vector, and thus 1/ is the sum of A and the weight of Wl . Hence, we see that any highest weight vector of weight 1/ contains a nonzero component of the form Vo Q9 W for a fixed highest weight vector Vo E fA and W E (fj')/l" If we could find more than n/l' (f 1-') linearly independent highest weight vectors, then we could evidently obtain a nontrivial linear combination in which the terms of the form Vo Q9 W add up to zero. But this linear combination would be a highest weight vector of weight 1/, which is a contradiction. D PROOF.
Let us discuss the application of this result in a simple special case. Take 9 = 5(( n, q, let A be an arbitrary dominant integral weight and put /-l = Wl = el. Then f /l is the standard representation en of g. The weights of en are el, ... , en and each of these has multiplicity one. We conclude that if mV(f A @e n ) > 0, then 1/ = A + ei for some i = 1, ... , nand mV(fA Q9 en) ::; 1. Thus, fA Q9 en is simply reducible, i.e. it splits into a direct sum of pairwise non-isomorphic irreducibles. Moreover, we can see immediately that this splitting into irreducibles can be realized using the Casimir element n from 2.2.15. There we have seen that on the isotypical component with highest weight A + ej, the Casimir element acts by the scalar
IA + ejl2
+ 2(A + ej, 8)
=
IAI2
+ 2(A, 8) + lejl2 + 2(A, ej) + 2(ej, 8).
Since all ej have the same length, these scalars being the same for i < j would imply (e; - ej, A + 8) = O. This is impossible, since ei - ej is a positive root and A + 8 lies in the interior of the dominant Weyl chamber. Hence, we see that, whatever irreducible components really show up, the Casimir element has different eigenvalues on different components. 2.2.17. Infinitesimal character. Let us return to the general question of decomposing a representation into isotypical components. We have indicated in 2.2.15 that elements of the center Z of the universal enveloping algebra U(g) act by scalars on irreducible representations of U(g). Rather than considering a single element of Z, we will next look at the action of the whole center. Denoting by x(a) the number by which an element a E Z acts, one obtains a map X : Z -> e, which clearly is a homomorphism of unital algebras. This homomorphism is called the infinitesimal character or the central character of the representation V. As we shall see immediately, the center is fairly large, so the infinitesimal character contains a good amount of information, even for infinite-dimensional representations. If V is a general representation of U(g) (or equivalently of g), then we say that V has an infinitesimal character if all elements of Z act by scalars on V. Then one gets a homomorphism X : Z -> e as above, which is the infinitesimal character of
2.2. COMPLEX SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
193
V. One should be aware of the fact that in spite of the similar name, the properties of the infinitesimal character are quite different from the properties of the character to be discussed below. Now there is a result of Harish-Chandra, which at the same time gives a complete description of the center Z of U(g) and a description of all unital homomorphisms Z -+ C, so one knows all possible infinitesimal characters in advance. We only outline this and refer to [Kn96 , V.5] for details. To formulate the result, note first that the inclusion of a Cartan subalgebra ~ ~ 9 gives an inclusion of the universal enveloping algebra U(~) into U(g). Since ~ is abelian, U(~) = S(~), the symmetric algebra. Noting that S(~) can be viewed as the algebra of polynomials on ~*, so the action of the Weyl group W of 9 on ~* induces an action on S(~) by (w. p)(¢) = p(w- 1 • ¢). Consider the subalgebras n± formed by all positive, respectively, negative root spaces. The Poincare-Birkhoff-Witt theorem immediately implies that as a vector space U(g) is isomorphic to the direct sum of S(~) and U(g)n+ + n_U(g). For u E U(g) we denote by Uo the component in S(~) according to this decomposition. Next, consider the linear map ~ -+ S(~) defined by H ~ H - 8(H)1, where 8 is the lowest form of g. This induces an algebra homomorphism r : S(~) -+ S(~), and we define the Harish-Chandra map, : U(g) -+ S(~) by,(u) = r(uo). Now Harish-Chandra's theorem reads as follows. 2.2.17 (Harish-Chandra). (1) The Harish-Chandra map, : U(g) -+ restricts to an isomorphism of algebras from the center Z of U(g) to the subalgebra S(~)W of polynomials which are invariant under the action of the Weyl group. (2) For a linear functional>. : ~ -+ C, the composition of the induced algebra homomorphism S(~) -+ C with the Harisch-Chandra map restricts to a homomorphism X>. : Z -+ C of unital algebras, and any such homomorphism is of this form. Finally, for >., J.t E ~* we have X>. = XIL if and only if there is an element w in the Weyl group such that w(>.) = J.t. THEOREM
S(~)
Our main use of the infinitesimal character is that it provides obstructions to the existence of homomorphisms between Verma modules. In view of Theorem 1.4.10, this leads to obstructions against the existence of invariant differential operators. From the definition of the Harish-Chandra map it follows easily that a module generated by a single highest weight vector of highest weight >. has infinitesimal character X>.+,s, so, in particular, this holds for the Verma module Mb(>')' Clearly, a U(g)-homomorphism (or equivalently a g-homomorphism) Mb(>') -+ Mb(J.t) can exist only if both modules have the same infinitesimal character. By Harish-Chandra's theorem, this is the case if and only if there exists a wE W such that w(>. + 8) = J.t + 8, or J.t = w(>. + 8) - 8 =: w· >.. This action of the Weyl group on weights is called the affine action. 2.2.18. Formulae for multiplicities, characters and dimensions. We now switch to the question of getting additional information on the irreducible representations of a complex semisimple Lie algebra g. This is also very helpful for dealing with general representations. A particularly important question is to determine the multiplicities of weights in irreducible representations. In view of Proposition 2.2.16 this has consequences for the decomposition of tensor products. It can also be helpful for deciding irreducibility, compare with 2.2.13.
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2. SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
A basic example of a formula for the multiplicity of a weight in an irreducible representation is the Freudenthal multiplicity formula. It expresses the multiplicity nJL(r.,,) in terms of the multiplicities of weights which are higher than p,. By symmetry of the weights under the Weyl group, it suffices to determine the multiplicities of dominant weights, so one may use the formula to recursively compute all multiplicities. Explicitly, the Freudenthal multiplicity formula reads as (2.4)
(2(,\ -
p"
p, + J)
+ 11,\ - p,11 2 )nJL (rA) = 2
L L (p, + ka, a)n/l+ko(rA), oEt.+ k21
where J is the lowest form of the Lie algebra 9 from 2.2.15. If p, =i ,\ is a dominant weight of r A, then ,\ - p, is a linear combination of simple roots with nonnegative integral coefficients. This immediately implies that (,\ - p" p, + J) > 0, so we see that the numerical factor in the left-hand side of (2.4) is positive. The Freudenthal multiplicity formula may be proved by analyzing the action of the Casimir element n on r A' Since this is a g-homomorphism, it maps the weight space (rA)/l to itself, and one may look at the trace of the action of n on this weight space. On the one hand, one knows that n acts by a scalar, which expresses this trace as a multiple of the dimension n/l(r A ) of the weight space. On the other hand, for a positive root a one may look at the corresponding subalgebra 50 ~ 51(2, C). This subalgebra naturally acts on the sum of weight spaces for weights of the form p, + na, and one may compute the trace analyzing this representation. Details of the proof can be found in [FH91, §25.1]. It is possible to capture the multiplicities of all weights of r A into a closed expression called the Weyl character formula, however, in a rather involved way. Fix a choice of Cartan subalgebra fJ C g. Consider a representation V of 9 on which the elements of fJ act simultaneously diagonalizable. Then V splits into the direct sum of weight spaces VA' If each of these weight spaces is finite-dimensional, then one says that V has a character. If this is the case, then one defines the character char(V) : fJ* ~ Z by char(V)('\) = dim(VA)' It follows easily from the definition that if V has a character and W C V is a subrepresentation, then Wand V jW have a character and char(V) = char(W) +char(VjW). More generally, in a finite exact sequence of representations having characters, the alternating sum of the characters is zero. If V is finite-dimensional, then of course V has a character and since char(V) has finite support, one may view it as an element of the group ring Z[fJ*] of the abelian group fJ*. The multiplication on Z[fJ*] is given by the convolution of functions, i.e. fg(¢) = L>t>H'=4> f('l/J)g('l/J'). Denoting by e4> E Z[fJ*] the function which is one on ¢ and zero on all other elements of fJ* , one gets e4> e>t> = e4>+l/J. By construction, the elements e4> form a linear basis of Z[fJ*J, so we may view elements of Z[fJ*] as expressions f = L4>EI)* a4>e4> with a4> E Z and only finitely many a4> nonzero. In particular, the element eO is a unit element in Z[fJ*]. From the definition of the convolution it is obvious that for finite-dimensional representations V and W one obtains char(V 0 W) = char(V) char(W). To formulate the first version of the Weyl character formula, we define for each weight ,\ E fJ;) an element AA E Z[fJ*] by AA := LWEW sgn( w )eW(A), where W denotes the Weyl group of g. Denoting by J the lowest form of g, the Weyl character formula states that (2.5)
A,s char(r A) = AAH·
2.2. COMPLEX SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
195
Weyl's original proof of this character formula involves what is nowadays called Weyl's unitary trick, i.e. that any complex semisimple Lie algebra has a compact real form. The representation theory of this compact real form is equivalent to the representation theory of 9 (see Remark 2.3.2) and passing to the group level, the character formula is proved using the Peter-Weyl theorem and integration; see [FH91, §26.2]. A purely algebraic proof can be based on the Freudenthal multiplicity formula. Expressing char(r A) using the Freudenthal formula, one shows that Ao char(r A) is nonzero only on elements in the Weyl orbit of ,\ + 8. Evidently, Ao char(r A) is alternating under the action of the Weyl group, and its value on ,\ equals 1, which then implies the character formula. Details of this proof can be found in [FH91, §25.2]. We will prove the character formula as a consequence of Kostant's version of the Bott-Borel-Wei! theorem in 3.3.9. It is not obvious that (2.5) determines char(rA)' To see this, one has to extend the group ring Z[~*]. Let Q+ be the set of all linear combinations of positive roots with nonnegative integral coefficients. Denote by Z{~*) C Zl)* the set of those functions, whose support is contained in a finite union of sets of the form '\0 - Q+ = {,\o - ¢ : ¢ E Q+}. Then it is easy to see that for f,g E Z{~*) and ¢ E ~ *, there are only finitely many pairs '¢, '¢' E ~ * such that ¢ = '¢ +,¢', f ('¢) ¥- 0, and g(,¢') ¥- O. Hence, the convolution of two elements of Z{~*) makes sense and one shows that the result again lies in Z{~*). Thus, the convolution makes Z{~*) into an associative commutative ring with unit eO. We continue to use the notation f = L 0, since X, Y E u. This reduces the construction of a Cartan involution to the construction of a compact real form of the complexification whose involution commutes with the involution corresponding to g. LEMMA 2.3.2. Let g be a real Lie algebra, 0 a Canan involution of g, and a any involutive automorphism of g. Then there is an inner automorphism 'Ij; E Int(g), such that 'lj;O'lj;-1 commutes with a. PROOF. Since 0 is a Cartan involution, B9 is a positive definite inner product on g, and by assumption 1700 is an automorphism of g. Now B9(aOX, Y) = -B(aOX,OY). Since a is an involutive automorphism of g, invariance of the Killing form implies that this equals -B(OX,aOY) = B 9(X, aOY). Hence, 170 : g -+ g is a symmetric linear map, so the square 4> := 170170 is positive and thus diagonalizable with positive eigenvalues. Then we can form 4>r for all r E JR just by taking rth powers of the eigenvalues, and these maps form a one-parameter group, i.e. 4>r+8 = 4>r 0 4>8 for all r, s E R The fact that 4> is an automorphism is equivalent to the fact that the Lie bracket on g maps the product of the A-eigenspace and the j.t-eigenspace of 4> to the Aj.t-eigenspace. This condition then holds for 4>r for all r, so each of these maps is an automorphism of g, too. Since r 1-+ 'lj;r is a smooth curve through the identity, it has values in Int(g). By construction 4>-1 = 017017, and thus 4>0 = 04>-1. This can be equivalently expressed as the fact that 0 maps the A-eigenspace of 4> to the A-1-eigenspace, and hence 4>rO = 04>-r for all r E R On the other hand, by construction 4> commutes with 170, which implies that 170 preserves the eigenspaces of 4>, so 4>r 170 = a04>r for all r E JR. Now we put 'Ij; := 4>1/4, and compute
('Ij;O'lj;-1)a = 'lj; 20a = 'Ij;-24>Oa = 'Ij;-2aO= aO'lj;-2 = a('Ij;O'lj;-1), which completes the proof.
o
THEOREM 2.3.2. Let g be a real semisimple Lie algebra. (1) There exists a Cartan involution 0 on g. (2) If 0 and Of are Cartan involutions on g, then there is a 4> E Int(g) such that
Of
=
4>04>-1.
PROOF. (1) Let ge be the complexification of g, and let Uo C ge be a compact real form. Let a and 7 be the involutions of ge corresponding to the real forms g and uo, respectively. Then we know from above that 7 is a Cartan involution for the real Lie algebra ge. By the lemma, there is an automorphism 4> E Int(gc) such that 4>74>-1 commutes with a. The fixed point set of 4>74>-1 is just 4>(Uo), so this is a compact real form of ge. We have already observed above, that, together with the fact that 4>r74>-1 to g is a Cartan involution.
204
2.
SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
(2) By the lemma, we find an inner automorphism ¢, such that ¢O¢-l commutes with 0'. One immediately verifies that ¢O¢-l is a Cartan involution of g, so to complete the proof it suffices to show that if 0 and 0' are commuting Cartan involutions of g, then 0 = 0'. If 0 and 0' commute, then their eigenspace decompositions are compatible, so any element X E 9 stabilized by 0 splits into the sum Y + Z of two elements stabilized by 0 such that 0' (Y) = Y and 0' (Z) = - Z. But then 0::; Bo(Z,Z) = -B(Z,Z) and 0::; BOI(Z,Z) = -B(Z,-Z), which is possible only for Z = O. Thus, we see that the +l--eigenspace of 0 is contained in the +l--eigenspace of 0', so by symmetry the +l--eigenspaces coincide. Similarly, we conclude that the -1 eigenspaces coincide, which concludes the proof. 0 COROLLARY 2.3.2. If 9 is a complex semisimple Lie algebra and u, u' C 9 are compact real forms, then there is an inner automorphism ¢ E Int(g), such that ¢(u) = u'. In particular, all compact real forms of 9 are isomorphic, so the classification of compact semisimple Lie algebras is equivalent to the classification of complex semisimple Lie algebras, and thus given by Dynkin diagrams. PROOF. Let rand r' be the conjugations with respect to the two compact real forms. Then these are Cartan involutions on the underlying real Lie algebra glR of g. By the theorem, there is an element ¢ E Int(glR) such that r' = ¢r¢-l. Since the group Int(glR) is generated by the maps ead(A) for A E 9 = glR, we see that Int(glR) = Int(g). Hence, the fix point set u' of ¢r¢-l is the image under ¢ of the fix point set u of r. 0 REMARK 2.3.2. (1) Let 9 be a real semisimple Lie algebra, 0 a Cartan involution on g, and consider the inner product Bo on g. For X, Y, Z E g, we get
Bo(ad(OX)Y, Z) = -B([OX, YJ, OZ) = B(Y, [OX, OZ)). Since 0 is an automorphism of g, the last term coincides with -Bo(Y, [X, Z)), which shows that ad(OX) is the negative of the adjoint map of ad(X). In particular, choosing any basis of g, the image of 9 under the adjoint representation is a Lie subalgebra of g((g), which is closed under transposition, and in this picture the Cartan involution 0 is given by the negative transpose. Hence, Proposition 2.1. 7 implies that any subalgebra of 9 which is stable under a Cartan involution is reductive. Conversely, suppose that 9 C g((n, JR) is a Lie subalgebra, such that X E 9 implies X t E g. Then O(X) = -xt defines an involutive automorphism of g. In the associated decomposition 9 = t E9 p, the factor t consists of all skew symmetric elements of g, while p consists of all symmetric elements of g. For X E t and YEp, the map ad(X) 0 ad(Y) clearly maps t to p and p to t, so 9 = t E9 P is an orthogonal decomposition with respect to the Killing form. Finally, one easily verifies that B is negative definite on t and positive definite on p, so this is a Cartan decomposition and hence 0 is a Cartan involution. In particular, this gives examples of Cartan involutions on many of the classical real Lie algebras. (2) The fact that a complex semisimple Lie algebra has a (up to isomorphism) unique compact real form is the basis for an approach to representation theory of complex semisimple Lie algebras known as Weyl's unitary trick. For a real semisimpIe Lie algebra 9 with complexification ge, there is a bijective correspondence between complex representations of 9 and ge, induced by restriction, respectively, complex linear extension. Applying this correspondence once more to a compact
2.3.
REAL SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
205
real form u of gl(;, we end up with a bijection between complex representations of 9 and u. Passing to the simply connected groups, we get a relation between smooth representations of a real semisimple Lie group, holomorphic representations of a complex semisimple Lie group and smooth representations of a compact Lie group. Many problems are much easier to treat in the compact setting, using for example the existence of invariant inner products. Also, the original proof of the Weyl character formula from 2.2.18 is based on the unitary trick and the use of integration on compact groups. See [FH91, §26.2J for a discussion of the unitary trick and the resulting proof of the Weyl character formula. 2.3.3. Cartan decomposition on the group level. There is an analog of the Cart an involution and the Cartan decomposition on the group level. Let 9 be a real semisimple Lie algebra, and let G be a connected Lie group with Lie algebra g. Let () be a Cartan involution on 9 and let 9 = t EB P be the corresponding Cartan decomposition. The main point about the Cartan decomposition on the group level for our purposes is that there is a nice subgroup KeG corresponding to the Lie subalgebra t c g, which is a maximal compact subgroup in most situations of interest. THEOREM 2.3.3. Let 9 be a real semisimple Lie algebra, () a Cartan involution on g, 9 = t EB P the corresponding Cartan decomposition, and let G be a connected Lie group with Lie algebra g. Then there is a unique automorphism 9 : G --+ G with differential (). The fixed point group K := {g E G : 9(g) = g} eGis a connected closed subgroup of G, which contains the center Z (G) and has Lie algebra t. The subgroup K is compact if and only if Z(G) is finite and in that case K is a maximal compact subgroup of G . Moreover, the mapping (g, X) f-+ 9 exp( X) defines a diffeomorphism K x p --+ G. SKETCH OF PROOF. (See [Kn96, VI, Theorem 6.31J for details.) One starts by considering the case G = Int(g) of the adjoint group. In that case, we may work in the group GL(g), respectively, in the Lie algebra g[(g), using the positive definite inner product Bo on 9 to define adjoints. As we have noticed in 2.3.2 above, in this picture ()(X) = -X*, so it is natural to try to define 9(g) = (g*)-I. Using invariance of the Killing form, one easily verifies that for 9 E Aut(g) and X,Y,Z E 9 one gets Bo([g*(X),g*(Y)J,Z) = Bo(g*([X,Yj),Z), which shows that Aut(g) is closed under adjoints. Applying this to 9 E Int(g), we conclude that g*g is a positive definite map lying in Aut(g). As in the proof of Lemma 2.3.2, this implies that (g*gt for r E R is a one-parameter group of automorphisms of g. In particular, this one-parameter group is contained in G = Int(g), so G is closed under adjoints, 9(g) = (g*)-1 is a well-defined involutive automorphism of G, and clearly the derivative of 9 is (). On the other hand, the one-parameter group (g* 9 must be of the form exp(rX) for some X E g. By definition 9(g*g) = (g*g)-I, which implies ()(X) = -X, and thus X E p. The fixed point group K of 9 is a closed subgroup of G and clearly it has Lie algebra t. By construction, 9 E K if and only if g* = g-l, so K is the intersection of G with the orthogonal group of the inner product Bo, which implies that K is compact. Next, ¢(k,X) := kexp(X) defines a smooth map K x p --+ G. If 9 EGis arbitrary, as above, we find an element X E P such that exp r X = (g* 9 for all
t
t
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2. SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
r E lR and we put p:= exp(~X). Thenp E G satisfiesp* = p and putting k = gp-l, we see that k*k = p-lg*gp-l = id. Thus, we get 9 = kp = kexp ~X and k E K, whence ¢ is surjective. On the other hand, if 9 = kexp(X), then g*g = exp2X, so we see that X is uniquely determined as half the generator of the one-parameter subgroup (g* g) r. Then k = 9 exp( - X) is uniquely determined, too, so ¢ is bijective. The exponential map restricts to a diffeomorphism from ponto exp(p), and this easily implies that ¢ is a diffeomorphism. Connectedness of G together with the diffeomorphism K x p --+ G implies that K is connected. Since G has trivial center, we only have to prove that K is maximally compact to complete the proof for the adjoint group. If K' ::J K is a subgroup of G, which properly contains K, then it must contain at least one element of the form exp X, for a nonzero element X E p. But then it must contain the one-parameter subgroup exprX = exp(Xt. From above we know that this is a group of positive operators, so it has unbounded eigenvalues, which shows that K' cannot be compact, so the proof for the adjoint group is complete. If G is a general connected Lie group with Lie algebra g, then put G = Int(g), e the involutive automorphism of G constructed above and K its fixed point group. Then we have a covering rr : G --+ G, and we put K := rr-1(K). Via rr, the group G acts transitively on G/ K with isotropy group K, so we get a diffeomorphism G / K ~ fl./ K. From G ~ K x p we immediately conclude that G / K is simply connected, and since G / K is simply connected it follows that K is connected, and by construction the Lie algebra of K is t. Again by construction, K is a closed subgroup of G and since Z (G) is exactly the fiber of rr over the unit element, we see that K contains Z(G) and is compact if and only if Z(G) is finite. Now rr x id : K x p --+ K x P is a covering, and the map ¢G(k, X) = k exp(X) is a covering of the map ¢G(k,X) = kexp(X). We know that ¢G is a diffeomorphism, and one easily verifies that ¢G is bijective, so we conclude that ¢G is a diffeomorphism, too. Next, let G be the universal covering of G and KeG the corresponding subgroup, which we already know to be the analytic subgroup with Lie algebra e. From 1.2.4 we know that there is a unique involutive automorphism G --+ G with differential (). By construction, K is contained in the fixed point group of which immediately implies that descends to an involutive automorphism e of G, which contains K in its fixed point group. On the other hand, if 9 exp X lies in the fixed point group of e, then one immediately verifies exp 2X = 0, whence X = 0, so K is exactly the fixed point group of e. Finally, if Z(G) is finite, then the fact that K is maximal compact in G immediately follows from the fact that K is maximal compact in G, so the proof is complete. 0
e:
e,
e
REMARK 2.3.3. The involution e : G --+ G is called the global Cartan involution, and the diffeomorphism K x p --+ G is called the global Carlan decomposition. In particular, this shows that G is homotopy equivalent to K, so (in the case of finite center) all the topology of a semisimple Lie group is already visible in the maximal compact subgroup. For example, SL(n, CC) is homotopy equivalent to SU(n), and SO(n,CC) and SL(n,lR) are homotopy equivalent to SO(n). The disadvantage of this decomposition is that it is not satisfactory from the point of view of the group structure of G, which is encoded in K and p in a nontrivial way. This drawback is removed by the Iwasawa decomposition to be discussed below.
2.3. REAL SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
207
2.3.4. Restricted roots. We next move to analogs of the root decomposition for noncompact real Lie algebras. Let 9 be a real semisimple Lie algebra with a Cartan decomposition 9 = t E9 P such that p =f {O} and corresponding Cartan involution O. From Remark 2.3.2 (1), we know that with respect to the inner product B(J we have ad(OX) = - ad(X)t. Hence, ad(X) is skew symmetric for X E t and symmetric for X E p. In particular, ad(X) is never diagonalizable (over the reals) for X E t and always diagonalizable for X E p. Let a c p be a maximal abelian subalgebra (which has to exist since p is finite-dimensional). Then the maps ad(A) for A E a form a family of commuting symmetric linear maps which is thus simultaneously diagonalizable. Moreover, the eigenvalue on a joint eigenspace depends linearly on A, and by symmetry different eigenspaces are automatically orthogonal to each other. Now for a linear functional A : a ~ ~, we denote by 9A the set of all X E 9 such that [A, X] = A(A)X for all A E a. The nonzero functionals A such that 9A =f {O} are called the restricted roots of 9 with respect to a. We denote the set of restricted roots by ~r' By construction, we obtain a decomposition of 9 into an orthogonal (with respect to B(J) direct sum 9 = 90 EEl EIhE~ .. 9A' If A is a restricted root, and X E 9A, then for A E a we get
[A, OX]
= O[OA, X] = -OrA, X] = -A(A)OX,
so we see that 0 restricts to a linear isomorphism between 9A and 9-A' In particular, A E ~r implies -A E ~r' On the other hand, we obviously have [9~, 9",] C 9Mw Finally, we can also easily describe the subalgebra 90. By definition a C 90, and from above we know that 0(90) c 90. Thus, 90 = (90 n t) EEl (90 n p), and clearly this decomposition is orthogonal with respect to Bo. By definition, a is maximal abelian in p, whence 90 n P = a. On the other hand, by definition m := 90 n t consists of all elements X of the subalgebra t such that [X, A] = 0 for all A E a, so this is exactly the centralizer Ze(a) of the subspace a in the t-module p. (Recall that by definition of a Cartan decomposition we have [t, p] c p, so P is at-module under the restriction of the adjoint action.)
Examples. (1) Let 9 be the underlying real Lie algebra of a complex semisimpIe Lie algebra. Then we know that the conjugation with respect to a compact real form u c 9 is a Cartan involution. Choose a Cartan sub algebra ~ c 9 and consider the compact real form u = i~o EEl EBaE~+ (~(Xa - X-a) EEl i~(Xa + X-a)) as constructed in 2.3.1. The Cartan decomposition we obtain has the form 9 = tEElp with t = u and p = iu. In particular, we have ~o c p, and the decomposition of 9 into eigenspaces for the adjoint action of ~o coincides with the root decomposition of the complex simple Lie algebra 9. From this, one readily concludes that ~o is a maximal abelian subalgebra of p. In particular, the restricted roots are exactly the restrictions of the roots to the real subspace ~o, and any restricted root space has real dimension two in this case. (2) Next, we consider various real forms of 5£(n, q. For the split real form 5£(n,~) we can simply use the subspace ~o of tracefree real diagonal matrices as the subspace a, and clearly the restricted roots in this case are exactly the restrictions of the roots of 5£(n, q to the subspace a. In particular, the restricted roots form an abstract root system of type A n - 1 for the split real form. Here, all restricted root spaces have real dimension one. For the compact real form 5u(n) there are no restricted roots since p = {O}. The next obvious real forms are the algebras 5U(p, q) with p + q = n of complex
208
2. SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
n X n-matrices which are skew Hermitian with respect to a Hermitian form of signature (p, q). Since su( q, p) is isomorphic to su(p, q), we may restrict to the case p ~ q. Since the case of su(P,p) is slightly special, we start by assuming p > q. To get a nice description of the restricted roots for these algebras, the simplest presentation is to view C n as C q EB C q EB C p - q with the Hermitian form defined by q
((x,Y, z), (x', y', z')) := ~(Xjyj j=l
p-q
+ Yjxj) + ~Zjzj. j=l
Clearly, the first part defines a Hermitian form of signature (q, q) on C 2q , while the second part defines a positive definite Hermitian form on Cp - q , so the whole form has signature (p, q). The special unitary algebra of this Hermitian form consists of all tracefree matrices M, such that M*.lJ = -.lJM, where.lJ is the block matrix ( n~o no0
~
Hp _ q
), with ITk denoting the k x k-identity matrix. A short computation
shows that we obtain all tracefree block matrices with blocks of size q, q and p - q of the form (
~ - ~. ~), with A E gl( q, q, C and E arbitrary complex q x (p- q)-
-EO -C' F
matrices, Band D in u(q) and F E u(p - q). The advantage of this presentation is that the Lie algebra is closed under conjugate transpose, which implies that putting e and p the subspace of skew-Hermitian, respectively, Hermitian matrices in g, we obtain a Cartan decomposition. In particular, the p-component consists of ~A where A * = A and B* = - B. Now consider all matrices of the form c· -c' 0 the q-dimensional subspace a of all real diagonal matrices contained in p, and for j = 1, ... ,q let ej : a ---+ ~ be the functional which extracts the jth diagonal entry. For a matrix A' such that (A')* = A' we get
(_AB
[(~'
(
-~, ~), ~ o
0
-E*
_Cc),
B -A* -C*
C)] ([A',A] E = -A'D - DA' F (A'E)*
A'B+BA' -[A',A]* -(A'C)*
A'C ) -A'E .
o
From this, one concludes that a c p is a maximal abelian subspace. Moreover, the A-block gives us restricted roots ei - ej for i oF j, the B-block leads to ei + ej for all i,j, from the C-block we get all ej, while the D-block gives -ei - ej and the E-block contains restricted root spaces for -ej. Thus, the restricted roots are exactly ±ei ±ej for i oF j as well as ±ej and ±2ej for j = 1, ... , q, so they form the non-reduced abstract root system (BC)q; see 2.2.8. Moreover, the real dimensions of the restricted root spaces all are equal to two, except for the restricted roots ±2ej, where the real dimension is one. The case of 9 := su(p,p) can be dealt with very similarly. Here the last block from above simply does not occur, so we may view 9 as the algebra of block matrices of the form (~_~.), with A E gl(p,q, Band D in u(P), but to assure that the matrix is tracefree, we have now to assume that the trace of A is real. This is closed under conjugate transpose, so we may again use the decomposition into skewHermitian and Hermitian part as the Cartan decomposition. The p-dimensional subspace a of real diagonal matrices contained in 9 lies in p and is maximal abelian in there. We get the functionals ei for i = 1, ... , q, as before and we may use the above computation for the adjoint action. We get the restricted roots ei - ej for i oF j (from the A-block), ei + ej for all i, j (from the B-block), and -ei - ej for all i,j (from the D-block). Hence, the restricted roots form an abstract root system of
2.3. REAL SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
209
type Cp , compare with 2.2.6(4). From the description one immediately concludes that for ±ei ± ej with i =I j, the restricted root space has real dimension 2, while the restricted root spaces for the restricted roots ±2ei are of real dimension one. (3) Replacing Hermitian forms by real bilinear forms, we can treat the orthogonal Lie algebra so(p, q) similarly. Again, we start by assuming p > q, and consider ~p+q = ~q EB ~q EB ~p-q, with the obvious real bilinear analog of the Hermitian form from (2), which has signature (p, q). The special orthogonal algebra of this inner product consists of all block matrices with blocks of sizes q, q, and p - q of the form (
~ _~t ~),
_Et _CtF
with A E
g[(n,~), B,D
E so(q), F E so(p - q) and C,E
arbitrary real q x (p - q)-matrices. This algebra is closed under forming transposes, and thus a Cartan decomposition is given by putting t' and p the subspaces of skew symmetric, respectively, symmetric matrices contained in g. As in (2) we get a q-dimensional commutative subalgebra a of real diagonal matrices contained in p, and we define the functionals el, ... ,eq : a ~ IR as before. For the commutator we get the analogous formula as in (2), but with adjoints replaced by transposes. This shows that a is a maximal abelian subspace of p and the restricted roots are ei - ej for i =I j from the A-block, and ±ej for j = 1, ... ,q from the C- and E-blocks. However, since the B- and D-blocks now consist of skew symmetric matrices (which have zeros on the main diagonal) these only lead to the restricted roots ±( ei + ej) for i =I j. Consequently, in this case the restricted roots form an abstract root system of type B q , and all restricted root spaces have real dimension one. The case of so(p,p) is closely parallel, but one only has the four blocks in the upper left part. The maximal abelian subspace a is again formed by real diagonal matrices, and the functionals ei for i = 1, ... ,p are defined as above. The discussion above shows that the restricted roots now are ±ei ± ej for i =I j, so they form an abstract root system of type Dp and all restricted root spaces are of real dimension one.
2.3.5. The Iwasawa decomposition. If 9 = t'EBP is a Cartan decomposition of a real semisimple Lie algebra g, then t' is a subalgebra of g. If we further choose a maximal abelian subspace a C p, then of course a also is a subalgebra of 9 and t'n a = {O}. Choose a notion of positivity in a* = L(a, IR) (see 2.2.5), and let ~;! be the corresponding subset of positive restricted roots. Define neg to be the direct sum of all positive restricted root spaces. By construction, neg is a nilpotent Lie subalgebra and a EB neg is a subalgebra with [a EB n, a EB n] = n. In particular, a EB neg is solvable. PROPOSITION 2.3.5 (Iwasawa decomposition of a real semisimple Lie algebra). In the notation above, we have 9 = t' EB a EB n. This decomposition is called the Iwasawa decomposition of g. PROOF. It only remains to show that t' and aEBn are complementary subspaces of g. If X E t' n (a EB n), then X = OX, and since a C p is stable under 0, we must have OX E a EB On. But we have observed in 2.3.4 that 0 maps the restricted root space g). to g_)., whence (a EB n) n (a EB On) = a. Hence, we conclude that X E t'n a = {O}. On the other hand, the restricted root decomposition reads as 9 = go EB n EB On, and again from 2.3.4 we know that go = a EB m, where m = Ze(a). Hence, given
210
2.
SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
an arbitrary element X E g, we find elements H E a, Xo Em and X A EgA for all A E D. r , such that X = H+ Xo + LAE~T X A. But this sum can be rewritten as
X = (xo +
L
(X-A
+ eX_A)) + H+ (
AE~;
and by construction the right-hand side lies in
L
(XA -
eX-A)) ,
AE~;
e+ a + n.
D
In the examples treated in 2.3.4, we can immediately see the form of the Iwasawa decomposition. In particular, for s[(n, q (respectively sl(n, JR.)), we obtain e= u(n) (respectively o(n)), a is the subalgebra of real diagonal matrices, and n is the subalgebra of complex (respectively real) strictly upper triangular matrices. As in the case of the Cartan decomposition, the Iwasawa decomposition admits an analog on the group level. Suppose that G is a connected real semisimple Lie group with Lie algebra g, and let 9 = e EB a EB n be an Iwasawa decomposition. Then we denote by A and N the analytic subgroups of G corresponding to the Lie subalgebra a and n, and we use the subgroup K ~ G from Theorem 2.3.3. Then we have: THEOREM 2.3.5 (Global Iwasawa decomposition). With G, K, A, and N as above, the multiplication (k, a, n) 1--4 kan is a diffeomorphism K x A x N --+ G. Moreover, the subgroups A and N of G are contractible. SKETCH OF PROOF. As in the case of the Cartan decomposition, we start with the case G = Int(g) ~ GL(g) of the adjoint group, and we consider adjoints with respect to the inner product Be on g. From Theorem 2.3.3 we know that in this case K is compact and consists of all maps in G which are orthogonal for Be. Using an orthonormal basis of 9 which is compatible with the restricted root decomposition 9 = go EB EIhE~r gA, we see that all elements of a act diagonally. Let us order the basis in such a way that the first elements correspond to the largest restricted roots (with respect to the fixed chosen ordering), elements from go are in the middle and elements corresponding to negative restricted roots are last. Then elements of n act by strictly upper triangular matrices. Consequently, A and N are analytic subgroups of the group of positive diagonal matrices, respectively, the group of upper triangular matrices with diagonal entries equal to one. Since both of these matrix groups are nilpotent and simply connected, their exponential maps are global diffeomorphisms. From this, one easily concludes that A and N are closed in GL(g) and thus in G, and simply connected. Moreover, this explicit description implies that (a, n) 1--4 an is a bijection from A x N onto a closed subgroup AN of G. This closed subgroup has Lie algebra a EB n, which implies that the map A x N --+ AN is a diffeomorphism. The image of (k, a, n) 1--4 kan is the product of the compact subset K with the closed subset AN of G, so it is closed in G. Moreover, since 9 = e EB a EB n, the map K x A x N --+ G has invertible differential in any point, so the image is open, too. Since G is connected, we conclude that K x A x N --+ G is a surjective local diffeomorphism. Finally, from the explicit description of the subgroups K and AN above, we immediately conclude that K n AN = {id}, which implies the theorem for G = Int(g). If G is a general connected Lie group with Lie algebra g, then we put G = Int(g) and we denote by K, A. and N the subgroups of G constructed above. As in the
2.3. REAL SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
211
case of the adjoint group, the map K x A x N - G has invertible differential in each point. Moreover, restricting the covering 11" : G - G to the subgroups A and N we get coverings, which have to be diffeomorphisms since A and N are simply connected. Thus, A and N are simply connected closed subgroups of G. Using the covering 11" and the results for G one easily verifies that K x A x N - G is bijective, which completes the proof. 0
2.3.6. Uniqueness of the Iwasawa decomposition. Next we have to show that the Iwasawa decomposition is independent (up to isomorphism) of the choices made in its construction. First, we need some additional information on the system of restricted roots. PROPOSITION 2.3.6. Let 9 be a real semisimple Lie algebra endowed with a Cartan decomposition 9 = t EEl P and a fixed maximal abelian subspace a c p. Then the system .a.r c a* of restricted roots is an abstract root system. The action of any element in the Weyl group of this root system can be realized by an inner automorphism of 9 which fixes t and a. PROOF. Since a C p, the restriction of the Killing form to a is positive definite (and coincides with the restriction of the inner product Bo, where 0 is the Cartan involution). In particular, for any restricted root A E .a.r , there is a unique element H>. E a such that A(H) = B(H, H>.) for all H E a. Choose a nonzero element E>. in the restricted root space g>., and consider its image OE>. under the Cartan involution. Then the bracket [E>.,OE>.] is contained in go. Evidently, we have O[E>.,OE>.] = - [E>., OE>.] , so this element lies in go n p = a. For H E a we get B([E>., OE>.] , H) = -B(OE>., [E>., H]) = A(H)B(E>.,OE>.), so [E>.,OE>.] = B(E>.,OE>.)H>.. and B(E>.,OE>.) = -Bo(E>.. E>.) < O. One immediately concludes that the elements E>., OE>. and H>. span a three-dimensional Lie subalgebra of 9 which is isomorphic to 5[(2, JR). Normalize the element E>. in such a way that B(E>.,OE>.) = I~~ (with the norm on a* induced by the Killing form). Consider the group Int(g) of inner automorphisms and let K C Int(g) be the (compact) subgroup corresponding to t. Since E>. + OE>. E t, we can form k>. = exp(~(E>. + OE>.)) E K. For an element H E a such that A(H) = 0, we have [E>., H] = 0 and [OE>., H] = O. Therefore, we get ad(~(E>. + OE>.)) (H) = 0, and thus Ad(k>.)(H) = ead(~(E),+OE),»(H) = H. On the other hand, to get the action of Ad(k>.) on the element H>.. observe that in 5[(2, JR) we have
(1 0)] (0 -11") [(11"0/2 11"/2) 0 '0 -1 - 11" 0 ' and applying the adjoint action once more, we obtain ( ~2 _~2 ) . One easily concludes that Ad(k>.)(H>.) = cos(1I")H>. = -H>.. Hence, Ad(k>.)(a) C a, and the adjoint action of k>. on a is the reflection in the hyperplane orthogonal to H>.. Transferring this to the dual, we see that the coadjoint action of k>. on a* is given by the reflection s>. in the hyperplane orthogonal to the restricted root A. If A(H) = 0 for some H E a and all A E .a.r. then [H, g>.] = 0 for all A, so H lies in the center of g. Since 9 is semisimple, this implies H = 0, and hence .a.r spans a*. For a restricted root A E .a.r , consider an element E>. E g>. and the corresponding subalgebra of 9 as constructed above. This subalgebra is isomorphic to 5[(2, JR) and acts on g, so we obtain an action of 5[(2, .(H) i= 0 for all >. E tl. r . Now we claim that a = Zg(H) n p. By the Iwasawa decomposition any element X E 9 can be written as Xo + Ho + E X.\ with Xo E t, Ho E a and each X.\ in a positive restricted root space. Then [H, X] = [H, Xo] + E.\ >.(H)X.\, the first summand lies in t, and the rest lies in n. Hence, we see that [H, X] = 0 implies X = Xo + Ho and if this, in addition, lies in 13, then X = Ho E a, which proves the claim. Now choose an element H E Ii on which all restricted roots with respect to Ii are nonzero. By Theorem 2.3.3, the subgroup K C Int(g) corresponding to t is compact. Hence, we can choose an element ko E K such that the function k r-? B(Ad(k)H, H) assumes a local extremum at ko. Then for any X E t, the map t r-? B(Ad(exp(tX)ko)H, H) assumes a local extremum at t = O. Writing Ad(exp(tX)ko) = Ad(exptX) Ad(ko), and differentiating with respect to tat t = 0, we see that 0 = B([X, Ad(ko)HJ, H) = B(X, [Ad(ko)H, H]) holds for all X E t. Since [Ad(ko)H, H] E t and the Killing form is negative definite on t, we conclude that [Ad(ko)H, H] = 0, whence Ad(ko)H E Zg(H) n 13 = a. Since a is abelian, we conclude that a C Zg(Ad(ko)H)np = Ad(ko)(Ii). Since both a and Ii are maximally abelian, we conclude that a = Ad(ko)(Ii). Since Ad(ko) preserves the subalgebra t, we may thus assume that Ii = a, and we are left with understanding the effect of the choice of positive restricted roots. By the lemma, we can apply the theory of abstract root systems as developed in
t
2.3.
REAL SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
213
Section 2.2.7 to ~r' In particular, the choice of a positive subsystem is equivalent to the choice of a simple subsystem. Starting from one simple subsystems one obtains all simple subsystems by the action of the Weyl group W = W(~r). Again, by the lemma, two such choices are related by an inner automorphism of 9 that fixes e and ~
0
2.3.7. Cartan subalgebras. If 9 is a real semisimple Lie algebra, then a Cartan subalgebra of g, is an abelian subalgebra ~ such that the complexification ~IC is a Cartan subalgebra of the complex semisimple Lie algebra glC. If, moreover, () is a Cartan involution on g, then a Cart an subalgebra ~ is called ()-stable if ()(~) c ~. If this is the case, then ~ = (~ n e) EB (~ n p), where 9 = e EB p is the Cartan decomposition corresponding to (). For a ()-stable Cartan subalgebra ~, the dimension of ~ n eis called the compact dimension of ~, and the dimension of ~ n p is called the noncompact dimension of~. A ()-stable Cartan subalgebra ~ ::::: 9 is called maximally compact (respectively maximally noncompact) if and only if its compact (respectively noncompact) dimension is maximal possible. In contrast to the complex case, Cartan subalgebras in real semisimple Lie algebras are not unique up to conjugation. Indeed, since the compact and noncompact dimensions can be read off the signature of the restriction of the Killing form, ()-stable Cart an subalgebras of different compact dimensions cannot be conjugate. This already shows up for the simplest possible example 9 = .5((2, JR). Since the complexification glC = .5((2, q is of rank one, any Cartan sub algebra ~ ::::: 9 must be of real dimension one. Now consider the subspaces ~1 of tracefree real diagonal matrices and ~2 of all matrices of the form (.!!t 6) with t E R The complexification of ~1 is just the space of tracefree complex diagonal matrices, i.e. the standard Cartan subalgebra of .5((2, q. Obviously, ~1 is ()-stable for the standard Cartan involution ()X = _xt and ~1 C p, so it is maximally noncompact. On the other hand, one easily verifies that the elements of ~2 act diagonalizably under the adjoint action, so this is a Cartan subalgebra, too. This Cartan subalgebra is obviously ()-stable and contained in e and thus maximally compact, whence 9 has two Cartan subalgebras which are not conjugate. THEOREM 2.3.7. Let 9 be a real semisimple Lie algebra endowed with a Cartan involution (), 9 = eEB p the corresponding Cartan decomposition, G a connected Lie group with Lie algebra g, and KeG the Lie subgroup with Lie algebra e from Theorem 2.3.3. (1) 9 has a Cartan subalgebra and any Cartan subalgebra of 9 is conjugate to a ()-stable Cartan subalgebra by an element ofInt(g). (2) Any two maximally compact (maximally noncompact) ()-stable Cartan subalgebras of 9 are conjugate by an element of K. (3) Up to conjugation by elements ofInt(g), there are only finitely many Cartan subalgebras of g. SKETCH OF PROOF. (1) To prove existence of a Cartan subalgebra, let a c p be a maximal abelian subspace, consider m = Zt(a), let t be a maximal abelian subalgebra of m, and put ~ = t EB a. We have to show that ~IC is maximally abelian and consists of semisimple elements. From 2.3.5 we know that in an appropriate basis of 9 the adjoint action of elements of a is diagonal, while elements of t act by skew symmetric matrices. All of these actions are diagonalizable in the complexification, so we see that for any element of ~IC the adjoint action is diagonalizable.
214
2. SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
To see that ~c is maximally abelian in gc it suffices to show that ~ is maximally abelian in g, which is easy. If ~ c g. is any Cartan subalgebra, let u c gc be the compact real form constructed from the Cartan subalgebra ~c ~ gc as in 2.3.1. Let u and r be the conjugations of gc corresponding to the real forms g and u. By construction, both of these involutions map ~c to itself, and r is a Cartan involution on gc. By Lemma 2.3.2, there is an element ¢ E Int(gc) such that the Cart an involution f = ¢r¢-l commutes with u. From the proof of this lemma we further know that ¢ = (urur) 1/4, which implies that ¢(~c) C ~c. Hence, f(~c) C ~c and since f commutes with u, we have f(g) C g. From the fact that f is a Cart an involution of gc one easily concludes that f restricts to a Cartan involution on g. This Cartan involution is conjugate to {} by an inner automorphism 1/J and since ~ is by construction i-stable, the Cartan subalgebra 1/J(~) is {}-stable. (2) Suppose that ~,~' are maximally noncompact (}-stable Cartan subalgebras in g. From the first part of the proof of (1) above, we conclude that ~ n p and ~' n p both have to be maximally abelian subspaces of p. From 2.3.6 we know that any two such subspaces are conjugate by an element of K, so we may, without loss of generality, assume that ~ n p = ~' n p. Writing a for the latter space, we see that ~ = a EEl t and ~' = a EEl 1', where t, l' c Ze(a) are maximally abelian. Thus, t and l' are maximally abelian subspaces in the Lie algebra of the compact group ZK(a), whose conjugacy is a classical result (see [Kn96, IV, Theorem 4.34]). The maximally compact case is less relevant for our purposes and can be dealt with similarly. (3) A similar argument as in the proof of (2) shows that a {}-stable Cartan subalgebra ~ ~ g is determined up to conjugacy by the subspace ~ n p (see [Kn96, VI, Lemma 6.62]). As above, we may assume that ~ n pea for a fixed maximal abelian subspace a c p. Finally, one proves that ~ n p must be the intersection of the kernels of some restricted roots (see [Kn96, VI, Lemma 6.63]), which leaves only finitely many possibilities. 0 2.3.8. Satake diagrams. Let g be a noncompact real semisimple Lie algebra endowed with a Cartan involution {} and let g = eEEl p be the corresponding Cartan decomposition. From Theorem 2.3.7 we know that there is a (}-stable maximally noncompact Cartan subalgebra ~ ~ g, and that a = ~ n p is a maximal abelian subspace of p. By definition, ~c ~ gc is a Cart an subalgebra, so we can consider the corresponding set .6. = .6.(gC, ~c) of roots. On the other hand, we have the system of restricted roots .6. r encoding the action of the elements of a c ~ C ~c. Since both the roots and the restricted roots describe eigenvalues of the adjoint action, we see that the restricted roots are exactly the nonzero restrictions of roots to a C ~c. Likewise, the restricted root spaces are given as g). = g n
ED
(gc)o.
0:01.=).
Since elements of a, respectively, t act by selfadjoint, respectively, skew symmetric maps, all roots are real on it EEl a. By definition g is a split real form of gc if and only if it contains a Cartan subalgebra on which all roots are real; see 2.3.1. Using Theorem 2.3.7 we see that this is the case if and only if m := Ze(a) = {a}. Let u be the conjugation of gc with respect to the real form g. For a E .6., we define u*a by u*a(H) := a(uH) for all H E ~c. Note that since u is conjugate
2.3.
REAL SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
215
linear, a*a is again complex linear, and identifying Ld~e, q with LlR(~' q, the map £1* coincides with complex conjugation. Since a is an automorphism of the real Lie algebra ge, one immediately concludes that for a root vector Eo. E (g q, using the presentation and the Cartan involution of this Lie algebra from Example (2) of 2.3.4. Thus, we view 9 as all tracefree block matrices with blocks of size q, q and p - q of the form
B C) E ( DA -A* -C* F ' -E*
with A E g((q, C), C and E arbitrary complex q x (p - q)-matrices, Band D in u(q) and F E u(p - q). This describes 9 as a subalgebra of its complexification s((p + q, C), and the Cart an decomposition 9 = e EB p is given by the splitting into skew-Hermitian and Hermitian part. We have also noticed already that the subset a of real diagonal matrices lying in 9 is a maximal abelian subspace of p. To complete this to a maximally noncompact Cartan subalgebra, we have to choose a maximal abelian subspace t of Zt(a). The subspace of purely imaginary diagonal matrices lying in 9 is a commutative subspace contained in this centralizer. Since this has dimension q + (p - q - 1) = p - 1, we see that it is a good choice for t. Thus, f) = t EB a is the subspace of all diagonal matrices contained in g, with the splitting given by imaginary and real part. The complexification f)c C s((p + q, C) is the standard Cartan subalgebra of all tracefree diagonal matrices. Hence, we know from 2.2.6(1) that the roots are all expressions of the form ei - ej for i =F j, where ei is the functional extracting the ith entry of a diagonal matrix. Obviously, the compact roots are exactly the roots ei - ej with i, j > 2q. Unfortunately, the standard ordering of roots is not appropriate for our purposes, so we have to consider a different ordering instead. To get such an ordering, we have to choose a basis of the space of real diagonal matrices, which starts with a basis of a. Denoting by Ei,j the elementary matrix, we define Hi := Ei,i - Eq+i,q+i for i = 1, ... , q and Hi := Ei,i - Ep+q,p+q for i = 2q + 1, ... ,p + q -1. With respect to the resulting ordering, the positive roots are ei - ej for i ~ q and i < j and for 2q < i < j and -ei + ej for q < i ~ 2q and i < j. A moment of thought shows that the resulting simple roots (ordered in such a way to get the usual Dynkin diagram for s((p + q, C)) are el - e2, ... , eq-l - eq, eq - e2q+l, e2q+l - e2q+2, ... , ep+q-l - ep+q, -e2q + ep+q, -e2q-l + e2q, ... , -e q +1 + e q +2. Hence, in the Satake diagram we have q white dots, followed by
p-q-l black dots and after that again q white dots, and it remains to determine the permutation of the white dots induced by a*. Keeping in mind that for i = 1, ... , q the (q + i)th entry of a diagonal matrix lying in f) is minus the conjugate of the ith entry and that a*QI~ = al~, we see that a*(ei - ei+l) = -eq+i + eq+i+l for i < q. Similarly, a*(eq - e2q+d = -e2q + e2q+1' The difference of this and -e2q + e p+q is the compact root e2q+1 - e p+q , so we see that our permutation exchanges the ith and the (p + q - i)th node for all i = 1, ... , q. To draw the Satake diagram it is better to fold it, and the result is G-.
! with q white roots in each of the two rows and p - q - 1 black roots in the middle. Note that in the case of su(p + l,p) we get a Satake diagram with only (an even
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2. SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
number of) white dots but connected by arrows describing the unique nontrivial automorphism of the underlying Dynkin diagram. Hence, the resulting Satake diagram differs (by these arrows) from the Satake diagram of the split real from s[(2p+ 1,JR). (4) The case g = su(P,p) for p ~ 2 can be dealt with similarly. We use the presentation, the Cartan decomposition and the maximal abelian subspace a C p as in example (2) of 2.3.4. So we view g as the algebra of all block matrices of the form (~_~.), with A E g[(p, C) having real trace, and B, D E u(p). The Cartan decomposition is the decomposition into skew-Hermitian and Hermitian part, while the maximally abelian subspace a is given by the space of real diagonal matrices contained in g. As a Cartan subalgebra ~ ::; g we may use the space of all diagonal matrices contained in g (which has the "correct" dimension 2p - 1 since the imaginary part of the trace of A has to vanish). As above, this leads to the usual Cartan subalgebra ~IC in the complexification glC = s[(2p, C), so we get the roots ei - ej for i =f j. Denoting the first p diagonal entries of an element of a by aI,' .. ,ap , any of the roots produces an expression of one of the forms ai - aj for i =f j, or ±(ai + aj) for arbitrary i, j, which implies that there are no compact roots in this case. To get an adapted ordering, we proceed as in (3) above, which leads to the positive roots ei - ej for i ::; P and i < j, and -ei + ej for p < i < j. The resulting simple roots (ordered in such a way that one gets the usual Dynkin diagram for the complexification) are el - e2, ... ,ep-I -ep, ep - e2p, e2p -e2p-1. ... , ep+2 - ep+I. To analyze the action of a* on the simple roots, we have to recall that a* amounts to conjugation on ~ C ~IC. Taking into account the form of ~, one immediately verifies that for i < p we get a*(ei - ei+d = ep+i+I - ep+i, while ep - e2p is fixed bya*. Thus, we see that a* is the unique nontrivial automorphism of the Dynkin diagram A 2p - I . To draw the Satake diagram, it is again better to fold it, and we obtain
[::1> (5) As we shall see from the classification of real semisimple Lie algebras in 2.3.11 below, the only real form of a complex Lie algebra of type Ae that we have not yet considered is g = s[( n, 1Hl) for n ~ 2. From 2.1. 7 we know that we may identify g with the Lie algebra u*(2n) of complex 2n x 2n-matrices of the block form ( _An ~) such that A, B E gl( n, C) and the real part of the trace of A vanishes. This subalgebra of sl(2n, C) is closed under conjugate transpose, so we may use the splitting into skew-Hermitian and Hermitian part as the Cartan decomposition. In particular, the p-component is given by those matrices, for which A is Hermitian and B is skew symmetric. Denoting elements of g as pairs (A, B) one easily verifies that for an element of the form (A', 0) with A' real (and thus tracefree), one gets [(A', 0), (A, B)] = ([A', A], [A', BD. Since p has skew symmetric matrices in the Bblock, this immediately implies that the space a of real diagonal matrices contained in g is a maximal abelian subspace of p (of dimension n - 1). From this description, one easily reads off that the restricted root system is of type A n - I with each restricted root space of real dimension four. We also conclude immediately that the space ~ of all diagonal matrices contained in g is a maximally noncompact Cartan subalgebra, whose complexification ~IC is the standard Cartan subalgebra of glC = s[(2n, C). Thus, we again obtain the
2.3. REAL SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
219
roots ei - ej for i f:. j. To get an appropriate subset of positive roots, we need a basis of ~ starting with a basis of a. In terms of elementary matrices, we define a basis of a by Hi := Ei,i - En,n + En+i,nH - E 2n ,2n for i = 1, ... , n -1. Independent of the extension of this to a basis of ~ we see that the roots ei - ej are positive if 1 ~ i < j ~ n, or n + 1 ~ i < j ~ 2n, or 1 ~ i ~ n and n + i < j ~ 2n. Also, the roots -ei + ej are positive for 1 ~ i ~ n and n + 1 ~ j < n + i for any extension. Thus, the only roots for which we have to decide about positivity are the roots ± (ei - en+i) for i = 1, ... ,n. It turns out that the more convenient choice is to complete the basis by Hi := -Ei - n+1,i-n+1 + Ei+1,i+1 for i = n, ... , 2n - 1. This forces -ei + en+i for i = 1, ... , n to be the remaining positive roots. From the construction it is clear that these last n roots are exactly the compact positive roots. Now -e1 + en+1 is the only positive root involving -el, so it has to be simple. Apart from -e2 + en+2 the only other positive root involving -e2 is e1 - e2, so one of these two roots must be simple. But e1-e2 = e1-en+2+(-e2+en+2), so we see that -e2 + en+2 must be simple. Similarly, one sees that -ei + en+i is simple for all i = 1, ... , n, so we have n compact simple roots. Next, el - en+2 and en+l - en+2 are the only positive roots involving -en+2, and one easily sees that el - en+2 is simple. Similarly, ei - en+i+l is simple for i = 2, ... , p - 2, which means that we have found all simple roots. Ordered in such a way that one gets the usual Dynkin diagram for the complexification, they are -el + en+1, e1 - en+2, -e2 + en+2, e2 e n+3, , •• , en-l - e2n, -en + e2n, so the Satake diagram has black and white dots alternating, starting and ending with a black dot. Finally, one immediately verifies that a*(ei -
en+i+l)
= -ei+l
+ en+i =
ei - en+i+l
+ (-ei + en+i) + (-ei+1 + en+i+d·
Since the last two roots are compact, the involutive permutation induced by a* is the identity and we obtain the Satake diagram ~ ... -----.-o-----e. 2.3.10. a-systems of roots. Since we will not need details in that direction, we only outline the classification of real simple Lie algebras. We follow the paper [Ar62], which describes a classification directly related to Satake diagrams. The classification consists essentially of two parts: On one hand, one has to show that two real semisimple Lie algebras are isomorphic if and only if their Satake diagrams are isomorphic (in the obvious sense). The other part is to describe the possible Satake diagrams. As in the complex case, one also needs the abstract version of the appropriate notion of a root system. This abstract concept is a so-called a-system of roots. By definition, this is given by an abstract root system A on a Euclidean vector space V (see 2.2.4 and 2.2.5) together with an involutive isometry a of V such that a(A) C A, i.e. a restricts to an involution of A. If A and 6. are a-systems on V, respectively, V with involutions a respectively iT, then an isomorphism of the a-systems is a linear isometry ¢ : V -+ V, which restricts to a bijection A -+ 6. and has the property that iT 0 ¢ = ¢ 0 a. A a-system of roots is called normal if for any root a E A, the element a( a) - a is not a root. Now let 0 be a real semisimple Lie algebra with complexification Oc endowed with a Cartan involution {} and a maximally noncompact (}-stable Cartan subalgebra ~ C O. Then we have the root system A = A (01(; , ~C), which by Theorem 2.2.4 is an abstract root system on the real span V of the elements of A. From
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2. SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
2.3.8 we know that the conjugation on gc with respect to the real form 9 preserves ~, which implies that it restricts to an involution of the space (~c)0 on which all roots are real. This involution is an isometry since the conjugation is an automorphism of the real Lie algebra gc, so it dualizes to an isometry a of V which preserves~. (In the notation of 2.3.8 this isometry was denoted by a* rather than a.) Thus, a real semisimple Lie algebra gives rise to a a-system of roots and by Lemma 2.3.8(1) these a-systems are always normal. To obtain this a-system, we have made two choices, namely the Cartan involution 0 and the maximally noncompact O-stable Cartan subalgebra~. But by Theorems 2.3.2 and 2.3.7 Cartan involutions and maximally noncompact Cartan subalgebras are unique up to conjugation, which immediately implies that the a-system of roots associated to 9 is uniquely determined up to isomorphism. The next step is the analog of the passage from a root system to a simple subsystem. To do this on the level of (abstract) normal a-systems of roots, one first has to prove some properties of normal a-systems that we have already verified in the case of a-systems coming from real semisimple Lie algebras. Since a is an involutive isometry it is selfadjoint, so the space V splits into a direct sum V = V+ EEl V_ of ±1-eigenspaces with respect to a. Now one defines ~c := ~ n V_ c ~ and observes that this is an abstract root system on the subspace of V_ spanned by ~c. On the other hand, we get a projection V -> V+ and we denote by ~r C V+ the set of nonzero elements in the image of ~ under this projection. (This is just the analog of passing from roots to restricted roots.) Now one can prove that if the original a-system is normal, then ~r is an abstract root system. Moreover, one can analyze the relation between ~ and ~r quite precisely on the abstract level, which is an important ingredient in the description of all possible Satake diagrams. Knowing these facts, we can now proceed as in 2.3.8 to call a subset ~ + C ~ of positive roots admissible if and only iffor any a E ~ + \~c we have a( a) E ~ +. Such orderings can, for example, be obtained as the lexicographic ordering with respect to a basis of V*, whose first elements form a basis of V';. Having chosen such an order, we can next pass to the corresponding simple subsystem and use part (2) of Lemma 2.3.8, whose proof works in the abstract setting without changes. We obtain an involutive permutation of the noncompact simple roots, and thus associate to any normal a-system of roots a Satake diagram. To see that a normal a-system of roots is completely determined by it's Satake diagram, one has to analyze the effect of the choice of the admissible positive system, which (as in the complex case) is done using the Weyl group: In the Weyl group W of ~ one has two obvious subgroups, namely the Weyl group We of ~e, and the subgroup W.,. of elements which commute with a. By construction, We is a subgroup of W.,. and it turns out that it is automatically a normal subgroup. On the other hand, since any element of W.,. preserves the decomposition V = V+ EEl V_, we get a homomorphism from W.,. to the Weyl group Wr of the restricted root system ~r' It turns out (see [Sa60, Appendix, Lemmas 1 and 2]) that this homomorphism is surjective with kernel We, so Wr ~ W.,./We. Moreover, by [Sa60, Appendix, Proposition A] the group W.,. acts transitively on the set of simple systems obtained from admissible positive systems. From this, one immediately concludes that two normal a-systems are isomorphic if and only if their Satake diagrams are isomorphic.
2.3.
REAL SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
221
Assuming that two real semisimple Lie algebras 9 and g have isomorphic Satake diagrams, the isomorphism of the underlying Dynkin diagrams is induced by an isomorphism of the complexifications ge and ge, respecting the Cartan subalgebras. Hence, we may assume that 9 and 9 are real forms contained in the same complex Lie algebra ge such that the Cartan subalgebras correspond to the same complex Cartan subalgebra and such that the Cartan involutions come from the same compact real form of ge. In this situation the analysis of the restricted root systems allows one to show that 9 and 9 are conjugate by an inner automorphism of ge; see [Ar62, Theorem 2.14). 2.3.11. The classification of real simple Lie algebras. Knowing that two real semisimple Lie algebras are isomorphic if and only if their Satake diagrams are isomorphic, it remains to describe which Satake diagrams arise from real simple Lie algebras. Recall that by Cartan's criterion ge is semisimple if and only if 9 is semisimple. Regarding simplicity, we have LEMMA 2.3.11. (1) The underlying real Lie algebra glR of a complex simple Lie algebra 9 is simple. (2) Let 9 be a real simple Lie algebra with complexification ge. Then ge is simple unless 9 is the underlying real Lie algebra of a complex simple Lie algebra, in which case ge = 9 EI1 g. PROOF. (1) Suppose that a C glR is a real ideal. Then a is semisimple, so a = [a, a) and thus a = [a, glR). Hence, X E a may be written as l:[Xj , Yj) for certain elements Xj E a and Yj E glR. Complex bilinearity of the bracket then implies that l:[Xj , iYj) = iX, but this element lies in [a, glR) = a. Thus, a is a complex ideal, so a = {O} or a = 9 by simplicity of g. (2) We have seen in 2.1.4 that the complexification of a complex Lie algebra 9 is isomorphic to 9 EI1 9 which implies that it is never simple. On the other hand, a Cart an involution on 9 (Le. the conjugation with respect to a compact real form) may be viewed as an isomorphism 9 - g, which completes the discussion of the complex case. Conversely, assume that 9 is real and a is a nontrivial (complex) ideal in ge. Denoting by a the conjugation of ge with respect to the real form g, the intersection an a( a) and the sum a + a( a) are ideals in ge, which are both stable under a and thus complexifications of ideals in g. Since a is nontrivial, the only possibility is that an a(a) = {O} and a + a(a) = ge in the second case. But this implies that ge = a EI1 a(a). By construction a is a complex Lie algebra and X I-t X + a(X) defines an isomorphism a-g. 0 From this we see that apart from the compact real forms of complex simple Lie algebras we have another obvious class of real simple Lie algebras namely the underlying real Lie algebras of complex simple Lie algebras. Eliminating these two trivial classes, it remains to discuss noncompact real forms of a given complex simple Lie algebra. Assume that 9 is a complex simple Lie algebra endowed with a Cartan subalgebra l) C 9 and a compact real form u C 9 such that the involution T corresponding to u leaves l) invariant. For example, one may take the compact real form obtained from l) as in 2.3.1. Since by part (2) of Theorem 2.2.2 and Corollary 2.3.2 any two Cartan subalgebras and any two compact real forms are conjugate, we conclude that any real form of 9 is conjugate to a real form gU C 9 such that the corresponding involution a commutes with T and preserves l).
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SEMISIMPLE LIE ALGEBRAS AND LIE GROUPS
Now considering the root system d = d(g,~) and the fixed involution T, we have to study isometric involutions a* of the real span V of d, which make (d, a*) into a a-system of roots, and are such that the dual map a : ~o ---+ ~o (the subspace on which all roots are real) commutes with the restriction of T. Then the a-system (d, a*) of roots is called normally extendible if and only if a extends to a conjugate linear involutive automorphism of the real Lie algebra g, such that the Cartan subalgebra ~ n gO" is maximally noncompact. Having given such an extension, gO" is a real form of g, such that the associated a-system of roots is (d, a*). Hence, it suffices to classify normally extendible a-systems (d, a*) starting from a reduced root system d with connected Dynkin diagram. The essential step in [Ar62] is that the problem of classifying normally extendible a-systems can be reduced to the case of restricted rank 1, where the restricted rank of a a-system d is defined as the rank of the associated restricted root system dr. In terms of Lie algebras, this means that one may reduce to the situation where the maximal abelian subspace a c p has dimension one. Essentially, this is done as follows: Consider the projection d ---+ d r onto the restricted root system. For a fixed restricted root A E dr, one denotes by d A C d the preimage of the set of all multiples of A under this projection. This is easily seen to be a a-system of roots on its real span, and it turns out that the Dynkin diagram of d A is connected. The theorem that allows the reduction to restricted rank one then states that a a-system d is normally extendible if and only if the corresponding systems d A are normally extendible for all restricted roots A. As we have noted in 2.3.10 above, one may obtain quite detailed information on the relation between d and d r in a general setting. Using this information, one may next determine explicitly the possible Satake diagrams of normal a-systems of roots of restricted rank one. Among these possible Satake diagrams, one may single out the normally extendible ones by a direct analysis. Knowing the Satake diagrams of normally extendible a-systems of restricted rank one, it is then rather easy to determine (case by case) the Satake diagrams corresponding to normally extendible a-systems for any connected Dynkin diagram. The complete result in the language of Satake diagrams is presented in Table B.4 in Appendix B. For any given real simple Lie algebra, one may determine the Satake diagram similarly as in Example (3) of 2.3.9, thus obtaining a complete list of non-isomorphic real simple Lie algebras. The result may be phrased as follows (see [Kn96, VI, Theorem 6.105]). THEOREM 2.3.11 (Classification of real simple Lie algebras). The following is a complete list of real simple Lie algebras and any two entries in this list are pairwise non-isomorphic: (1) The underlying real Lie algebras of complex simple Lie algebras of type An for n ~ 1, Bn for n ~ 2, en for n ~ 3, Dn for n ~ 4, E 6 , E 7 , E a, F4 , and G 2 • (2) The compact real forms of the complex simple Lie algebras from (1). (3) The classical matrix Lie algebras s[( n, JR) s[( n, JHI) su(p, q) so(p, q)
n~2 n~2
p p
~ q > 0, p + q ~ 3 > q > 0, p + q odd, p + q ~ 3
2.3. REAL SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
sp(2n, JR.) Sp(p, q) so(p, q) so*(2n)
223
n~3
> 0, p + q ~ 3 q > 0, p + q even, p + q
P~ q
p
~
~
8
n~5
(4) 12 exceptional noncomplex, noncompact simple real Lie algebras; see Table B.4 in Appendix B. Except for the last entry 50*(2n), we have met all the classical Lie algebras which occur in this classification in 2.1.7, 2.2.6, and 2.3.1. To describe 50*(2n), recall that for a usual complex or quaternionic Hermitian form the real part is symmetric and the imaginary part is skew symmetric. Now one may also consider Hermitian forms with skew symmetric real part and symmetric imaginary part. In the complex case, these are just purely imaginary multiples of usual Hermitian forms, so nothing new is obtained. This is not true over the quaternions, however. Indeed, it turns out that for each n there is an (up to isomorphism) unique such form on JH[n. The Lie algebra 50* (2n) is then defined as the space of those matrices in 5[(n,JH[) (see 2.1.7) which are skew-Hermitian with respect to this form. In particular, we can view 50*(2n) as a Lie subalgebra of s[(2n,Q, and it is easy to see that an explicit realization is given by
Evidently, this consists of skew symmetric matrices, so we can view it as a Lie subalgebra of so(2n,Q, which turns out to be the complexification of so*(2n). 2.3.12. Other methods of classification - Vogan diagrams. Following the original approach of Cartan (see [Carl4]), there are several ways to obtain the classification of real simple Lie algebras based on a maximally compact Cartan subalgebra rather than a maximally noncompact one. We will briefly outline the approach presented in the book [Kn96], in which the author also introduces a new diagrammatic representation of real semisimple Lie algebras by so-called Vogan diagrams. From the point of view of the classification, Vogan diagrams are probably simpler than Satake diagrams, since the possible Vogan diagrams are easier to describe (see below). On the other hand, several aspects of the structure of a real semisimple Lie algebra are encoded in the Satake diagram in a much more transparent way than in the Vogan diagram, which is why we prefer to use Satake diagrams. Let us start with a real semisimple Lie algebra 9 endowed with a Cartan involution () and a maximally compact (}-stable Cartan subalgebra ~ :::; g. Let gc be the complexification of 9 with Cartan subalgebra ~c, and consider the associated set ~ of roots. Denoting by ~ = tEB a the splitting of ~ according to (), as before all roots are real on it EB a. The fact that ~ is maximally compact implies that there are no real roots, i.e. no root vanishes identically on t. Next, one observes that the Cartan involution () acts on the set of roots. Since () acts by 1 on t and by -Ion a, we conclude that it fixes the imaginary roots, i.e. those which vanish on a, and it defines an involutive permutation of the other roots (called "complex roots"). The appropriate positive subsystems ~ + C ~ now are those induced by a basis of (~c)o = it EB a, which starts with a basis of it. This choice and the fact that no root vanishes on t ensures that ()(~ +) C ~+ holds for the corresponding positive system, which implies that () defines an involutive permutation of the simple roots.
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Finally, for an imaginary root (i.e. one that vanishes on a) one sees as in 2.3.8 that the corresponding root space lies either in tc or in Pc (and this time both situations are really possible). Now the Vogan diagram associated to (g,~, a +) is defined as the Dynkin diagram of gc with the two element orbits of () on the simple roots indicated by arrows and an imaginary root painted if its root space lies in Pc. Note that the action of () must be the restriction of an automorphism of the Dynkin diagram. Thus, Vogan diagrams are made up of the same ingredients as Satake diagrams. One shows that if two triples (g,~, a +) and (g,~, ~ +) lead to the same Vogan diagram, then the Lie algebras g and g must be isomorphic. Next, one introduces abstract Vogan diagrams as Dynkin diagrams endowed with an automorphism and some fixed points of the automorphism painted, and proves that any such abstract Vogan diagram comes from some triple (g,~, a +) as above. In this respect Vogan diagrams behave much nicer than Satake diagrams (for which the determination of the possible abstract diagrams is the main difficulty in the classification). However, various choices of a + (for fixed g and ~) may lead to different Vogan diagrams, and one is left with the problem to describe when two Vogan diagrams come from isomorphic real Lie algebras. This is solved by a theorem of Borel and de Siebenthal. One may choose the positive system in such a way that there is at most one painted vertex, and in the case of the trivial automorphism, one obtains further restrictions on the possible painted vertices (which are only relevant for the exceptional algebras). Given these restrictions, one can then push through the classification by a case-by-case analysis. 2.3.13. Relation to symmetric spaces. Much of the interest in the classification of real semisimple Lie algebras among differential geometers comes from the relation to symmetric spaces. Since this ties in nicely with the geometry of homogeneous space that we have studied in Section 1.4, we briefly outline the relation here. Recall that a Riemannian symmetric space is a connected Riemannian manifold (M,g) such that for each point x E M there exists an isometry CTx which has x as an isolated fixed point and satisfies CT~ = id. Since an isometry of a connected Riemannian manifold is determined by its value and its tangent map in one point (see 1.1.1), we see that we must have TxCT x = -idTxM and together with CTx(X) = x, this uniquely determines CT x. In particular, if "I is a geodesic in M starting at x, then CTx("(t» = 'Y( -t), whence CT x is called the geodesic reflection in x. The first observation now is that the group Isom( M) of isometries of M acts transitively on M. To see this, take two points x and y in M. Since M is connected, x and y may be joined by a broken sequence of geodesics, i.e. there are points x = Xl, X2, . .. ,Xn = y, elements t}, . .. ,tn-l E lR. and geodesics "11, • .. ,'Yn-l in M such that 'Yi(O) = Xi and 'Yi(ti) = Xi+l. Defining CTi to be CT-Yi(t;J2), one immediately concludes that CTi(X;.) = Xi+!. Thus, CTn-l 0 •• ·0 CTl is an isometry that maps x to y. In particular, choosing a point Xo E M, we may identify M with GjG xo , where G is the group of isometries of M (which is a Lie group by 1.5.11) and G xu is the isotropy subgroup of the point Xo. Since the isometry group acts t~ansitively and .M is connected, one easily concludes that the connected component of the identity in the isometry group still acts transitively on M. So let us assume that M = G I H is a connected symmetric space with G the connected component of the identity in Isom(M). Then H is the isotropy subgroup of a base point 0 E M, so, in particular, H is compact; compare with 1.4.4. Now consider the geodesic reflection CTo in the base point. Denoting bye the left action of
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G on M, consider the map a 0 0 £g 0 a 0 for some element 9 E G. This is an isometry of M and since £g lies in the connected component of the identity and a~ = id, there must be a unique element 8(g) such that £9(g) = a o 0 £g 0 ao' From this defining equation, one immediately verifies that 8 is an involutive automorphism of the Lie group G. Moreover, for hE H, we see that £9(h) stabilizes the point 0, so we have 8(h) E H. Taking the tangent map in 0, and using that Toa o = -id, we conclude that TO£9(h) = To£h, whence 8(h) = h for all h E H, so H is contained in the fixed point group of 8. The derivative () : 9 ---+ 9 of the automorphism 8 is an involutive automorphism of the Lie algebra 9 and since 8 restricts to the identity on the subgroup H, () has to restrict to the identity on the Lie algebra I) of H. On the other hand, the defining equation for e immediately implies that ao(£g(o)) = £9(g) (0), so we see that a o : G/H ---+ G/H is induced by the map 8 : G ---+ G. But this implies that the mapping g/I) ---+ g/I) induced by () is just Toa o = -id. On the other hand, since H is compact, there is an H-invariant complement m to the Lie subalgebra I) ::::; g, and clearly () must act as minus the identity on m. In particular, this implies that the fixed point group of 8 has Lie algebra I), so we conclude that H must lie between the fixed point group of e and its connected component of the identity. Conversely, let us assume that G is a Lie group, 8 : G ---+ G is an involutive automorphism, I) is the Lie algebra of the fixed point group of 8, and H ::::; G is a subgroup lying between the fixed point group of e and its connected component of the identity. Then the derivative () of 8 must be diagonalizable, and one easily verifies that the corresponding decomposition 9 = I) EEl minto ±l--eigenspaces for () is H-invariant. If there is an H-invariant inner product on m, then it extends to a G-invariant Riemannian metric on M = G/H. The involutive automorphism 8 descends to an involutive diffeomorphism a 0 : M ---+ M, and by construction Toa o = -id. Since 8 is an automorphism, we get a o 0 £g = £9(g) 0 ao. Thus, for a point x = gH E M, we get T;Jpo = To£9(g) 0 Toa o 0 Tx£g-l. Since the righthand side of this is the composition of three mappings preserving inner products, we conclude that a o is an isometry, so it defines a geodesic reflection in the point oEM. For x = gH as above, ax = £g 0 a o 0 £g-l defines a geodesic reflection in x, so M = G / H is a symmetric space. Hence, we have arrived at a Lie theoretic description of Riemannian symmetric spaces. The theory of real Lie groups that we have developed immediately gives rise to an important class of examples: Suppose that G is a noncompact connected semisimple Lie group, 8 : G ---+ G is a global Cart an involution and K ::::; G is the fixed point group of 8; see 2.3.3. (Recall that K is automatically connected in this case.) Then the derivative () of 8 is a Cartan involution, and the decomposition of 9 into eigenspace is the Cartan decomposition 9 = t EEl p. The Killing form is positive definite on p and K -invariant (since it is even G-invariant), so by 1.3.3 we have found all ingredients necessary to make G / K into a (noncompact) symmetric space. Similarly, one can construct compact symmetric spaces starting in the situation above with the compact real form u = t + ip of the complexification gl(: of g. Further, one shows that all examples of Riemannian symmetric spaces such that the connected component of the isometry group is semisimple can be obtained in that way, which shows that the classification of symmetric spaces of that type is essentially equivalent to the classification of real semisimple Lie algebras.
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2.3.14. Basics on representations of real Lie algebras. We conclude this chapter by discussing representations of real semisimple Lie algebras. We start with some generalities on the relation between representations of a real Lie algebra and of its complexification. Recall from 2.1.3 that for a real Lie algebra 9 a complex representation is given by a homomorphism from 9 to the Lie algebra 91c(V) of complex linear endomorphisms of a complex vector space V. Since this is a complex Lie algebra, we know from 2.1.4 that such a homomorphism uniquely extends to a complex linear homomorphism from the complexification ge to 91c(V). This homomorphism defines a complex representation of the complex Lie algebra ge. Conversely, given a complex representation of ge on V, we can of course restrict the corresponding homomorphism p : ge -+ g[c(V) to the Lie subalgebra 9 C ge, thus obtaining a complex representation of g. Since p by definition is complex linear, these two constructions are inverse to each other and thus define a bijection between complex representations of the real Lie algebra 9 and of the complex Lie algebra ge. Suppose that p : ge -+ glc(V) is a complex representation, and let W C V be a complex subspace, which is invariant under the action of the Lie subalgebra 9. Since any Z E ge can be written as Z = X +iY for X, Y E 9 we get p(Z) = p(X)+ip(Y), so W is invariant under the action of ge. In particular, V is irreducible as a complex representation of 9 if and only if it is irreducible as a complex representation of ge. Hence, the complex representation theory of 9 and ge are completely equivalent. Since for a real Lie algebra 9 the term "complex representation" only means that any element acts by a complex linear map, one may view a complex representation of 9 as a real representation enduwed with an invariant complex structure. For this, denote by J : V -+ V the linear map defined by multiplication by i = A. Then J2 = J 0 J = -id, and the fact that elements of 9 act by complex linear maps is equivalent to J being a 9-homomorphism. Conversely, a real representation endowed with an invariant complex structure can be viewed as a complex representation. For a complex representation V of 9 let J : V -+ V be the complex structure. Then -J : V -+ V defines a 9-invariant complex structure, too. Let us denote by V the real vector space underlying V endowed with the complex structure -J. Then we have a natural complex representation of 9 on V, which is called the conjugate representation of V. Notice that the identity of V is an isomorphism between the real representations underlying V and V, but as we shall see below V and V are not isomorphic as complex representations in general. Note also, that if we extend V and V to representations of ge, then the identity map is not compatible with the actions of all elements of ge, since their actions are defined by complex linear extension. Fixing the real form g, forming the conjugate representation gives rise to an operation on complex representations of ge: One first restricts a representation to 9, then forms the conjugate, and extends back to ge. Suppose that p : 9 -+ 9[(V) is a real representation of g. Then we can form the complexification Ve = V ®IR C. Since any linear endomorphism of V uniquely extends to a complex linear endomorphism of Ve, we obtain a complex representation Pe : 9 -+ 91c(Vc) (which then of course extends to the complexification ge as above). Now there is a natural real structure R : Ve --+ Ve on Ve defined by R(v ® z) := v ® z for v E V and z E C or equivalently R(v + iw) = v - iw for
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v, wE V eVe. By definition R2 = RoR = id, R is conjugate linear and compatible with the action of g. If W c V is a g-invariant subspace, then We c Ve is a complex subspace which is invariant under the actions of 9 and ge. In particular, if Ve is an irreducible complex representation, then the real representation V is irreducible, too. The converse assertion is not true in general: 2.3.14. Let 9 be a finite-dimensional real Lie algebm. (1) For a complex representation W of g, there is a real representation V of 9 such that W ~ Ve if and only if there is a g-invariant real structure R on W. If this is the case, then R defines an isomorphism W ~ W. (2) Let V be an irreducible real representation of g. Then either Ve is an irreducible complex representation or there exists an invariant complex structure J on V and Ve ~ V EEl if is the decomposition into irreducible components. PROPOSITION
PROOF. (1) We have seen above that for a real representation V, the complexIDeation Ve admits an invariant real structure R. Since R is conjugate linear, we may view it as an isomorphism Ve ---t Ve of complex representations. Conversely, assume that W is a complex representation and R : W ---t W is a g--equivariant conjugate linear isomorphism such that R2 = id. Since R2 = id the real vector space W splits into the direct sum of the ±l--eigenspaces of R and we denote by V c W the +1--eigenspace. Since v EVe W if and only if R( v) = v and R is g--equivariant, the real subspace V C W is g-invariant. Since R is conjugate linear, multiplication by i maps V to the -l--eigenspace and vice versa. Hence, the two eigenspaces have the same dimension, and W = V EEl iV is naturally isomorphic to Ve as a g-representation. (2) Suppose that W C Ve is a proper complex subspace, which is g-invariant. Let R be the natural real structure on Ve. Then R(W) is a g-invariant complex subspace, too, so also W n R(W) and W + R(W) are such subspaces. Since the latter two subspaces evidently are invariant under R, we conclude from (1) that they are the complexifications of their intersections with V eVe. Since by construction these intersections are g-inva.riant and V is irreducible, this is only possible if W n R(W) = {O} and W + R(W) = Ve. Hence, we get Ve = W EEl R(W) as a g-representation and both summands have trivial intersection with V eVe. Restricting the projection onto W to V, we obtain a g--equivariant linear isomorphism V ---t W, which shows that V admits an invariant complex structure and W is irreducible. As above, R(W) ~ W, and the last claim follows. D
From part (1) of this proposition we see that general complex representations do not arise as complexifications of real representations. Indeed, any representation arising as a complexification must by isomorphic to its conjugate representation. This condition is not sufficient, however, and we have to discuss this next. Let us assume that W is a complex irreducible representation of 9 such that W ~ W as a g-representation. An isomorphism of the two representations may also be viewed as a g--equivariant, conjugate linear isomorphism
. = >. if and only if ai = an-i for all i = 1, ... , n - 1. If p + q = n is odd, then none of the fundamental weights Wi is self-conjugate, and we can write any self-conjugate weight as E ai (Wi +Wn-i) where i = 1, ... , n 21 and the ai are nonnegative integers. Since Wn-i = Wi, we know from above that .:(g, Wi + Wn-i) = 1, so .:(g, >.) = 1 for any self-conjugate weight >.. If p + q = n is even, then the fundamental weight W n /2 is self-conjugate, and each self-conjugate weight >. can be written as a linear combination of W n /2 and the weights Wi + Wn-i for i = 1, ... , ~ - 1. As above, the latter weights have index one, so the index of >. depends only on the coefficient of W n /2. The fundamental representation V n / 2 with highest weight Wn/2 is An/2cn . Now the wedge product An/2cn Q9 An/2cn _ Ancn defines a nondegenerate g-invariant bilinear form on Vn / 2 , which is symmetric if n = 4k and skew symmetric if n = 4k + 2. The resulting isomorphism V n / 2 - V;/2 can be composed with the isomorphism V;/2 - Vn / 2 coming from the induced Hermitian inner product. It is easy to see that the result is an invariant real structure on Vn / 2 for n = 4k and an invariant quaternionic structure for n = 4k + 2. Consequently, any self-conjugate weight has index +1 for n = 4k, while for n = 4k + 2 the index is +1 if the coefficient on W n /2 is even and -1 if this coefficient is odd. (3) We conclude the discussion with the real form 9 := s[(n, IHI) of s[(2n, C). Here the situation is simple, since the standard representation C 2n ~ JH[n has an invariant quaternionic structure. Looking at the exterior powers we see that Wi = Wi for all i = 1, ... ,2n-1 and .:(g,Wi) = (_l)i. Thus, any dominant integral weight is self--conjugate, and expanding such a weight>. as >. = E aiwi, we see that .:(g,>.) = (_1)a 1 +aa+··+a 2 n - 1 •
Part 2
General theory
CHAPTER 3
Parabolic geometries This chapter is devoted to the definition and study of the fundamental properties of parabolic geometries. They are defined as Cartan geometries of type (G, P) for semisimple Lie groups G and parabolic subgroups P. The usual definition of parabolic subgroups is (via the notion of parabolic subalgebras) based on the structure theory of semisimple Lie algebras. Alternatively, parabolic subalgebras may also be described in terms of so-called Ikl-gradings of semisimple Lie algebras. Starting from the latter description, the basic theory of parabolic geometries can be developed using almost exclusively the elementary theory of semisimple Lie algebras as presented in Section 2.1. This is the point of view taken in Section 3.1. Analyzing Ikl-gradings and the corresponding subgroups, one is lead to a sequence of structures underlying a parabolic geometry, the weakest of which is called an infinitesimal flag structure. The basic question addressed in Section 3.1 is to what extent a parabolic geometry is determined by the underlying infinitesimal flag structure. To obtain results in this direction, we have to impose the technical condition of regularity, which avoids particularly bad types of torsion. On the other hand, to single out a unique (up to isomorphism) parabolic geometry with a fixed underlying infinitesimal flag structure one has to assume that the Cartan connection satisfies a normalization condition. It is one of the key features of parabolic geometries that there is a conceptual choice for a normalization condition, and the normalization conditions for all parabolic geometries can be described in a uniform way. Under a cohomological restriction on the Ikl-graded Lie algebra it is then shown that passing to the underlying infinitesimal flag structure induces an equivalence of categories between regular normal parabolic geometries and regular infinitesimal flag structures. This is the central result of Section 3.1. If the cohomological condition is not satisfied, we describe how a more complicated underlying structure is equivalent to a regular normal parabolic geometry. A brief discussion of complex parabolic geometries (the holomorphic version of the theory) and of an alternative approach to parabolic geometries via abstract tractor bundles concludes Section 3.1. To understand the possible Ikl-gradings of a given semisimple Lie algebra 9 and to deal with examples efficiently, one has to invoke the structure theory of semisimple Lie algebras as developed in Sections 2.2 and 2.3. Section 3.2 starts by showing that both in the real and complex case the subalgebras obtained from Ikl-gradings are exactly the parabolic subalgebras in the sense of representation theory. Since the possible parabolic subalgebras (up to conjugation) can be read off the Dynkin diagram, respectively, the Satake diagram of g, we· get a complete overview over possible gradings.
233
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Next, we study the homogeneous models of parabolic geometries, the generalized flag manifolds. Using representation theory, we obtain various realizations of these homogeneous spaces. The structure theory also leads to an efficient way of dealing with irreducible representations of a parabolic subalgebra and their relation to irreducible representations of g. The last ingredient coming from the structure theory is related to the Weyl group W of g. Any parabolic subalgebra in 9 gives rise to a subset of W, which can be naturally viewed as an oriented graph. This is called the Hasse graph or Hasse diagram of the parabolic and it is a central ingredient in the theory of parabolic geometries. In the end of Section 3.2 we describe algorithms to determine the Hasse diagram explicitly. We show how the integral homology and cohomology of generalized flag varieties can be described in terms of the Hasse diagram. Section 3.3 contains a complete proof of Kostant's version of the Bott-BorelWeil theorem. For any parabolic subalgebra in g, this theorem describes the Lie algebra cohomology of the nilradical of the parabolic with coefficients in the restriction of an arbitrary finite-dimensional irreducible representation of g. The description is in terms of the Hasse diagram, so we get an algorithm for computing the cohomologies explicitly. In particular, we get complete information on the cohomological conditions used in Section 3.1. We show how to deduce the classical Bott-Borel-Weil theorem, which describes the cohomology of the sheaf of holomorphic sections of an irreducible homogeneous vector bundle on a generalized flag variety from Kostant's version. Finally, we also obtain a short proof of the Weyl character formula. 3.1. Underlying structures and normalization We start this section by discussing Ikl-gradings of a semisimple Lie algebra g. The nonnegative parts of such gradings form distinguished Lie subalgebras of 9 and hence give rise to distinguished Lie subgroups in any Lie group with Lie algebra g. As we shall see later, these are exactly the parabolic subalgebras, respectively, subgroups in the sense of representation theory. Given a semisimple Lie group G and a parabolic subgroup PeG, parabolic geometries of type (G, P) are then defined as Cartan geometries of the given type. Following the ideas from Section 1.5, we find several geometric structures underlying a parabolic geometry, the weakest of which is called an infinitesimal flag structure. One ingredient of an infinitesimal flag structure is a filtration of the tangent bundle of the manifold carrying the parabolic geometry. Requiring this filtration to be compatible with Lie bracket of vector fields leads to the notion of regularity for infinitesimal flag structures and parabolic geometries. It is easy to describe the set of all parabolic geometries having a fixed underlying regular infinitesimal flag structure. This description suggests necessary properties for a normalization condition, which singles out one of these parabolic geometries. We then construct a normalization condition with the required properties using a bit of Lie theory. Assuming normality, one can project the Cartan curvature to a certain quotient, thus obtaining the harmonic curvature of the geometry. Using the Bianchi identity, we prove that the harmonic curvature still is a complete obstruction against local flatness. Understanding the algebraic properties of the normalization condition leads to the central results of this section, which concern existence and uniqueness of a
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regular normal parabolic geometry inducing a given regular infinitesimal flag structure. Under a cohomological condition, this leads to an equivalence of categories between regular normal parabolic geometries and regular infinitesimal flag structures. Without assuming this condition, we still get an equivalence to a slightly more complicated underlying structure. Next, we discuss complex parabolic geometries, which are the holomorphic version of the theory. We characterize complex parabolic geometries among real parabolic geometries via their curvature. This is used to prove an equivalence between parabolic geometries and underlying structures in the holomorphic category. The last part of the section is devoted to the alternative description of parabolic geometries in terms of abstract tractor bundles and tractor connections. It should be mentioned here that there is an alternative description of the geometric structures underlying a regular parabolic geometry. In his pioneering work, N. Tanaka called them --structures of type m". Together with further alternative constructions for the canonical Cartan connections, this approach is outlined in Appendix A.
"Gt
3.1.1. Filtrations. The basic data needed for the definition of a parabolic geometry are a semisimple Lie algebra endowed with a certain type of grading and a Lie group with that Lie algebra. However, the grading on the Lie algebra is rather an auxiliary object, while the main structure is the filtration associated to this grading. Relations between filtered objects and the associated graded objects will play a central role in the theory, so for the convenience of the reader we collect here some basic facts about filtered vector spaces, filtered Lie algebras and filtered vector bundles. A filtered vector space is a vector space V together with a sequence {Vi: i E Z} of subspaces Vi c V, such that Vi :::) Vi+ 1 for all i E Z, UiEZ Vi = V and niEZ Vi = {a}. We will usually deal with the case of finite-dimensional spaces and finite filtrations, which we will write as V = Vi :::) Vi+! :::) ... :::) V k- 1 :::) V k. Using this notation, it is always assumed that Vi = V for all € < j and Vi = {a} for all € > k. The trivial filtration on a vector space V is given by Vi = V for i S 0 and Vi = {a} for i > O. From a filtration {Vi} of a vector space V, one can canonically construct a graded vector space gr(V) = EBiEZgri(V) by putting gri(V) := Vi/Vi+l for all i E Z. The graded vector space gr(V) is called the associated graded to the filtered vector space V. In the case of a finite filtration V = Vi :::) ... :::) Vk, we obtain a finite grading gr(V) = gri(V) EB· •• EB grk (V) by omitting the zero summands gri(V) for i < j and i > k. An important point to note here is that although the name "associated graded" Inight suggest this, for a filtered vector space (V, {Vi}) with associated graded gr(V) there is neither a natural linear map from V to gr(V) nor a natural linear map in the opposite direction. The only natural linear maps available are the canonical projections Vi ---+ gri(V) = Vi /Vi+l. One may, however, construct a linear isomorphism between V and gr(V) by making choices. Indeed, choosing for each i E Z a subspace Vi c Vi which is complementary to the subspace Vi+l C Vi, the restriction of the canonical projection induces a linear isomorphism Vi ---+ gri(V), The inverses of these linear isomorphisms give rise to a linear map gr(V) = EBi gri(V) ---+ V, which is easily seen to be a linear isomorphism. Choosing such a
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linear isomorphism corresponds exactly to choosing a grading V = EBiEZ Vi which induces the given filtration, i.e. which has the property that Vi = EBj~i Vj. While at this stage the distinction between a filtered vector space and the associated graded vector space may seem artificial, it immediately becomes important if there is some additional structure on V. Suppose, for example, that we have given a representation of a Lie group G or a Lie algebra 9 on V which preserves the filtration, i.e., which is such that the action of any element preserves each of the subspaces Vi c V. Then each Vi is a subrepresentation of V. In particular, we get an induced representation on gri(V) = Vi jVHI, and hence a representation on gr(V). While the vector spaces V and gr(V) are linearly isomorphic, the representations on V and gr(V) are far from being equivalent in general. For example, by Theorem 2.1.1 any complex representation V of a solvable Lie algebra 9 admits a g-invariant filtration such that each of the components gri (V) of the associated graded is one-dimensional. In particular, [g, g] then acts trivially on gr(V). Many natural constructions with vector spaces can be carried out in the filtered setting. If (V, {Vi}) is a filtered vector space and W C V is a linear subspace, then we get a filtration on W by putting Wi := W n Vi. By definition, the inclusion W '-----+ V is a filtration preserving linear map. Since Wi is a subspace in Vi, its image in gri(V) = Vi /VHI is a linear subspace isomorphic to Wi /(VHI n Wi) = gri(W), so we can naturally view gr(W) as a subspace of gr(V). Moreover, we clearly get a filtration on the quotient V/W by defining (V/W)i to be the image of Vi in VjW. This simply means that (V/W)i = Vi/(W n Vi) = Vi/Wi. Now the natural projection Vi -+ (V/W)i maps VHI to (V/W)HI and thus induces a surjective linear map gri(V) -+ gri(V/W), The kernel of this map consists exactly of those elements which have a representative in Wi C Vi, and thus is given by gri(W), In conclusion, we get a natural isomorphism gr(V/W) = gr(V)/ gr(W) and gri(V/W) = gri(V)/ gri(W), Consider two filtered vector spaces (V, {Vi}) and (W, {Wj}) and a linear map f: V -+ W. We say that f has homogeneity;::: j for some jEll, if f(V i ) c WHj holds for all i. In particular, f has homogeneity;::: 0 if and only if it is filtration preserving. The maps of homogeneity ;::: j form a linear subspace L(V, W)j c L(V, W) and we obtain a filtration of L(V, W). For ¢ E L(V, W)j, we have ¢(Vi) C WHj and ¢(VHI) C WHj+l, so for each i we get an induced map gri(V) -+ grHj(W), These fit together to define a linear map grj (¢) : gr(V) -+ gr(W), which is homogeneous of degree j. By definition, grj(¢) = 0 if and only if ¢ E L(V, W)j+I, so we get an injective linear map gr(L(V, W)) -+ L(gr(V), gr(W)), which is compatible with the gradings. Choosing linear isomorphisms V --+ gr(V) and gr(W) -+ W as above, we see that this map is also surjective, so gr(L(V, W)) ~ L(gr(V),gr(W)) with the grading by homogeneous degree. All of this easily extends to multilinear maps and also to the subspaces of symmetric and antisymmetric k-linear maps Vk -+ W. Notice that passing from ¢ E L(V, W)j to grj(¢) has functorial properties. More precisely, if Z is another filtered vector space and 'I/J E L(W, Z)k, then obviously 'l/Jo¢ E L(V, Z)j+k and grj+k('I/J 0 ¢) = grd'I/J) 0 grj(¢)' Specializing to the case W = JK with the trivial filtration, we get a natural filtration on the dual V* of a filtered vector space. This means that (V*)i is the annihilator of V-HI. From above, we know that gr(V*) ~ L(gr(V), JK) = (gr(V))* with the grading by homogeneous degrees of maps, i.e. gTi(V*) = (gr_i(V))*,
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Finally, if (V,{Vi}) and (W,{Wi}) are filtered vector spaces, then for all i,j we have the subspace Vi ® Wi in the tensor product V ® W. We define (V ® W)k to be the sum of all components Vi ® Wi such that i + j = k. One can show in general, that this makes V ® W into a filtered vector space and that grk(V ® W) =
EB
gri(V) ® grj(W).
i+i=k
We will only need this for finite-dimensional vector spaces, for which it follows easily from the above considerations using that V ® W can be identified with the dual of the space of bilinear maps V x W -+ K This easily extends to tensor products of more than two factors. In the case of tensor powers of a single filtered vector space we get induced filtrations on the symmetric and alternating powers. In particular, we get a natural filtration on the kth exterior power AkV of a filtered space V, and the associated graded is naturally isomorphic to Ak gr(V), with the grading characterized by the fact that for Vi of degree ji, the element Vl A ... A Vk has degree i l + ... + ik.
Filtered vector bundles. Everything we have said about filtered vector spaces extends without essential changes to (finite-dimensional) filtered vector bundles over a smooth manifold. A filtered vector bundle over a smooth manifold M is a smooth vector bundle p : E -+ M together with a sequence {Ei : i E Z} of smooth subbundles such thatEi::) Ei+l and such that there are io < jo E Z such that Ei = E for i ~ io and Ei = M (the zero vector bundle) for i > jo. Given such a filtered bundle, we may form the quotient bundles gri(E) := Ei / Ei+l and the associated graded vector bundle gr(E) = EBiEzgri(E). As before, we ignore the zero summands, so gr(E) = grio(E) E9 ... E9 grio(E). Choosing a covering of M by open subsets over which all the subbundles Ei are simultaneously trivial, we see that we may actually view E as a bundle modelled on a filtered vector space (V, {Vi}), i.e. there are vector bundle charts'IjJ: p-l(U) -+ U X V, for E such that p-l(U) nEi = 'IjJ-l(U x Vi), so we have charts compatible with the filtration. Then it is natural to view the bundle gr(E) as being modelled on gr(V) and in charts as above the natural projections Ei -+ gri(E) simply correspond to the natural projections Vi -+ gri(V). Notice, however, that for a vector bundle E modelled on a vector space V, choosing a filtration on V usually does not lead to a filtration of E. This works only, if the transition functions of a given vector bundle atlas have values in filtration preserving endomorphisms of V. Of course, there is neither a natural bundle map from E to gr(E) nor a natural bundle map from gr(E) to E. However, as in the case of vector spaces, there is a distinguished class of bundle isomorphisms between E and gr(E). Since any exact sequence of vector bundles splits, one can always find a smooth subbundle Ei in the filtration component Ei C E such that Ei = Ei E9 Ei+l. The natural projection Ei -+ gri(E) of course induces an isomorphism Ei ~ gri(E), and putting these together we get an isomorphism E ~ gr(E). Also, the natural constructions work exactly as in the case of vector spaces, so we have canonical filtrations on sub bundles , quotient bundles and the dual bundle of a filtered vector bundle, as well as on tensor products of filtered vector bundles, and bundles of linear and multilinear bundle maps between filtered vector bundles. The descriptions of the associated graded bundles is analogous to the case of vector spaces.
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Filtered Lie algebras. Finally, we want to consider algebraic structures on filtered vector spaces. We only discuss Lie algebras, but other types of algebra structures can be treated similarly. A filtered Lie algebra is a Lie algebra (g, [ , ]) together with a filtration {gi : i E Z} of the vector space 9 such that for all i, j E Z we have [gi, gil C gi+i. In particular, for each i ~ 0, the subspace gi egis a Lie subalgebra, and for each i > 0 the subspace gi is an ideal in the Lie algebra gO. Moreover, if the filtration is finite (in the positive direction), i.e. if there is a j E Z such that gi = {O}, then the subalgebra gl egis nilpotent. Next, consider the associated graded vector space gr(g). The Lie bracket may be viewed as a filtration preserving map 9 ® 9 ~ g, so we get an induced mapping on the associated graded, i.e. a map gr(g) ® gr(g) ~ gr(g). By construction, this is skew symmetric, and using the functorial properties of the maps induced on the associated graded vector space, one easily verifies that it satisfies the Jacobi identity, thus making gr(g) into a graded Lie algebra. We will mostly deal with the case where the filtration on 9 actually comes from a grading, i.e. we start with a graded Lie algebra 9 = gk EB ..• EB gl for some k < i E Z such that [gi, gj] c gi+j (where gn = {O} for n < k or n > i) and define gi = E9i~i gj. Clearly, [gi, gil C gi+i, so this makes 9 into a filtered Lie algebra, which is isomorphic to the associated graded gr(g) as a Lie algebra. However, even in this simple situation it will be important to carefully distinguish between the filtration and the associated grading. One way to obtain a filtration on a Lie algebra is to choose a filtration of a representation space. Indeed, suppose that : 9 ~ L(V, V) is a representation of a Lie algebra g, and that {Vi: i E Z} is a filtration on the vector space V. Then we have the induced filtration on L(V, V) and thus on the linear subspace (g). Explicitly, A E gi if and only if A(Vi) c Vi+i for all j E Z. Then the filtration {gi : i E Z} automatically makes 9 into a filtered Lie algebra: For A E gi, B E gi and v E Vk, we have [A, B]·v = A·B·v-B·A·v, and by definition both summands lie in Vi+j+k, and thus [A, B] E gi+i. Similarly, a grading V = E9 Vi on a representation space for 9 induces a grading on the Lie algebra g, by defining gj as the space of those elements A E 9 such that for each i E Z and v E Vi we have A . v E Vi+i' The same computation as above shows that for A E gi and B E gi we get [A, B] E gi+i, whence this makes 9 into a graded Lie algebra. These constructions are compatible with the passage from a grading to a filtration. Indeed, if a grading on 9 is induced by a grading of the representation space V, then the associated filtration on 9 comes from the associated filtration on V. 3.1.2. Ikl-graded semisimple Lie algebras. DEFINITION 3.1.2. Let 9 be a semisimple Lie algebra and let k > 0 be an integer. A Ikl-grading on 9 is a decomposition 9 = g-k EB· .. EB gk of 9 into a direct sum of subspaces such that
• [gi, gil C gi+j, where we agree that gi = {O} for Iii> k, • the subalgebra g_ := g-k EB ••• EB g-1 is generated (as a Lie algebra) by g-l, • g-k #- {O} and gk #- {O}.
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By definition, if £I = g-k EEl ... EEl gk is a Ikl-grading, then I' := go EEl ... EEl gk is a subalgebra of £I, and 1'+ := £II EEl ... EEl gk is a nilpotent ideal in p. Similarly, the subalgebra £1_ = g-k EEl ... EEl £I-I is nilpotent by the grading property. By the grading property, go C £I is a subalgebra, and the adjoint action makes each £Ii into a go-module, such that the bracket [ , ] : £Ii ® gj --+ gHj is a £10homomorphism. The central object for the further study will be the pair (£1,1'), while go is an auxiliary object, which is usually easier to deal with. We will always have to distinguish carefully between go-invariant data and p-invariant data in the sequel. What makes life more complicated is that go = 1'/1'+, so go naturally is a quotient of p. Hence, we can view any go module at the same time as a p-module with trivial action of 1'+. Since the object of main interest is the subalgebra 1', it is clear that the grading of £I (which of course is not p-invariant) will be of minor importance, while the main object is the associated filtration £I = g-k ~ g-k+1 ~ ... ~ gk defined by £Ii := ffir,':i gj. Then by the grading property (£I, {£Ii}) is a filtered Lie algebra and by definition I' = gO and 1'+ = £II. In particular, any filtration component £Ii C £I is a I' submodule. Hence, the quotient gr'i (g) = £Ii I gHI naturally is a p-module, and by the filtration property 1'+ acts trivially on this quotient. Hence, gri(g) is simply £Ii with the go-action trivially extended to p. Thus, gr(g) becomes a p-module with trivial action of 1'+. Let us collect some basic properties of £I in the following: PROPOSITION 3.1.2. Let £I = g-k EEl ... EEl gk be a Ikl-graded semisimple Lie algebra over K = lR or C and let B : £I x £I --+ K be a nondegenerate invariant bilinear form; see 2.1.5. Then we have:
(1) There is a unique element E E £I, called the grading element, such that [E, X] = jX for all X E gj, j = -k, ... , k. The element E lies in the center of the subalgebra go :::; g. (2) The Ik I-grading on £I induces a Ik i I-grading for some k i :::; k on each ideal s C g. In particular, £I is direct sum of Ik i I-graded simple Lie algebras, where k i :::; k for all i and k i = k for at least one i. (3) The isomorphism £I --+ £1* provided by B is compatible with the filtration and the grading of g. In particular, B induces dualities of go-modules between £Ii and g-i, and the filtration component £Ii is exactly the annihilator (with respect to B) of £I-HI. Hence, B induces a duality of p-modules between gig-HI and £Ii, and in particular between £III' and 1'+. (4) For i < 0 we have [gi+1, £I-I] = £Ii. If no simple ideal of £I is contained 'in go, then this also holds for i = O. (5) Let A E £Ii with i > 0 be an element such that [A, X] = 0 for all X E £I-I' Then A = O. If no simple ideal of £I is contained in go, then this also holds for i = O. PROOF. (1) Consider the map D : £I --+ £I which is defined by D(X) = jX for X E gj, j = -k, ... , k. Since [£Ii, gj] C gHj, we immediately conclude that D([X, Y]) = [D(X), Y] + [X, D(Y)] for all X, Y E £I, i.e. D is a derivation. By part (2) of Corollary 2.1.6, semisimplicity of £I implies that there is a unique element E E £I such that D(X) = [E, X]. Decomposing E = E_ k +·· ·+Ek with Ei E £Ii, we get 0 = [E,E] = 2:~=_k[E,Ej] = 2:~=_kjEj, whence E = Eo E go. By definition [E, A] = 0 for all A E go, so E lies in the center of go.
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(2) Let 5 C 9 be an ideal. By (1), the grading components gi are the eigenspaces for ad(E), where E denotes the grading element. The projections onto these eigenspaces can be written as polynomials in ad(E). Any such polynomial maps the ideals to itself. Hence, if XES decomposes as X = X-k + ... + X k according to the grading, then each Xj lies in s. This implies that we get an induced grading on s. The second statement follows since by Corollary 2.1.5, 9 can be written as a direct sum of simple ideals. (3) Invariance of B implies that B([E, X], Y) = -B(X, [E, Y]) for all X, Y E g. For X E gi and Y E gj, we get 0 = (i + j)B(X, Y), whence B(X, Y) = 0 unless i + j = O. Nondegeneracy of B now immediately implies that it.s restrictions to go x go and gj x g_j for j = 1, ... , k are nondegenerate. Hence, the isomorphism 9 --7 g* induced by B is compatible wit.h the grading. The result.ing dualit.ies are go-equivariant by invariance of B. The compatibilit.y with the grading also implies that the restriction of B to 0-H1 X Oi vanishes, so B induces a bilinear form gig-HI x gi --7 K Now gig-HI is linearly isomorphic to g-k E9 ... E9 g-i and hence has the same dimension as Oi = Oi E9 ... E9 Ok. Nondegeneracy of B hence implies that we get a duality between the two spaces, which is compatible wit.h t.he p-actions by invariance of B. The final statement is just. the special case i = 1. (4) The condition that g_ is generated by g-l immediately implies the statement for i :s; -2. Since [E,X] = -X for X E g-l, we get t.he statement for i = -1. Finally, using the fact that 0_ is generated by 0-1, one immediately verifies that [01, g-l] E9 EBi#O gi is an ideal in g. By part (2), there is a complementary ideal which has to be contained in go, so the last part follows. (5) Let B be the Killing form of g, which is nondegenerate and invariant; see 2.1.5. For any Z EO-HI and X E g-l we get 0 = B([A,X],Z) = B(A, [X,Z]), so A is orthogonal with respect to the Killing form to [g-l,g-i+1J, so the result follows from (4) and (3). 0 REMARK 3.1. 2. The proofs of parts (1) and (3) show that any grading on a semisimple Lie algebra 0 must be symmetric around zero, i.e. it must have the form O-k E9 ... E9 gk for some k. EXAMPLE 3.1.2. Using the structure theory of semisimple Lie algebras, one gets a nice description of all possible Ikl-gradings. We will discuss this in detail in Section 3.2, so we just describe some examples that we have met already. (1) Put 0 = so(p + 1, q + 1). In our study of conformal structures in 1.6.3 we have met the decomposit.ion 9 = 0-1 E9 00 E9 gl with g-l ~ ~p+q, gl ~ ~(p+q)* and 00 = co(p,q), the conformal algebra of signature (p,q), so this gives an example of a Ill-grading. Notice that the adjoint action of 00 on g-l is exactly the standard action of co(p, q) on ~p+q which is the basis for the relation of this grading to conformal geometry. This example can be nicely understood in the setting of filtrations and gradings on a Lie algebra induced by filtrations and gradings on a representation space. The representation involved here is the standard representation V = ~p+q+2. The filtration on V has the form V :> VO :> VI, and it is given by choosing a nullline VI (which is spanned by the first basis vector in the presentation of 1.6.3) and putting V O := (VI).1.. In particular, we see immediately that p is exactly the st.abilizer of the null line VI in this case. To come to t.he grading, one simply has to
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choose a null line V-I such that the inner product is nondegenerate on V-I E9 VI. In the presentation of 1.6.3, the null line V-I is spanned by the last basis vector. Having chosen V-I. one defines Vo := (v_d.L n VO and VI = VI to get the grading V = V-I E9 Vo E9 VI. Naively, one would expect that this leads to a grading of the form 9 = g-2 E9 ... E9 g2; but it turns out that g±2 = {O}, so we obtain the grading 9 = g-1 E9 go E9 gl· (2) Consider a ][(-vector space V of dimension p + q and the Lie algebra 9 = sl(V). Choosing a subspace VI c V of dimension p, we get a filtration of the form V = VO :) VI, which induces a filtration 9 = g-1 :) gO :) gl on 9 as described in 3.1.1. Choosing a complementary space Vo such that V = Vo E9 VI with VI = VI (whence dim(Vo) = q) the procedure from 3.1.1 induces a Ill-grading on g. Putting V = ][(p+q, VI the span of the first p-basis vectors and Vo the span of the last q basis vectors, we get a nice block presentation of this Ill-grading. Writing elements of sl(p + q,][{) as block matrices (~ g) with blocks of size p and q, the block corresponding to C has degree -1, the block corresponding to A and D has degree zero and the block corresponding to B has degree one. In particular, p is the stabilizer of the subspace ][{P C ][{p+q. The special case p = 1 is exactly the situation which occurred in the discussion of the projective sphere in 1.1.3. This example can be easily generalized using more complicated filtrations, respectively, block decompositions. The extremal case is to consider the filtration ][(n :) ][(n-l :) ... :) ][(2 :) ][{, respectively, the grading ][{n = ][( E9 ... E9 lK. In matrix terms, this means that in g":= sl(n,][{) one assigns to the entry in the ith row and the jth column degree j - i. One obtains an In - II-grading, for which p is the subalgebra of tracefree upper triangular matrices and p+ is the subalgebra of strictly upper triangular matrices. (3) A complex analog of the construction in example (1) above leads to a 121grading on the Lie algebra 9 = su(n+ 1,1) which underlies CR-geometry; compare to 1.1.6. Take a complex vector space V of dimension n + 2 endowed with a Hermitian form ( , ) of signature (n + 1, 1) and choose a null line VI C V. Putting VO := (Vl).L, we get a filtration V :) VO :) VI of V by complex subspaces. This comes from the grading V = V- 1E9VoE9V1 given by choosing a null line V-I such that the Hermitian form is nondegenerate on V-I E9 VI and putting Vo := V On (V_l).L and VI = VI. Via the construction from 3.1.1 this gives rise to a 121-grading on g, and the associated filtration has the property that p = gO is exactly the stabilizer of the null line VI. Again, this may also be nicely presented by a block decomposition of matrices: Put V = Cn +2 and the Hermitian form (z,w) := ZotVn+1 + Zn+1WO + E'}=1 ZjWj, which is of signature (n + 1, 1). This form is chosen in such a way that we may take VI to be spanned by the basis vector eo and V-I to be spanned by en +1' Then the Lie algebra 9 is the space of all tracefree matrices M such that M*Jr = -JrM, where Jr = R~ ~) with Hn the n x n unit matrix. A short computation shows
(g
that M has t: :e :f the form
(~
A-¥.
:~(a)Hn _i~*)
with A E su(n), a E C, -a X E cn, Z E C n*, and x, Z E JR. By construction, the entry ix has degree -2, X has degree -1, a and A have degree zero, Z has degree one, and iz has degree two. ~x
-x
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242
3.1.3. The group level. Consider a Ikl-graded semisimple Lie algebra 9 = g-k EB ... EB gk and a Lie group G with Lie algebra g. Our next task is to study subgroups of G corresponding to the Lie subalgebras go c 13 c g. Recall that for a subset A c g, the normalizer NG(A) of A in G is defined as {g E G : Ad(g)(A) C A}. As before, we denote by {gil the filtration induced by the Ikl-grading. LEMMA
3.1.3. (1) The Lie subalgebras go C 13
c
9 can be characterized as
go ={X E 9 : ad(X)(gi) C gi for all i = -k, ... , k}, 13 ={X E 9 : ad(X)(gi) C gi for all i = -k, ... , k}. (2) Let G be a (not necessarily connected) Lie group with Lie algebra g. Then P := n~=-k NG(gi) eGis a closed subgroup with Lie algebra p. PROOF. (1) In both statements, the inclusion C is clear. Conversely, take X E 9 and decompose it as X-k+" ,+Xk according to the grading. For the grading element E E go c 13 we obtain ad(X)(E) = kX_k + ... + X-I - Xl - ... - kXk. Consequently, if ad(X)(go) C go, then Xi = 0 for i #- 0 and hence X E go. Likewise, if ad(X)(p) C 13 we must have Xi = 0 for all i < 0, and hence X E p. (2) For each i, the subset gi egis closed, and hence the normalizer NG(gi) is a closed subgroup of G, so PeG is a closed subgroup. Since Ad(exp(X)) = ead(X), we see that the Lie algebra of P is formed by all elements X E 9 such that ad(X)(gi) C gi for all i. Now the result follows from (1). 0
DEFINITION 3.1.3. Let 9 = g-kEB" ·EBgk be a k-graded semisimple Lie algebra and let G be a Lie group with Lie algebra g. (1) A parabolic subgroup of G corresponding to the given Ikl-grading is a subgroup PeG which lies between n:=-k NG(gi) and its connected component of the identity. (2) Given a parabolic subgroup PeG we define the Levi subgroup Go c P by
Go := {g E P: Ad(g)(gi) C gi for all i = -k, ... , k}. Note that by definition, any parabolic subgroup PeG is closed and has Lie algebra p. Further, Go C P is an intersection of normalizers of closed subsets, and hence a closed subgroup of P. From the lemma and its proof we see that Go C P corresponds to the Lie subalgebra go C p. The names "parabolic subgroup" and "Levi subgroup" are motivated by the relation to the structure theory of semisimple Lie algebras which will be discussed in Section 3.2. THEOREM 3.1.3. Let 9 = g-k EB··· EB gk be a k-graded semisimple Lie algebra, and let G be a Lie group with Lie algebra g. Let PeG be a parabolic subgroup for the given grading and let Go c P be the Levi subgroup. Then (gO, Z) 1-+ go exp(Z) defines a diffeomorphism Go x 13+ - P, and
(gO, ZI, . .. , Zk) is a diffeomorphism Go x gl
X •.• X
1-+
go exp(ZI) ... exp(Zk)
gk - P.
PROOF. Both maps are obviously smooth, and their tangent maps in (e, 0) (respectively (e, 0, ... ,0)) are just the identity go x 13+ - 13, respectively, go x ... X gk - p. More generally, since for all elements from 13+ the adjoint action is a nilpotent endomorphism of 13, and thus has only zero eigenvalues, the tangent map of the exponential mapping in any point of 13+ is injective; see [KMS, Corollary
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4.28]. This implies that both maps are diffeomorphisms locally around any point of the form (e, Z) (respectively (e, Zl, ... ,Zk). Since both maps are equivariant for the left multiplication of Go, we conclude that they both are diffeomorphisms locally around each point in their domain of definition. Hence, we only have to show that the maps are bijective. We first show that the second map is surjective. Given 9 E P, consider the adjoint action Ad(g) : 9 -+ g. This is an automorphism of the filtered Lie algebra g, so it induces a linear map ¢o = gro(Ad(g)) on the associated graded greg), which is homogeneous of degree zero; see 3.1.1. Explicitly, for X E gj the element ¢o(X) is the gj-component of Ad(g)(X) E gj. One immediately verifies that ¢o is a Lie algebra homomorphism, and an inverse to ¢o can be constructed in the same way starting from Ad(g-l). Hence, ¢o is an automorphism of the graded Lie algebra g. By construction, for an element Y E gj we have Ad(g)(Y) - ¢o(Y) E gi+l. Put ¢l := ¢oloAd(g) and consider Y E gj. By construction, ¢l(Y)-Y E gj+1. In particular, for the grading element E, we have E - ¢1 (E) E 91, and we denote by ZI the gl-component ofthis element. Then ¢l(E) is congruent to E-Zl modulo g2. Moreover, since ZI E gl, we get ad( -Zt)(E-Zl) = Zl, so ad( -Zl)2((E-Zl)) = 0, and thus Ad(exp( -Zt))(E - Zl) = ead(-zd(E - Zd = E. Putting ¢2 = Ad(exp(-ZI)) 0 ¢1 we see that ¢2(E) is congruent to E modulo g2. Further, for each Y E gj the element ¢2(Y) is congruent to Y modulo gj+1. Inductively, we find elements Zi E gi and automorphisms ¢i of 9 of the form ¢i = Ad(exp(-Zi-l)) 0 ¢i-l, such that ¢i(E) is congruent to E modulo gi, and ¢i(Y) is congruent to Y modulo gi+l for each Y E gj. The automorphism ¢k+l then by construction satisfies ¢k+l (E) = E. Hence, for Y E gj we see that [E,¢k+l(Y)] = ¢k+l([E, Y]) = i¢k+l(Y), so ¢k+l(Y) E gj. But by construction, ¢k+l(Y) is congruent to Y modulo gj+1, so ¢k+l(Y) = Y. Thus, we can write the identity map as Ad(exp( -Zk))
0'"
0
Ad(exp( -Zt}) 0 ¢OI
0
Ad(g).
This shows that ¢o is the adjoint action of go := gexp(-Zk)· .. exp(-Zl)' By definition, this implies go E Go and 9 = goexp(ZI) ···exp(Zk) as required. By the Baker-Campbell-Hausdorff formula, we may rewrite exp(ZI) ... exp(Zk) as exp(Z) for some element Z E 13+, so we have proved surjectivity for both maps. To prove injectivity, note first that for Z E 13+ and Y E gj, the element Ad(exp(Z))(Y) is congruent to Y modulo gj+1. Hence, for Y E gj we can recover Ad(go)(Y) as the gj-component of Ad (gO exp(Z)) (Y). Knowing Ad(go), we also get Ad(exp(Z)). Next, we can determine the lowest nonzero homogeneous component of Z as the lowest nonzero homogeneous component of Ad(exp(Z))(E) - E E 13+. Step by step we may then determine the higher homogeneous components of Z in the same way, and once we have determined Z we can also recover go. Hence, we have proved that the first map is injective and thus a diffeomorphism. For the second map, one may recover Ad(go) from Ad(go exp(Zl)'" exp(Zk)) as above. Splitting off Ad(go) we can recover Zl as the negative of the gl-component of Ad(exp(Zl)" ·exp(Zk))(E). Knowing this, we step by step compute all the Zj, and thus also go. 0 The second part of this theorem immediately suggests how to define further subgroups of P. Namely, we define P+ c P to be the image of 13+ under the
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exponential mapping. By part (2) of the theorem, exp : p+ ----> P+ is a global diffeomorphism and P+ c P is a closed nilpotent subgroup. For Z E p+ and 9 E G we have gexp(Z)g-l = exp(Ad(g)(Z)), and for 9 E P we have Ad(g)(Z) E p+, which shows that P+ C P is a normal subgroup. Finally, again by part (2) of the above theorem PI P+ 9:' Go. In particular, this shows that P is the semidirect product of the subgroup Go and the nilpotent normal vector subgroup P+. This can be easily generalized by replacing p+ = gl by gi for i ~ 2. As we shall see later on, g2 = [p+,p+], and more generally gi is the ith power ofp+ for all i = 1, ... , k, so we denote the exponential image of gi by Then exp : gi ----> pi is a global diffeomorphism, pi c P is a closed normal nilpotent subgroup. In particular, for any i = 2, ... , k we may consider the quotient group PI Pi, which is the semi direct product of the subgroup Go and the nilpotent normal vector subgroup P+I pi c PI pi·
pi.
3.1.4. Definition and basic properties of parabolic geometries. DEFINITION 3.1.4. A parabol'ic geometry is a Cartan geometry of type (G, P), where G is a semisimple Lie group and PeG is a parabolic subgroup corresponding to some Ikl-grading of the Lie algebra 9 of G. We will use the terminology "parabolic geometry of type (G, P)" in this situation. We can easily deduce some basic properties of parabolic geometries by applying the theory developed in Section 1.5. These properties are determined by properties of the corresponding Klein geometry (see 1.4.1), so we have to look at the homogeneous models G I P. PROPOSITION 3.1.4. Let 9 be a Ikl-graded semisimple Lie algebra, G a Lie group with Lie algebra g, and PeG a parabolic subgroup corresponding to the Ikl-grading. Then we have: (1) The Klein geometry (G, P) is infinitesimally effective if and only if no simple ideal of 9 is contained in 90. Assuming this condition, the kernel K of this Klein geometry is a discrete normal subgroup of Go and any morphism between parabolic geometries of type (G, P) is determined by its base map up to multiplication by a locally constant function with values in K. (2) The Klein geometry (G, P) is naturally split using the subalgebra g_ c 9 as a complement to p. However, it is very far from being reductive: For any subspace n c 9 which is complementary to p, the p-module generated by n is the direct sum of all simple ideals of 9 which are not contained in go. (3) Parabolic geometries of type (G, P) do not admit natural linear connections on the tangent bundle. Hence, they do not admit natural pseudo-Riemannian metrics either. PROOF. (1) Recall from 1.4.1 that the kernel K of the Klein geometry (G, P) is the subgroup of G consisting of all elements which act as the identity on G I P. It is the largest normal subgroup of G which is contained in P. Infinitesimal effectivity means that K is discrete, or equivalently that there is no ideal in 9 that is contained in p; see 1.4.1. Now if 5 c 9 is an ideal, then by Corollary 2.1.5, 5 is itself semisimple, and it inherits a Iki I-grading by part (2) of Proposition 3.1.2. But then 5 C P is only possible if 5 C go, and the first statement follows. If no simple ideal is contained in go, then the normal subgroup KeG is discrete. In particular, for X E 9 and k E K we must have exp( -tX)k exp( tX) = k,
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since the left-hand side is a smooth curve in K. Differentiating at t = 0, we get Ad(k)X = X, so K lies in the kernel of the adjoint action and hence in Go. The statement on morphisms now follows directly from Proposition 1.5.3. (2) By definition, the subalgebra 9_ is complementary to p. Let us next determine the p-submodule generated by 9_. For X, Y E 9_ and Z E p, the Jacobi identity gives [X, [Z, Y]] = [[X, ZJ, Y] + [Z, [X, Y]]. Splitting [X, Z] into a 9_- and a p-component, we see that the first summand 011 the right-hand side splits into the sum of an element of g_ and the bracket of an element of p with an element of 9_, so we conclude that [X, [Z, Y]] is contained in the p-module generated by 9_. Inductively, this implies that this module is stable under the adjoint action of 9_, and thus it is an ideal in 9. As above we conclude that this implies that the p-submodule of 9 generated by 9_ is the sum of all simple ideals of g, which are not contained in go. Now let us assume that n C 9 is a linear subspace complementary to the subalgebra p. Since the projection onto an eigenspace of an operator can be written as a polynomial in the operator and the grading element E is contained in p, we see that the g_ -component of any element of n is contained in the p-module generated by n. Since n is complementary to p, this implies that g_ is contained in the pmodule generated by n, so from above we conclude that this module contains the sum of all simple ideals of 9 which are not contained in 90. This completes the proof of (2). (3) It suffices to show that there is no G-invariant linear connection on the tangent bundle T(GjP). Let us denote by Ad and ad the actions of P and p on gjp induced by the adjoint action. According to Theorem 1.4.7 we have to prove that there is no P-equivariant map q, : 9 --+ L(9jp, 9jp) (which we write as X I--!- q, x) such that q, A = ad(A) for all A E peg. Equivariancy of q, can be written explicitly as q, Ad(g)(X) = Ad(g) 0 q, x 0 Ad(g-l) for all X E 9 and 9 E P. Assume that such a map cp does exist, and choose a nonzero element Z E 9k. Then ad(Z)(9) c p, which implies that Ad(exp(Z)) = idg/~. We claim that there is a nonzero element A E go with ad(A) = O. By part (5) of Proposition 3.1.2, there is an element X E 9-1 such that 0 =I- [Z, X] E gk-1. On the other hand, equivariancy of q, implies that q,Ad(exp(Z))(X) = q,x. For k = 1, we get Ad(exp(Z))(X) = X + [Z,X] + 4[Z, [Z,X]], and ad([Z, [Z,X]]) = 0 since [Z, [Z,X]] E gl. Thus, q,Ad(exp(Z))(X) = q,x + ad([Z,Xl), and since [Z,X] E 90, this is an element as required. For k > 1, we get Ad(exp(Z))(X) = X + [Z, X], and thus again ad([Z, Xl) = O. (We shall see later that this is already a contradiction, but this needs more detailed structure theory for Ikl-graded Lie algebras.) Since ad([Z,X]) = 0 implies Ad(exp([Z,X])) = id, we can repeat the argument with Z replaced by [Z, X], finding an element Xl E g-1 such that [[Z, X], Xl] =I- 0, but since ad([[Z, X], Xl]) is the lowest homogeneous component of cp Ad(exp([Z.X]))(Xd - q, x, it must be the zero map. Iterating this argument we obtain a nonzero element A E go with ad(A) = 0 as claimed. But for any element Y E 9-1 we then get ad(A)(Y + p) = [A, Y] + p, since A E go. This implies that [A, Y] = 0 for all Y E g-1, which contradicts part (5) of Proposition 3.1.2 applied to the direct sum of all simple ideals of 9 which are not contained in go. 0 We will usually assume that none of the simple ideals of 9 are contained in 90, thus ensuring infinitesimal effectivity. Indeed, any such ideal can be simply left out
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without essential changes. Note, however, that the Klein geometry (G, P) will not be effective in general. In fact, it is very desirable to have non--effective geometries to deal with spin-structures and similar structures.
3.1.5. The underlying infinitesimal flag structure. As described in Sections 1.5 and 1.6, Cartan geometries (satisfying some normalization condition) are sometimes determined by weaker underlying structures, and these are the cases of most interest. The prototypical example of this situation are conformal structures. As described in 1.6 the conformal frame bundle together with its soldering form can be recovered from an appropriate Cartan geometry and if this geometry is normal, then it can in turn be uniquely constructed from the conformal frame bundle. Therefore, it is natural to look for structures underlying parabolic geometries. There is a whole family of such structures which can be constructed step by step from one another. However, in most situations the weakest and simplest underlying structure is already equivalent to a normal parabolic geometry, and we discuss only this weakest structure at this point. The more complicated underlying structures will be described in 3.1.15 below. Let us fix a Ikl-graded semisimple Lie algebra 9 = g-kEB·· 'EBgk, a Lie group G with Lie algebra g, and a parabolic subgroup PeG corresponding to the grading. We continue to use the notation of 3.1.2 and 3.1.3. Consider a parabolic geometry (p: 9 ~ M,w) of type (G,P). From 3.1.3 we know that we have the reductive subgroup Go and the nilpotent normal subgroup P+ of P, which decompose P as a semidirect product. Since P acts freely on g, the same is true for P+, so we can form the orbit space go := 9 / P+. By construction, the projection P factors to a smooth map Po : go ~ M. If U c M is open such that there is a principal bundle chart 'l/J : p-l(U) ~ U X P, then 'l/J is equivariant for the principal right action, so it factors to a diffeomorphism pC;l(U) = p-l(U)/p+ ~ U x (P/P+). This is obviously equivariant for the right action of Go, so we conclude that Po : go ~ M is a smooth principal bundle with structure group P / P+ ~ Go. On the other hand, the inclusion of Go into P leads to local smooth sections of the projection 9 ~ go, so this is a principal bundle with structure group P+. The second step is to descend parts of the Cartan connection w to the bundle go. To formulate this, we need some more observations. Let us return to the principal bundle p : 9 ~ M and consider the filtration 9 = g-k ::) g-k+1 ::) ... ::) gk ::) {O} from 3.1.2. Since the Cartan connection w induces an isomorphism T9 ~ 9 x g, we see that for each i = -k, ... , k we get a smooth subbundle Tig := w- 1 (gi) of Tg. This defines a filtration Tg = T-kg ::) ... ::) Tkg of the tangent bundle Tg. Moreover, since the filtration {gil is P-invariant, equivariancy of w implies that each of the subbundles Tig is stable under the principal right action, i.e. Tr9(Tig) c Tig for all 9 E P and all i = -k, ... , k. Since w reproduces the generators of fundamental vector fields, we further conclude that for i 2': 0 the subbundle Tig is spanned by the fundamental vector fields with generators in gi C g. In particular, TOg is the vertical bundle of p : 9 ~ M, while Tlg is the vertical bundle of 9 ~ go. Since the filtration of Tg is stable under the principal right action it can be pushed down to go and to M, so we obtain filtrations Tgo = T-kgo ::) ... ::) Togo and TM = T-k M ::) ... ::) T- 1 M by smooth subbundles. By construction, the tangent maps to all the bundle projections are filtration preserving and Togo is exactly the vertical bundle of Po : go ~ M.
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Finally, observe that once one has a filtration of the tangent bundle of a manifold, it makes sense to consider partially defined differential forms, i.e. sections of a bundle of the form L(Ti M, V), where V is some finite-dimensional vector space or a vector bundle over M. For i = -k we get T-k M = T M and sections of L(T- k M, V) are V -valued differential forms on M. Using this we formulate: PROPOSITION 3.1.5. Let (p : g -7 M, w) be a parabolic geometry of type (G, P) corresponding to the Ikl-grading 9 = g-k $ ... $ gk of the Lie algebra 9 of G. Let (Po : go -7 M) be the underlying Go -principal bundle. Then for each i = -k, ... , -1, the Cartan connection w descends to a smooth section of the bundle L(Tigo, gi). For each u E go and i = -k, ... , -1 the kernel of w?(u) : T~go -7 gi is T~+1go, and each w? is equivariant in the sense that for 9 E Go we have (rg)*w? = Ad(g-I) 0 w?
wp
PROOF. Let us denote by 71" : g -7 go the natural projection. For a point Uo E go, some i = -k, ... , -1 and a tangent vector ~ E T~ogo choose a point u E g with 71"( u) = Uo and a tangent vector E Tug such that T7I" . = ~. By construction of the filtrations, E T~g and thus w(e) E gi = gi $ ... $ gk. Define w?(~) to be the gccomponent of w(e). Fixing the choice of u, two possible choices for differ by an element in the kernel of T u7l", and we have observed that this kernel equals TIg. Hence, the difference of the values of w lies in g1, and therefore does not influence the gi-component. On the other hand, any other possible choice for the point in g is of the form u . 9 for some 9 E P+, and from 3.1.3 we know that 9 = exp(Z) for some Z E 1'+. Given a lift E Tug of ~, also Turg . E Tu.gg is a lift of~. But then equivariancy of w implies that w(u . g)(Turg . = ead(-Z)(w(u)(~)). Since Z E 1'+ we know that ad(Z)(gi) c gi+1, so again the gi-component remains unchanged, and we get a well-defined map Tgo -7 gi. Once we know that w? is well defined, we can locally write it as follows: Choose a local smooth section a of 71" : g -7 go, and consider the gccomponent of (a*w)ITig. By definition, this maps ~ E Tuogo to the gi-component of w(a(uo)) (Tuoa . ~), so it coincides with wp(~). Hence, we see that each w? is smooth. By construction, w?(~) vanishes if and only if ~ admits a lift such that w(e) E gi+1, which is equivalent to E Ti+1g and thus to ~ E Ti+1g 0 • Finally, note that since P+ c P is a normal subgroup, it follows immediately that the projection 7r : g -7 go is Go--equivariant. Hence, given a tangent vector ~ E T~ogo, a lift E T~g and 9 E Go, we see that T(rg) . is a lift of T(r 9 ) . ~ (where we denote the principal right action on both bundles by r). Thus, equivariancy of w implies 0 equivariancy of each wp.
e
e
e
e
e
e)
e
wp :
e
e
e
e
3.1.6. Infinitesimal flag structures. There is an abstract version of the structures found as underlying a parabolic geometry in the last subsection. Fix a semisimple Lie group G, a Ikl-grading of the Lie algebra 9 of G, and a parabolic subgroup PeG as before. Then we have the Levi subgroup Go c P, and the normal subgroup P+ c P. Consider a smooth manifold M with a illtration TM = T-k M :J T- k+1 M :J ... :J T- I M such that the rank of Ti M equals the dimension of gi /1' for all i = -k, ... , -1. Let p : E -7 M be a principal fiber bundle with structure group Go. Then one gets a filtration of the tangent bundle TE of the form T E = T-k E :J ... :J TOE by letting TOE be the vertical bundle and TiE :=
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(Tp) -1 (Ti M) for i < O. By construction, the map Tp : T E - t T M is filtration preserving and each of the subbundles Ti E is stable under the principal right action. DEFINITION 3.1.6. (1) An infinitesimal flag structure of type (G, P) on a smooth manifold M is given by:
(i) A filtration TM = T-k M ::J ... ::J T-1 M ofthe tangent bundle of M such that the rank of Ti M equals the dimension of gi /p for all i = -k, ... ,-1. (ii) A principal Go-bundle p: E - t M. (iii) A collection (J = ((J-k, ... , (J-d of smooth sections (Ji E r(L(Ti E, gi)) which are Go-equivariant in the sense that (rg)*(Ji = Ad(g-1) 0 (Ji for all 9 E Go, and such that for each u E E and i = -k, ... , -1 the kernel of (Ji(U) : T!E - t gi is T~+1 E c T!E.
(2) Let M and M be smooth manifolds endowed with infinitesimal flag structures ({TiM},p: E - t M,(J) and ({TiM},p: E - t M,e) of type (G,P). Then a morphism of infinitesimal flag structures is a principal bundle homomorphism ~ : E - t E which covers a local diffeomorphism f : M - t M such that T f is filtration preserving and ~*ei = (Ji for all i = -k, ... ,-1. Notice that in part (2) the condition that T f : T M - t T M is filtration preserving implies that T~ : T E - t T E is filtration preserving, so the section e i of L(Ti E, gi) pulls back to a section ~*ei of L(Ti E, gi). In this language, the results of 3.1.5 say that any parabolic geometry (p : g - t M,w) of type (G,P) gives rise to an underlying infinitesimal flag structure ({TiM},po : go - t M,w O) of type (G,P). From the construction it follows immediately that any morphism of parabolic geometries descends to a morphism of infinitesimal flag structures. Hence, we obtain a functor from the category of parabolic geometries to the category of infinitesimal flag structures of the same type. One of the main aims of this section will be to show that under weak assumptions this functor restricts to an equivalence between appropriate subcategories. We can clarify the geometric meaning of an infinitesimal flag structure. Fix a filtration T M = T-k M ::J ... ::J T-l M such that the rank of Ti M equals the dimension of gi /p for all i = -k, ... , -1. Then the rank of gri(TM) = Ti M/Ti+1 M equals the dimension of gi, so we can use g_ = g-k $ ... $ g-1 as the modelling vector space for the graded vector bundle gr(T M). On the other hand, via the adjoint action the group Go acts on g_ and the action preserves the grading, so we get a homomorphism Go - t GLgr(g_). Hence, it makes sense to talk about a reduction to the structure group Go of the bundle gr(TM). PROPOSITION 3.1.6. Let 9 = g-k E& ••• $ gk be a Ikl-graded semisimple Lie algebra, let G be a Lie group with Lie algebra g, PeG a parabolic subgroup corresponding to the grading, and Go c P the Levi subgroup; see 3.1.3. (1) An infinitesimal flag structure of type (G, P) on a smooth manifold M is equivalent to a filtration T M = T-k M ::J ... ::J T- 1M of the tangent bundle of M such that for each i the rank ofTi M equals the dimension of gi /p and a reduction of the structure group of the associated graded bundle gr(T M) to the structure group Go with respect to the homomorphism Ad : Go - t GLgr(g_). (2) Suppose that no simple ideal of 9 is contained in go. Consider two infinitesimal flag structures ({TiM},p : E - t M,(J) and ({TiM},p : E - t M,e) of type (G, P). Let ~1. ~2 : E - t E be two morphisms of infinitesimal flag structures
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249
covering the same base map 1 : M --+ M. Then there is a locally constant map ¢: M --+ K such that iI>2(U) = iI>1(U)· ¢(p(u)). Here, K c Go is the subgroup 01 all elements g E G such that Ad(g) = id g • PROOF. (1) Let us fix a filtration {TiM} of TM such that the rank of TiM equals the dimension of gi Ip. Then a reduction of gr(T M) to the structure group Go is given by a principal Go-bundle p : E --+ M together with a homomorphism iI> from E to the graded linear frame bundle P := GLgr(g_,gr(TM)) of gr(TM), which induces the identity on M and is equivariant for the homomorphism Ad : Go --+ GLgr(g_). To prove (1), we have to show that such a homomorphism is equivalent to a collection () = ((}i) of sections as required in the definition of an infinitesimal flag structure. This is analogous to the case of G-structures discussed in 1.3.6. Given an infinitesimal flag structure ({TiM},p : E --+ M,(}), take a point u E E. For each i we have the linear map (}i(U) : T~E --+ gi. By definition, the kernel of this map is T~+l E, so for dimensional reasons it has to descend to a linear isomorphism T~EIT~+l E --+ gi. The tangent map Tup induces a linear isomorphism T~EIT~+l ~ T~MIT~+l M, where x = p(u). Hence, we may interpret (}(u) = ((}-k(U), ... , 0-I(U)) as a linear isomorphism gr{TxM) --+ g_, whose inverse defines an element of P x • Hence, we obtain a smooth fiber bundle homomorphism iI> : E --+ p, which covers the identity on M. Equivariancy of the form () reads as O(u· g) = Ad(g-l) 0 O(u), which is exactly the condition required for a principal bundle homomorphism. For the other direction, we observe that the bundle P = GLgr(g_,gr{TM)) carries a natural analog of the soldering form described in 1.3.5. Since 7r : P --+ M is a principal bundle, we can lift the filtration of T M to a filtration TP = T-kp ~ ... ~ TOp as above. Having this filtration at hand, it is obvious how to obtain an analog 8 = (8-k, ... , 8_ 1) ofthe soldering form, where each 8 i is a smooth section of L(Tip, gi): For x EM, a point in P x is a linear isomorphism ¢ : g_ --+ gr(TxM) which is homogeneous of degree zero. Given a tangent vector ~ E TJP, we have T",7r . ~ E T~M. Denoting by [Tu7r· ~l E gri(TxM) its equivalence class, we define 8i(~) := ¢-I([Tu7r· W E gi· By construction, this is smooth, the kernel of 8 i (u) is T~+lP, and we get equivariancy in the sense that (rtP)*8 i = 1/J-l o8 i for each
1/J E GLgr(g_). Given a reduction of structure group iI> : E
--+
P we see that by construction
TiI> : T E --+ TP is filtration preserving, so we can form the pullback iI>*8 = (iI>*8_k' ... ' iI>*8_ 1) of the generalized soldering form. One immediately checks that ({TiM}, E --+ M, iI>*8) is an infinitesimal flag structure, and this construction is inverse to the one described above. (2) By definition, 1 is a local diffeomorphism, and for an open subset U c M such that 1 : U --+ I(U) is a diffeomorphism, iI>1 and iI>2 are principal bundle isomorphisms p-l(U) --+ p-l(f{U)). Since the locally constant maps ¢ can be pieced together, we may assume that 1 is a diffeomorphism and thus iI>1 and iI>2 are isomorphisms of principal bundles, and replacing iI>2 by iI>;1 0 iI>1, we may assume that we deal with an automorphism iI> of the infinitesimal flag structure on M that covers the identity. By definition, this means that there is a smooth map ¢ : E --+ Go such that iI>(u) = U· ¢(u). For u E E and ~ E TuE, we get iI>*Oi(U)(~) = Oi(U· ¢(u))(TuiI>· ~). As in the proof of Proposition 1.5.3 one shows that TuiI> . ~ equals the sum of
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e
Tur(u) . and some vertical vector field. Since each Oi vanishes on vertical fields, we conclude that CP*Oi(U)(e) = (r(u))*Oi(U)(e) = Ad(¢(u)-1)(Oi(u)(e)). For cp to be a morphism of infinitesimal flag structures we must have CP*Oi = Oi for all i = -k, ... , -1, which implies that the adjoint action of ¢(u) on 9i is the identity for all i = -k, ... , -1. Since for j > 0, the Go-module 9j is dual to 9-j, we conclude that Ad(¢(u)) also is the identity on all 9j with j > O. Thus, Ad(¢(u)) also acts as the identity on the submodule [91.9-1] C 90, which coincides with 90 by part (4) of Proposition 3.1.2 since no simple ideal of 9 is contained in 90. Consequently, the function ¢ has values in K, which has trivial Lie algebra and hence is a discrete subgroup of Go, so ¢ must be locally constant. 0
Note that part (1), in particular, implies that for an infinitesimal flag structure ({TiM},p: E ---4 M,O) we have a natural isomorphism gri(TM) ~ E xGo 9i. Explicitly, this identification is induced by the map E x 9i ---4 gri (T M) defined by (u, X) 1-+ [Tup, e], where E T~ E is any tangent vector such that Oi (e) = X and [] denotes the class in gri(TM) = T i M/Ti+ 1M.
e
EXAMPLE 3.1.6. Since in all other cases we need restrictions on infinitesimal flag structures to characterize the relevant examples, here we stick to the simplest case of a Ill-grading. Let 9 = 9-1 $90$91 be a Ill-graded semisimple Lie algebra, G a Lie group with Lie algebra 9, PeG a parabolic subgroup for the given grading and Go, P+ c P the usual subgroups. In this case, the situation becomes very simple, since the filtration degenerates to T M = T- 1 M. Hence, from above we conclude that infinitesimal flag structures of type (G, P) are simply reductions of structure group of T M to the group Go. Here, we view M as being modeled on the vector space 9-1 and the reduction is with respect to the homomorphism Ad : Go ---4 GL(9-d. Also, morphisms of infinitesimal flag structures simply are morphisms of the reductions. Let us look at two special cases in a little more detail: (1) Consider the Ill-grading on the Lie algebra 9 = so(p+ 1, q+ 1) discussed in 1.6.3 and in Example 3.1.2 (1). Then 9-1 ~ 1R,P+q, 90 ~ co(p, q) and 9 1 ~ lR(p+q)*, and the adjoint action of 90 on 9-1 is the standard action. Put G = PO(p + 1, q + 1) and let PeG the stabilizer of an isotropic line. Then by part (2) of Proposition 1.6.3, Go = CO(9-1) ~ CO(p, q) via the adjoint action. Consequently, an infinitesimal flag structure of type (G, P) in this case is a first order CO(p, q)structure, and hence is equivalent to the choice of a conformal class of pseudoRiemannian metrics of signature (p, q) on M. We will discuss other possible choices of groups in 4.1.2. The prolongation procedure presented in 1.6.4 and 1.6.7 shows that an infinitesimal flag structure of type (G, P) uniquely determines a Cartan bundle endowed with a normalized Cart an connection. Hence, in this case, there is an equivalence of categories between normalized Cartan geometries and infinitesimal flag structures. (2) We shall soon see that surprisingly a similar correspondence between normal Cart an connections and certain infinitesimal flag structures exists for almost all parabolic geometries. There are, however, two series of exceptions, one of which corresponds to a Ill-grading: Consider 9 = sl( 1+n, lR) with the Ill-grading obtained from the block decomposition (- t:(A) !), with A E 91(n, lR), ¢ E lR n* , and v E IRn, as in Example 3.1.2 (2), i.e v corresponds to 9-1. A to 90 and ¢ to 91. The adjoint action of A E 90 on v E 9-1 is immediately seen to be given by (A + tr(A))v. It is easy to see (and we will do this explicitly in 4.1.5) that choosing G := PSL(n+l, lR)
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251
and P the stabilizer of the line spanned by the first basis vector, we get Go = GL(g-l) ~ GL(n,~) via the adjoint action. Hence, in this case an infinitesimal flag structure contains no information, so there is no hope to get a Cartan connection from it. We will discuss the right notion of underlying structures for this parabolic geometry in 3.1.15 below. 3.1. 7. Regularity. Regularity is a restriction on infinitesimal flag structures (and hence on parabolic geometries), which is of crucial importance in the theory. It expresses a tight relation between the two ingredients of an infinitesimal flag structure, the filtration of the tangent bundle and the reduction to the structure group Go. Let ({TiM},p : E -+ M,() be an infinitesimal flag structure of some fixed type (G, P). We have seen in 3.1.6 that gri(TM) ~ E xGo gi and thus gr(T M) ~ E x Go g_. Via this identification, the Lie bracket on g_ (which is preserved by the adjoint action) induces a bilinear bundle map
{ , } : gr(TM) x gr(TM)
-+
gr(TM),
which is compatible with the grading, Le. {gri(TM),grj(TM)} c gri+j(TM). This makes gr(T M) into a bundle of nilpotent graded Lie algebras modelled on g_. Under additional assumptions a similar structure is already intrinsic to the filtration {Ti M}. Assume that this filtration is compatible with the Lie bracket in the sense that for ~ E f(Ti M) and'r/ E f(Tj M) the Lie bracket [~, 'r/J is a section of Ti+jM. (We follow the usual convention that TiM = TM for all e::; -k.) Notice that this condition is automatically satisfied if the filtration is of length at most 2. Then for each i = -k, ... , -1 let us denote by qi : Ti M -+ gri(T M) the natural quotient map, and consider the operator f(TiM) x f(TjM) -+ f(gri+j(TM» defined by (~,'r/) I--> qi+j([~,'r/]). For a smooth function f E COO(M,~) we have [~,f'r/J = (~. f)", + J[~,'r/J. Since i ::; -1, we see that TjM c Ti+j+1M, so the first term lies in the kernel of qi+j' Hence, we conclude that the mapping defined above is bilinear over smooth functions, so it is induced by a bilinear bundle map TiM x TjM -+ gri+j(TM). Moreover, if ~ E Ti+1M or", E Tj+1M then [~,,,,J E Ti+j+1 M, so again this lies in the kernel of qi+j' Thus, our map further descends to a bundle map gri(TM) x grj(TM) -+ gri+j(TM). Taking these maps together, we obtain a bundle map £. : gr(TM) x gr(TM) -+ gr(TM), which is compatible with the gradings. Since £. is induced by the Lie bracket of vector fields, it follows immediately that it makes each fiber gr(TxM) into a nilpotent graded Lie algebra. DEFINITION 3.1.7. (1) A filtered manifold is a smooth manifold M together with a filtration T M = T- k M :::::l ••• :::::l T- 1M of its tangent bundle by smooth subbundles, which is compatible with the Lie bracket in the sense that [~,,,,J E f(Ti+j M) for any ~ E r(Ti M) and 'r/ E f(Tj M). (2) For a filtered manifold (M, {Ti M}) the tensorial map
£. : gr(T M) x gr(T M)
-+
gr(T M)
induced by the Lie bracket of vector fields as described above is called the (generalized) Levi bracket. For x EM, the nilpotent graded Lie algebra (gr(TxM),£.x) is called the symbol algebra of the filtered manifold at x. The bundle (gr(TM), £.) of nilpotent graded Lie algebras obtained in this way is called the bundle of symbol algebras.
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(3) An infinitesimal flag structure ({TiM},E - M,O) is called regular if (M, {Ti M}) is a filtered manifold and the algebraic bracket { , } : gr(T M) x gr(T M) - gr(T M) coincides with the Levi bracket C. (4) A parabolic geometry is called regular if the underlying infinitesimal flag structure constructed in Proposition 3.1.5 is regular. Suppose that (M, {Ti M}) and (M, {Ti M}) are filtered manifolds and that
f : M - M is a local diffeomorphism such that T f : T M - T M is filtration preserving. Then for each x E M, the tangent map Txf induces a linear map gr(Txf) : gr(TxM) - gr(Tf(x)M). From the fact that the pullback of vector fields is compatible with the Lie bracket, one immediately concludes that this map pulls back Cf(x) to cx, so it is an isomorphism between the symbol algebras. Hence, the symbol algebras are the basic invariants of a filtered manifold, which replace the tangent space for ordinary manifolds. In general, the isomorphism type of the symbol algebra may change from point to point, so (gr(T M), C) is not necessarily locally trivial as a bundle of Lie algebras. If we suppose that this is the case, however, then we get a smaller canonical frame bundle for gr(TM). Assume that the bundle of symbol algebras is locally trivial with typical fiber a nilpotent graded Lie algebra a. Then there is a natural frame bundle for gr(TM) with structure group the group Autgr(a) of all automorphisms of the graded Lie algebra a. The fiber of this frame bundle in x E M is the set of all Lie algebra isomorphisms from a to the symbol algebra at x. If the filtration {TiM} is part of a regular infinitesimal flag structure of type (G, P), then the bundle of symbol algebras is isomorphic to (gr(TM), { , }) as a bundle of Lie algebras, so it is locally trivial with typical fiber g_. Moreover, the adjoint action actually defines a homomorphism Go - Autgr(g-), which is infinitesimally injective if no simple ideal of 9 is contained in go. The definition of regularity means that the construction of Proposition 3.1.6 gives a reduction of structure group of the natural frame bundle of gr(TM) corresponding to Ad : Go Autgr(g_). Thus, we make the following: OBSERVATION 3.1. 7. A regular infinitesimal flag structure of type (G, P) on a smooth manifold M is equivalent to: • A filtration {TiM} of the tangent bundle which makes M into a filtered manifold such that the bundle of symbol algebras is locally trivial and modelled on the nilpotent graded Lie algebra g_ . • A reduction of structure group of the natural frame bundle of gr(T M) with respect to Ad: Go - Autgr(g_). A similar characterization holds for morphisms, i.e. they are equivalent to filtration preserving local diffeomorphisms which are compatible with the Go-structure on the associated graded bundles to the tangent bundles.
Thus, regular infinitesimal flag structures provide examples of the natural analog of first order G-structures in the setting of filtered manifolds. This makes their geometric interpretation easy. It should be mentioned at this point that in many cases Ad : Go - Autgr(g-) is an isomorphism; see 4.3.1. In these cases, a regular infinitesimal flag structure is only given by an appropriate filtration of the tangent bundle. This leads to highly interesting examples of parabolic geometries. For later use, we characterize regularity of an infinitesimal flag structure in terms of the frame form O.
3.1.
UNDERLYING STRUCTURES AND NORMALIZATION
253
PROPOSITION 3.1.7. Let ({TiM},p: E ---t M,(}) be an infinitesimalfiag structure such that (M, {Ti M}) is a filtered manifold. Then the structure is regular if and only if for all i, j < 0 such that i + j ? -k and all sections ~ E r(Ti E) and 1J E nTj E) we have (}Hj([~, 7]]) = [Oi(~)' OJ (7])J. PROOF. If we choose ~ and 1J to be projectable, then [~, 7]J lifts the bracket of the projections. Since (M, {Ti M}) is a filtered manifold, this implies that [~, 7]] E nTHj E). For general sections ~ and 7] and a point x E E, the class of [~, 7]](x) in TxE/T~+j E depends only on ~(x) and 17(X). Hence, we conclude that we always have [~, 7]] E nTHj E), so OHj([~, 7]]) makes sense. Since TiE and Tj E are contained in T H i+ 1 E, we see that OHj([~, 7]](x)) depends only on ~(x) and 7](x). Since the same is evidently true for the other side of the equation we see that (}i+j([~, 7]]) = [(}i(~)' OJ (7])] holds for arbitrary vector fields ~ and 7] if and only if it holds for projectable fields. But if ~ and 7] project to sections ~ E r(Ti M) and 7] E nTj M), then the functions Oi(~) represents the section qi(~fE ngri(TM)) aIi"d likewise for OJ (7]). Hence, the right-hand side of the equation-represents {qi (~), qj (7])}. On the other hand, since [~, 7]] projects to [~, 17], we conclude that the function (}Hj([~, 7]]) represents qHj([{,!l]) = £(qi({),qj(!l))' 0 REMARK 3.1.7. (1) Since the components 0i of a frame form are only partially defined, their exterior derivative is not well defined in general. However, if (M, {TiM}) is a filtered manifold, then from the proof of the proposition we see that also (E, {Ti E}) is a filtered manifold. In particular, for i,j < 0 such that i + j ? -k we can use the standard formula for the exterior derivative to define d(}i+j on Ti Ex Tj E. Alternatively, we can extend 0Hj arbitrarily to a 9Hrvalued one form on E and observe that the restriction of the exterior derivative of this form is independent of the choice of extension. In this language, the equation in the proposition can be written as dOHj(~, 7]) + [(}i(~)' OJ (7])] = O. EXAMPLE 3.1.7. Let us briefly describe the geometric interpretation of infinitesimal flag structures and the geometric relevance of regularity in an important example. Details of this example will be discussed in 4.2.4. Consider the Lie algebra 9 = .5u(n + 1,1) endowed with the J2J-grading from Example (3) of 3.1.2, so we have the following description of g: {
(~zx
A-
~ f~(a)lln _i~*): A E .5u(n),a E e,x E en, Z E en*,x,z E JR.}. -X
-a
From this block form, one immediately sees that the bracket 9-1 x 9-1 ---t 9-2 (which completely describes 9_) is given by [X, Y] = -X·Y + y* X. This is twice the imaginary part of the standard Hermitian inner product on en. In particular, we have [iX, iY] = [X, Y], and the bracket is nondegenerate, i.e. [X, Y] = 0 for all Y implies X = O. On the other hand, go ~ .5u(n) E9 e and the action of the semisimple part .5u(n) on 9-1 ~ en is the standard representation. Put G = PSU(n+ 1,1), the quotient of SU(n+ 1,1) by its center, and let P be the stabilizer of the isotropic line spanned by the first basis vector. Then it turns out that the adjoint action identifies the subgroup Go C G with the group of all pairs (¢}, ¢2) of linear isomorphisms ¢1 : g-1 ---t 9-1 and ¢2 : 9-2 ---t 9-2 such that
3. PARABOLIC GEOMETRIES
254
¢1 is complex linear and [¢l(X), ¢l(Y)] = ¢2([X, Y]) for all X, Y E g-l. (This implies that ¢2 is uniquely determined by ¢1, but this is not important here.) To obtain an infinitesimal flag structure of type (G, P), we have to start with a smooth manifold M of dimension 2n + 1, and the filtration of the tangent bundle T M we need in this case is simply given by a subbundle T- 1 MeT M of real rank 2n. The associated graded of the tangent bundle then has the form gr(T M) = gr_2(TM) EB gr_1 (TM) and gr_2(TM) = TM/T-l M is a real line bundle, while gr -1 (T M) = T-1 M. From the description of Go given above it is easy to see that a reduction to the structure group Go of gr(T M) is equivalent to an almost complex structure Jon T- 1 M and a skew symmetric bundle map { , } : T- 1 M x T- 1 M ~ gC2(TM), which is (in appropriate trivializations) the imaginary part of a definite Hermitian form. Therefore, an infinitesimal flag structure in this case is given by a complex subbundle of rank n plus the choice of the tensorial map { , }. In the language of CR-structures, a rank n complex subbundle in a manifold of dimension 2n + 1 is called a hypersurface type almost CR-structure of CR-dimension n. However, in this situation the tensorial map { , } is an additional ingredient, which is not deeply related to the almost CR-structure. The situation changes completely, if one imposes the regularity condition. If the filtration TM :::> T- 1 M is part of a regular infinitesimal flag structure, then the Levi bracket C : T- 1 M x T- 1 M ~ T M /T- 1 M has to be equal to the bracket { , }. In particular, C has to be nondegenerate, which is equivalent to the fact that T- 1 MeT M is a contact structure. In terms of CR-geometry, this means that the almost CR-structure is nondegenerate. Since { , } comes from the imaginary part of a Hermitian form, we further conclude that C(Je, J",) = C(e, "') for all In CR terms, this means that the CR-structure is partially integrable. Under this assumption, it turns out that C is the imaginary part of the classical Levi form, and the Levi form being definite means that the almost CR-structure is strictly pseudoconvex. Summarizing, we see that a regular infinitesimal flag structure of type (G, P) is equivalent to a strictly pseudo convex partially integrable almost CR-structure. This is also easily expressed in terms of filtered manifolds. A filtration such that any symbol algebra is isomorphic to g_ (as a real Lie algebra) is just a contact structure. The additional reduction to Go C Autgr(g_) is equivalent to a complex structure on T- 1 M, which makes M into a partially integrable almost CR-manifold. The morphisms are equivalent to filtration preserving local diffeomorphisms such that the restriction to T-l M of each tangent map is complex linear, i.e. to local CR diffeomorphisms.
e, ",.
3.1.8. A characterization of regular parabolic geometries. Our next goal is to characterize regularity of parabolic geometries in terms of their curvature. Recall from 1.5.1 that the curvature is the basic invariant for any Cartan geometry. For a parabolic geometry (p : g ~ M, w) it can be either encoded in the curvature form K E 2(g, g) or in the curvature function K : g ~ L(A2g_, g). By definition, K(e, "') = m.v(e, "') + [wee), w(",)], while K(U)(X, Y) = K(w~l(X), w~l(y)). Recall further that K is horizontal and both K and K are equivariant in an appropriate sense; see 1.5.1. We will obtain the characterization of regularity from a more general result, which will be very useful for the geometric interpretation of the torsion of parabolic
n
3.1. UNDERLYING STRUCTURES AND NORMALIZATION
255
geometries in the sequel. The tools used in this proof are closely related to normal Weyl structures, which will be discussed in 5.1.12. PROPOSITION 3.1.8. Let 9 = g-k EB ... EB gk be a Ikl-graded semisimple Lie algebra, G a Lie group with Lie algebra g, PeG a parabolic subgroup corresponding to the grading, and Go c P the Levi subgroup. Let (p : g ---. M, w) be a parabolic geometry of type (G, P) with curvature function K : g ---. L(A2g_, g), u E g any point, and put x = p(u) EM. Then there is an open neighborhood U of x E M and a linear extension operator TxM ---. X(U), written as ~ 1-+ ~, which is compatible with all structures on TM obtained from the Cartan connection wand has the following property: For~, 'rJ E TxM let X, Y E g_ be the unique elements such that Tup· W~l(X) = ~ and Tup· w~l(y) = 'rJ. Then
PROOF. Note first that the elements X and Yare well defined since two lifts of a tangent vector on M differ by some vertical vector and g_ is complementary to p. For any element Z E g_ consider the vector field w-1(Z) and its flow FI~-l(Z)(u) which is defined for sufficiently small t. Then there is a neighborhood V of 0 in g_ on which the map 1, we get the bracket on T- 1M x gri(T* M) from invariance of B, which implies that for ~ E T- 1M, ¢ E gri(T*M) and"., E gr_i+1(TM) (note that by assumption -i + 1 < 0), we get {~, ¢}(".,) = -¢( {C".,}). To get the bracket T- 1M x gr1 (T* M), one has to compute explicitly the trilinear map 9-1 x 9-1 X 91 ---- 9-1 defined by (X, Y, Z) I-t [[X, Z], Y], since this is exactly the endomorphism of 9-1 induced by [X, Z] E 90. The fact that 9_ is generated by 9-1 implies that as a Lie algebra gr(T M) is generated by T- 1M. Using the Jacobi identity, we can then compute all brackets on gr(TM) x gr(T* M). Hence, we are only missing the brackets defined on gr(T* M) x gr(T* M), which now follow from invariance of B. For i,j > 0, ¢ E gri(T* M), 1/J E grj(T* M), and ~ E gr_i_j(TM), we have {¢,1/J}(O = ¢({1/J,O)· 3.1.10. Normalization. With the information about the adjoint tractor bundle at hand, we may now return to the task of finding a normalization condition which leads to a unique parabolic geometry with fixed underlying regular infinitesimal flag structure. So we start with a regular parabolic geometry (p : g ---- M, w) of type (G, P), and consider the underlying infinitesimal flag structure (Po : go ---M, wO) introduced in 3.1.5. We want to study the set of Cartan connections W E n1 (9,9) which induce the same infinitesimal flag structure. To do this, we start by analyzing the affine structure on the space of Cartan connections on a given principal bundle.
3.1. UNDERLYING STRUCTURES AND NORMALIZATION
259
By definition, the difference w - w of two Cartan connections is an element of 01(9,g). Since both Cartan connections reproduce generators of fundamental vector fields, this form has to vanish upon insertion of a vertical tangent vector, so it is horizontal. Further, since both wand ware P-equivariant, so is their difference. By Corollary 1.2.7 we can interpret ~ := w- w as a one-form on M with values in AM. Conversely, a one-form ~ E Ol(M, AM) can be interpreted as a horizontal equivariant g-valued one-form on 9. As such, we can add it to a given Cartan connection w to obtain a g-valued one-form w on 9 which is equivariant and reproduces the generators of fundamental vector fields. Hence, wis a Cartan connection, provided that it restricts to a linear isomorphism on each tangent space. The set of all f E L(g/p, g), such that id + f 0 IT is a linear isomorphism of g, is an open neighborhood of zero, and one immediately verifies that it is Pinvariant. Passing to associated bundles, this invariant open subset gives rise to an open subbundle of L(TM,AM), and w is a Cartan connection if and only if ~ is a section of this open subbundle. Thus, the space of all Cartan connections on the bundle 9 is an affine space modelled on the space of smooth sections of that open subbundle of L(TM, AM). We will simply write w- w = ~, respectively, w= w + ~ to indicate this affine structure over a space of one-forms with values in the adjoint tractor bundle. Notice that if we have a linear map f : g/p ---? 9 which is homogeneous of degree ~ 1, i.e., which has the property that f(gi /p) C gi+l for all i < 0, then id + f 0 IT is homogeneous of degree ~ 0, and the induced linear map on gr(g) is the identity. In particular, id + f 0 IT is a linear isomorphism. Hence, if a one-form ~ with values in AM is actually a section of the filtration component L(TM, AM)l, then for any Cartan connection w the construction above again leads to a Cartan connection. We need a final piece of information. From 3.1.1 we know that the filtrations on TM and AM induce a filtration of the vector bundle L(AiTM,AM) for any ( = 0, ... ,dim(M). The sections of this bundle are the (-forms with values in the adjoint tractor bundle. The filtration on this bundle is given by homogeneous degree, i.e. T E L(AiTM,AM)m if and only if T(TilM, ... , Ti€M) C Ail+···+ie+mM for all iI, ... , ii < 0. We also know from 3.1.1 that the associated graded bundle gr(L(AiTM, AM)) is given as L(Af gr(TM),gr(AM)) with the grading induced by homogeneous degrees. Hence, gr(L(AiTM, AM)) may be viewed as the associated bundle to 90 corresponding to the Go-module L(Aig_,g), so this depends only on the underlying infinitesimal flag structure. On the other hand, Go-equivariant maps between these modules give rise to well-defined bundle maps. The spaces L(A2g_, g) are the chain spaces in the standard complex for computing the Lie algebra cohomology of g_ with coefficients in 9 (viewed as a g_module via the adjoint action); see 2.1.9 for the definition of Lie algebra cohomology. Moreover, since the adjoint action of Go on g_ and on 9 is by Lie algebra homomorphisms, the Lie algebra differential L(Aig_, g) ---? L(AHlg_, g) as defined in formula (2.2) in 2.1.9 is a Go-homomorphism. Thus, we get induced bundle maps gr(L(AiTM, AM)) ---? gr(L(A'-+lTM, AM)) for all {~ 0. By construction, they are given by the usual formula for the Lie algebra coboundary, but with all Lie
a:
a:
260
3. PARABOLIC GEOMETRIES
brackets replaced by the algebraic bracket, i.e.
e
84>(f,0, ... ,f,e)
:= ~) _l)i {f,i, 4>(f,0,
... ,~, ... ,f,e)}
i=O
i<j where the hats denote omission. From this formula, it is obvious that 8 preserves homogeneities, i.e. if 4> is homogeneous, then also 8(4)) is homogeneous of the same degree. Using this, we can now formulate: PROPOSITION 3.1.10. Let (p : g -+ M,w) be a regular parabolic geometry of type (G, P), let W E n1 (9, g) be another Carlan connection and put CI> := W - W E n 1(M,AM). (1) The Cartan connections wand w induce the same filtration of T M if and only if CI> E n1(M, AM)O and they induce the same underlying infinitesimal flag structure if and only if CI> E n1 (M, AM) 1 . (2) Suppose that CI>(Ti M) c AiH M for some fixed f ;::: 1. Then the difference K - ", of the curvatures of the two Carlan connections maps TiM x Tj M to Ai+iHM. The induced section greeK -",) of gre(L(A2TM, AM)) is given by 8(gre(CI»), where gre(CI» E r(gre(L(TM,AM))) is the section induced by CI>. PROOF. (1) Since for i < 0 we may characterize TiM as the image under Tp of Tig = w- 1(gi) and both wand W restrict to linear isomorphisms on each tangent space, we see that the fact that winduces the same filtration as w is equivalent to the fact that w(Tig) c gi for all i < O. Since w- w by definition vanishes on TOg, we see that wand w induce the same filtration if and only if their difference maps each Tig to gi. From the construction in 3.1.5 it is clear that wand winduce the same infinitesimal Hag structure if and only if their difference maps Tig to gi+l for all i < 0. From the definition of CI>, one immediately concludes that this is equivalent to CI>(TiM) c Ai+1M. (2) We start by working in the picture of forms on g. In terms of the function 4> : g -+ L(g/p, g) representing CI>, we have w= w + 4> 0 w, or more precisely,
w(u)(f,) = w(u)(f,) + 4>(u)(w(u) (f,)). From the definition of the exterior derivative, one concludes that this implies dW(f" TJ) = dw(f" TJ) + d4>(f,) (w(TJ)) - d4>(TJ)(w(f,)) + 4>(dw(f" TJ))· On the other hand, [w(f,),w(TJ)] = [w(f,),w(TJ)] + [4>(w(f,)),w(TJ)] + [w(f,),4>(w(TJ))] + [4>(w(f,)),4>(w(TJ))]· Thus, we obtain
(3.1) K(f" TJ) - K(f" TJ) =d4>(f,)(w(l1)) - d4>(TJ)(w(f,)) + 4>(dw(f" TJ)) + [4> (w(f,)) , w(TJ)] + [w(f,), 4>(w(TJ))] + [4>(w(f,)) , 4>(w(TJ))]· The relation between CI> and ¢ reads as CI>(Tp . f,) = w- 1(4)(w(f,))). Thus, the condition on homogeneity of CI> exactly means that for f, E Tig and TJ E Tig with i,j < 0, we have 4>(w(f,)) E gi+t and 4>(w(TJ)) E giH. On the other hand, since 4>(Tn g) c gnH, the same is true for d4>(f,) and d4>(TJ). In particular, the first two terms in the right-hand side of (3.1) have values in gi H and giH, respectively. By regularity, dw(f" TJ) E gi+j, which together with the above implies that the next three terms in (3.1) have values in gi+iH, while the last term has values in gi+i+2e. Since i,j < 0 and f > by assumption, this implies that the difference
°
3.1. UNDERLYING STRUCTURES AND NORMALIZATION
i((e,T/) - K(e,T/) has values in class of
gi+j+i,
261
and its class in gri+j+i(g) coincides with the
¢(dw(e,T/)) + [¢(w(e)), w(T/)]
+ [w(e), ¢(w(T/))].
By regularity, dw(e, T/) is congruent to -[w(e), w(T/)] modulo gi+j+1, and hence the first summand is congruent to -¢([w(e),w(T/m modulo gi+j+i+1. On the level of the associated graded to the tangent bundle, the bracket [ , ] on g corresponds to the algebraic bracket on gr(AM) and gr(TM). Now by construction, the function g -+ gr(L(A2(g_), g)) corresponding to gr((i~-/ S2}, S3) -
K(K(S1> S2), S3) -
DS3
(K(Sl' S2))).
eye!
The right-hand side of this expression is a trilinear map AM x AM x AM -+ AM. By our assumptions on K, the first two summands are visibly homogeneous of degree 2:: £, while the third summand is homogeneous of degree 2:: 2£ > £. Finally, for the last summand, K(S1>S2) is by assumption a section of Ai 1 +i2 +iM, so also DS3 (K(Sl' S2)) is a section of that subbundle, and all terms coming from these
3. PARABOLIC GEOMETRIES
266
swnmands are homogeneous of degree> f. Hence, computing gr e of this expression, we only have to consider the first two terms in each summand of the cyclic swn. Since the algebraic bracket on AM induces the algebraic bracket on gr(AM) (and thus also on gr(TM)), passing to gre we simply get 0=
L ({grl(I\;)(6,6),6} + grl(I\;)({6,6},~3)). eyel
Using skew symmetry of gre(I\;) and the algebraic bracket, one immediately sees that the right-hand side equals -a(gre(I\;))(~1.~2,~3). If we now assume that the parabolic geometry in question is normal (and regular), then by definition a*(I\;) = O. Since a* is compatible with homogeneities, we get 0= gre(a*(I\;)) = gro(a*)(gre(I\;)), so we see that gre(I\;) is a section of the subbundle ker(gro(a*)) nker(a) = ker(O). On the other hand, the homogeneous component of degree f of I\;H by construction equals the image of gre(I\;) E r(ker(gro(a*))) in the quotient modulo the image of a*, which implies that this component coincides with grt(I\;). Inductively, starting from f = 1, this shows that I\;H = 0 implies I\; = O. 0 This result is remarkable in various respects. First, it reduces the question of local flatness from considering the curvature I\; to considering the harmonic curvature I\;H, which is much simpler to understand. Second, it immediately leads to an overview on what the essential curvature quantities for any parabolic geometry are. As we shall see in Section 3.3, there is a simple algorithm to compute the cohomology H2(g_, g) explicitly as a go-representation. Moreover, one even gets complete information on how this cohomology (viewed as ker(O)) sits within the space L(A2g_, g). In most cases, H2(g_, g) consists offew components only, so one gets quite effective information on possible obstructions to local flatness.
3.1.13. Existence of normal Cartan connections. We are ready to take the first step towards proving existence of normal parabolic geometries with a given underlying regular infinitesimal flag structure. By normalizing a given Cartan connection, we prove that any regular infinitesimal flag structure that comes from some parabolic geometry also comes from a normal parabolic geometry. This uses only the algebraic properties of the normalization condition as described at the end of 3.1.10. PROPOSITION 3.1.13. Let (p : 9 ---t M,w) be a regular parabolic geometry with curvature I\; E n2(M,AM) and suppose that a*I\; E nl(M,AM)t for some f 2: 1. Then there is a normal Cartan connection wE n1 (Q,g) such that w- wE n1(M,AM)t. In particular, there is always a normal Cartan connection w which induces the same underlying infinitesimal flag structure as w. PROOF. We show that we can find a Cartan connection wsuch that w- w E nl(M,AM)l and such that the curvature ii, of w satisfies a*ii, E n1(M,AM)l+1. Inductively, this implies that we can find a normal wsuch that w-w E n1 (M, AM)t. So let us assume that a*I\; E n1(M,AM)l for some fixed f 2: 1. Then we have an induced section grt(a*I\;) of the bundle L(gr(TM), gr(AM)) , whose value in each point is homogeneous of degree f. The bundle map
a* : A2T* M ® AM
---t
T* M ® AM
is compatible with homogeneities, so it induces a bundle map between the filtration components in degree f. In particular, this implies that we find an element 'ljJ E
3.1. UNDERLYING STRUCTURES AND NORMALIZATION
n2(M,AM)£ such that 8*'IjJ
267
= 8*K. Hence, we obtain
gr£(8*K)
= gr£(8*'IjJ) = gro(8*)(grl('IjJ)),
so grl!(8*K) is a section of the subbundle im(gro(8*)) C L(gr(TM),gr(AM)). The Hodge decomposition (3.2) in 3.1.12 for n = 2 shows that the subbundle im(8) C L(A 2 gr(T M), gr(AM)) is complementary to the sub bundle ker(gro (8*)). Hence, gro(8*) restricts to a bundle isomorphism im(8) -+ im(gro(8*)). Thus, we can find a smooth section ¢ of L(gr(TM),gr(AM)) whose value in each point is homogeneous of degree f such that gro(8*)(8¢) = grl(8* K). Next, we can find a one-form cP E nl(M,AM)( such that gr((cp) = ¢. As we have observed in 3.1.10, interpreting cP as a horizontal equivariant form on g, we can form a Cartan connection w= w - CPo By part (1) of Proposition 3.1.10, wand w induce the same underlying infinitesimal flag structure. On the other hand, from part (2) of the same proposition, we see that the curvature K, of w has the property that K,- K E n2(M,AM)l and the induced section gr£(K,-K) of L(A2 gr(TM),gr(AM)) is given by -8(gr£(cp)) = -8¢. Now 8*K, = 8*K + 8*(K, - K) E nl(M,AM)t',
and we compute gr£(8*K,) = gr((8*K)
+ gro(8*)(gr£(K, -
K)) = gr£(8*K) - gro(8*) (8¢) = O.
This shows that 8*K, E nl(M,AM)t'+l, and the result follows. The last statement follows since regularity of w implies that K E n2(M, AM)l and thus 8* K E nl (M, AM) 1 . Hence, there is a normal Cartan connection wsuch that w-w E nl(M,AM)l, which implies that wand w induce the same underlying infinitesimal flag structure. 0 We have already mentioned an analogy between infinitesimal flag structures and first order G-structures in 3.1.6 and 3.1.7. To prove existence of normal parabolic geometries with a given underlying regular infinitesimal flag structure, we have to make this analogy more precise and consider the analog of connections on first order G-structures as discussed in 1.3.6. There we started with a Lie group H and an infinitesimally injective homomorphism H -+ GL(m, JR.). We can view this homomorphism as a representation of H on JR.m and form the semidirect product B = JR.m ~ H. This is a Lie group which contains H as a closed subgroup, BjH ~ JR.m and the resulting action of B on JR.m consists of all transformations generated by translations and the elements of H. A first order G-structure with structure group H is then a reduction of the linear frame bundle of an m-dimensional manifold M to this structure group. We have seen in 1.3.6 and Example 1.5.1 (ii) that principal connections on such G-structures are equivalent to Cartan geometries of type (B, H). In our current situation, we have to replace the homomorphism H -+ GL(m, JR.) by the homomorphism Go -+ Autgr(g-) induced by the restriction of the adjoint action. To take into account the nontrivial Lie algebra structure on g_, we have to replace JR.m by the (uniquely determined) simply connected Lie group G_ with Lie algebra g_. Via the left trivialization TG _ ~ G _ x g_ of the tangent bundle of G _, the grading of g_ induces a decomposition of TG _ into a direct sum of smooth subbundles. The group G _ endowed with these subbundles is sometimes called the Carnot group associated to the graded nilpotent Lie algebra g_. Since G _ is simply
268
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connected, any automorphism of 9_ integrates to a group automorphism of G _. Hence, we can view Aut gr (9-) as a group of automorphisms of G_ and use this to define a semidirect product of G_ and Go. However, in the parabolic case, a simpler direct description is available. Consider a Ikl-graded semisimple Lie algebra 9 = 9-k E9 •.• E9 9k, a Lie group G with Lie algebra 9-, and a parabolic subgroup PeG corresponding to the grading. Let Go c P be the Levi subgroup as defined in 3.1.3. Similarly, as in 3.1.3, we can defined a subgroup respecting the filtration which is opposite to {9 i }. More formally, consider L:= {g E G: Ad(g)(9i) C 9-k E9 ... E9 9i for all i = -k, ... , k}.
This is a closed subgroup of G and as in 3.1.3 one shows that it has Lie algebra pop := 9_ E9 90. By construction Go is a subgroup of L corresponding to the subalgebra 90 and we define pop to be the union of all those connected components of L which meet Go. This is called the opposite parabolic subgroup to P. As in the proof of Theorem 3.1.3, one shows that any element g E pop can be uniquely written as g = exp(X)go for X E 9_ and go E Go. (Getting a representation with go on the right rather than on the left can be realized by passing to inverses.) Moreover, the exponential map is a diffeomorphism from 9_ onto its image, which is a closed subgroup in pop isomorphic to G _. This identifies pop with a semidirect product of G _ and Go as required. In this setting, the concept of a Cartan geometry of type (POP, Go) makes sense. Note that the decomposition pOP = 9_ E9 90 is Go-invariant, so Cartan connections of this type decompose into a soldering form and a principal connection. THEOREM 3.1.13. Let 9 = 9-k E9 .•. E9 9k be Ikl-graded semisimple Lie algebra, G a Lie group with Lie algebra 9, PeG a parabolic subgroup corresponding to the grading and Go c P the Levi subgroup. Then any regular infinitesimal flag structure of type (G, P) on a smooth manifold M is induced by a normal parabolic geometry of type (G, P). PROOF. Let ({TiM},po : E ~ M,O) be a regular infinitesimal flag structure. In view of Proposition 3.1.13, it suffices to construct some parabolic geometry (p: 9 ~ M,w) which induces this structure. Making several choices, this can be done directly. First, choose any principal connection 'Y E nl(E, 90) on the principal Go-bundle Po : E ~ M. This splits TEas the direct sum of the vertical bundle V E and the horizontal subbundle HE = ker("t). For each u E E the map TuPo : HuE ~ TxM is a linear isomorphism, where x = Po(u). Consider the filtration TE = T-kE::J ... ::J TOE = V E from 3.1.6 and put Hi E := Ti En HE for i < O. By construction, each Hi E is a smooth subbundle and TuPo : HuE ~ TxM is an isomorphism of filtered vector spaces for each u E E. Next, for each i = -k, ... , -1 we have the smooth subbundle TiM C TM, and we may choose a smooth homomorphism 1ri : T M ~ Ti M of vector bundles which is a projection onto this subbundle. As we have noted above, for each u E E with Po(u) = x and each i < 0, the mapping TuPo : HuE ~ TxM is an isomorphism of filtered vector spaces. Using this, we can lift each 1ri to a smooth bundle map 7ri : T E ~ HiE, which is characterized by Tpo 0 7ri = 1ri 0 Tpo. In particular, 7ri is a projection onto the subbundle Hi E c T E. Moreover, denoting by r 90 the principal right action of go E Go we obtain T(r90) 07ri = 7ri 0 T(r90), so
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this projection is Go-equivariant. Having made these choices, we can now define 1 - 0= (O-k, ... ,O-l,OO) E n (E,g_ EBgo) by 00 = 'Y and Oi(~):= Oi(7l"i(~)) for each i < O. We claim that 0 is a Cartan connection of type (POP, Go) on E. If O(~) = 0, then 'Y(~) = 0 and hence ~ E HE. On this subbundle, Lk is the identity, and hence O-k(~) = 0, which means that ~ E T- k +1 E. This implies 7l"-k+1(~) = ~, so O-k+l(~) = 0 and hence E T- k+2E. Iterating this, we get ~ E TOE = VE, and since we have already seen that E HE, we get ~ = O. Hence, is injective and thus a linear isomorphism on each tangent space. For A Ego, the fundamental vector field (A lies in TOE and thus 7l"i((A) = 0 for all i < 0 and 'Y((A) = A, so 0 reproduces the generators of fundamental vector fields. Finally, the adjoint action of Go preserves the decomposition g-k EB ... EB go, so we can verify Go-equivariancy componentwise. For the component 00 = 'Y this holds by definition, while for the other components it follows immediately from invariance of 7l"i and equivariancy of Oi. Hence, we have proved our claim. Consider the inclusion pop '-....+ G. By construction, pop n P = Go and infinitesimally, we obtain the inclusion pop '-....+ 9 which induces a linear isomorphism pOP /go ---+ g/p. Following Example 1.5.16, we can use the inclusions i : Go ---+ P and a : pOP ---+ 9 to obtain the data for an extension functor as defined in 1.5.15, which maps Cartan geometries of type (POP, Go) to Cartan geometries of type (G, P). Applying this functor to (E, 0), we obtain the principal bundle g := E xGo P and a Cartan connection wE n1(g,g) which is characterized by the fact that j*w = 0, where j : E ---+ g is the inclusion. In particular, the Cartan connection w induces the given filtration on T M. Composing the projection g ---+ g/ P+ with j induces an isomorphism E ---+ g / P+ of principal Go-bundles. For u E E and ~ E TuE we can use j (u) E g and Tuj . ~ E Tj(u)g as representatives. But then by construction w(Tuj .~) = O(e), and if E Ti E, then the gi-component of this is just Oi(e). But this exactly shows that the underlying infinitesimal flag structure of (g ---+ M, w) is (E ---+ M,O). 0
e
o
e
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3.1.14. Uniqueness of normal Cartan connections. The remaining point to understand the relation between regular infinitesimal flag structures and normal regular parabolic geometries is the question of uniqueness of normal Cartan connections. One can expect uniqueness at best up to isomorphism. Namely, suppose that (p : g ---+ M, w) is a normal parabolic geometry and that "I]! : g ---+ g is an automorphism of the principal bundle g covering the identity on M. Then "I]!*w is a Cartan connection, and "I]! is a homomorphism from the parabolic geometry (g, "I]!*w) to (g, w). From the definition of the curvature one immediately concludes that the curvature of "I]!*w is just "I]!*K, where K is the curvature of w. Naturality of 8* then implies that 8* ("I]! *K) = "I]! * (8* /'l.) = 0, so also "I]! *w is a normal Cart an connection. If "I]! has the property that it induces the identity on the underlying infinitesimal flag structure (and we shall see in the proof of the following proposition that there are many automorphisms having this property), then we have two normal Cartan connections inducing the same underlying infinitesimal flag structure. We will soon show, however, that under a cohomological condition this is the only freedom left. To formulate this, we have to observe that the Lie algebra differential 8 : en (g_, g) ---+ e n+1 (g_, g) is homogeneous of degree zero for the natural gradings on the chain spaces. Consequently, each cohomology space Hn (g_, g) is naturally graded, i.e. it decomposes as Hn(g_, g) = Eel! Hn(g_, g)e according to
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homogeneous degrees of representative cocycles. Of course, we also have the associated filtration given by the subspaces Hn(g_,g)l:= ffim~tHn(g_,g)m. PROPOSITION 3.1.14. Let 9 = g-k E9 ... E9 gk be a Ikl-graded Lie algebra such that HI (g_, g)t = 0 for some l ~ 1. Let G be a Lie group with Lie algebra g, PeG a parabolic subgroup for the given grading, Go c P the Levi subgroup, and let (p: g --+ M,w) be a normal regular parabolic geometry of type (G, P). Suppose further that w E (21(g, g) is another normal Cartan connection such that for each i = -k, ... , -1 the difference w - w maps Tig to gi+e. Then there is an automorphism \lI of the principal bundle g, which induces the identity on the underlying infinitesimal flag structure such that \lI*w = w. PROOF. Putting ell := w- w, the assumption on w can be phrased as ell E (21(M,AM)t. By part (2) of Proposition 3.1.10, the difference K - K of the curvatures of wand w lies in (22(M,AM)l, and grt(K - K) = 8(gre(eIl)). Moreover, since both w and w are normal, 8*(K - K) = 0, and passing to the associated graded, this implies 0 = grt(8*(K - K)) = gro(8*)(gre(K - K)). Formula (3.2) from 3.1.12 shows that the subspaces ker(gro(8*)) and im(8) of L(A2 gr(TM),gr(AM)) have zero intersection, so we see that grl(K - K) = 8(grl(eIl)) = O. Since by assumption H 1 (g_, g)e = 0, this implies that gre(eIl) must lie in the image of 8. Let us first assume that l > k. Then A eM = 0 and hence grt (ell) = 0, so ell E (21(M,AM)l+I. Repeating the argument finitely many times and using that (21 (M, AM)i = 0 for i > 2k, we obtain ell = 0 and thus w = w. It remains to discuss the case l :s; k. Since gre(eIl) lies in the image of 8, we can choose a section 't/J of A eM such that 8(gre('t/J)) = - gre(ell). The section 't/J corresponds to a smooth function g --+ ge that we denote by the same symbol, which satisfies 't/J(u· g) = Ad(g-I)('t/J(u)) for all u E g and g E P. Now we define a smooth map \lI : g --+ g by \lI(u) := U· exp('t/J(u)). Equivariancy of't/J implies that exp('t/J(u· g)) = g-1 exp('t/J(u))g, which in turn implies \lI(u· g) = \lI(u) . g, so \lI is an automorphism of the principal bundle g which covers the identity on M. Thus, \lI*w E (21(g,g) is a Cartan connection, which is normal by construction. For a point u E g and a tangent vector e E T~g, consider
\lI*w(u)(e) = w(u· exp('t/J(u))) (Tu\ll . e). We can compute the right-hand side as in the proof of Proposition 1.5.3 as
Ad(exp(-'t/J(u)))(w(u)(e)) + o(expo't/J)(u)(e), where 0 denotes the left logarithmic derivative; see 1.2.4. By assumption, 't/J has values in gl, which is a Lie subalgebra of 9 since l ~ 1. Thus, exp o't/J has values in the corresponding subgroup of P, and o(expo't/J) has values in gt. Hence, we conclude that \lI*w(u)(e) is congruent to Ad(exp(-'t/J(u))(w(u)(e)) modulo gl C gi+l+l. Using Ad(exp(-'t/J(u))) = e-ad('IjJ{u)), we see that \lI*w(u)(e) is congruent to w(u)(e) - ['t/J(u), w(u)(e)] modulo gi+l+1. But w(u)(e) = w(u)(e) + eIl(u) (e), while in the second term, we may replace wby w without changing the class modulo gi+l+l . Thus, we see that
(\lI*w - w)(u)(e) == eIl(u)(e) - ['t/J(u),w(u)(e)] modulo gi+l+l. Hence, \lI*w - w E (21 (M, AM)t and griH((\lI*w - w)(u)(e))
= {grt('I/I),gri(en = -8(grt('t/J))(gri(e))·
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Hence, gre(W*w - w) = gre(tP) - 8(grl("p)) = 0 and W*W - w E nl(M,AM)l+l. Iterating this argument, we can make the difference to be in nl(M,AM)k+l and we have already dealt with this case before. 0 From this uniqueness result, we directly get the general theorem establishing the relation between parabolic geometries and infinitesimal flag structures in most cases, namely under the assumption that Hl(g_, g)1 = O. We will clarify the meaning of this condition completely in 3.3. There we will show that if 9 does not contain a simple summand isomorphic to 5((2), then we always have HI (g_, g)2 = O. Moreover, HI (g_, g)1 1= 0 happens only if 9 contains a simple factor that belongs to one of two specific series of simple graded Lie algebras. Geometrically, these two series correspond to classical projective structures (see 1.1.3, 4.1.5, and Example 3.1.6 (2) for the corresponding Ill-grading), respectively, contact projective structures (see 1.1.4 and 4.2.6). THEOREM 3.1.14. Let 9 = g-k E9 •.. E9 gk be a Ikl-graded semisimple Lie algebra such that none of the simple ideals of 9 is contained in go, and such that Hl(g_,g)1 = O. Suppose that G is a Lie group with Lie algebra g, and PeG is a parabolic subgroup corresponding to the grading with Levi subgroup Go c P. Then associating to a parabolic geometry the underlying infinitesimal flag structure and to any morphism of parabolic geometries the induced morphism of the underlying infinitesimal flag structures defines an equivalence between the category of normal regular parabolic geometries of type (G, P) and the category of regular infinitesimal flag structures of type (G, P). PROOF. We have noted in 3.1.6 that passing to the underlying infinitesimal flag structure of a parabolic geometry defines a functor, and by definition this functor preserves regularity. Thus, we get a well-defined functor from normal regular parabolic geometries to regular infinitesimal flag structures. From Theorem 3.1.13 we see that conversely given a regular infinitesimal flag structure, we can construct a normal regular parabolic geometry inducing it. Now we claim that for two normal regular parabolic geometries any morphism between the underlying infinitesimal flag structures uniquely lifts to a morphism of parabolic geometries. Having proved this, any construction like the one from Theorem 3.1.13 extends to a functor from regular infinitesimal flag structures to normal parabolic geometries, which evidently establishes the claimed equivalence of categories. So let us assume that (p : g --4 M, w) and (p : g --4 M, w) are two normal regular parabolic geometries. Let ({TiM},po : go --4 M,w O) be the underlying infinitesimal flag structure of (Q --4 M, w) and likewise for the other geometry. Assume that tPo : go --4 go is a morphism of infinitesimal flag structures covering f : M --4 M, i.e. a homomorphism of Go-principal bundles such that TtPo is filtration preserving and tPow? = w? for all i = -k, ... , -1. Choose open coverings {Ui : i E I} of M and {l\ : j E J} of M which trivialize the bundles p : g --4 M and p : g --4 M, respectively. Over each Ui we get isomorphisms p-l(Ui ) ~ Ui x P and Pol(Ud ~ Ui x Go, and the natural projection 7r : g --4 go in this picture is just given by the natural projection P --4 Go ~ P/P+ in the second factor. Consequently, there is a Go-equivariant section ai : POl(Ui ) --4 p- 1 (Ui) of the projection 7r, and similarly we get Uj: POl(Uj ) --4 p-l(Uj ). For indices i E I and j E J such that Uij := Ui n f- 1 (Uj ) 1= 0, we can define a map tP ij : P- 1 (Ui j) --4 p-l(f(Uij)) by tPij(a(u) . g) = u(tPo(u)) . g for
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u E pC;I(Uij ) and 9 E P+. Since (1 and jj are Go-equivariant, this is a principal bundle homomorphism and by construction irO M is a holomorphic principal P-bundle over the complex manifold M. (3) (p: g -> M, w) is a complex parabolic geometry of type (G, P) if and only if /1';0,2 = D and /1';1,1 = D, i.e. if and only if the curvature /I'; is of type (2, D). PROOF. (1) and (2): By the Newlander-Nirenberg theorem (see 1.2.10) integrability of an almost complex structure is equivalent to vanishing of the Nijenhuis tensor. By definition, the Nijenhuis tensor of an almost complex manifold is given by N(~, TJ) = [~, TJ]- [J~, JTJ] + J[J~, TJ] + J[~, JTJ]. Hence, N is just the (D,2) component of the Lie bracket of vector fields, which is tensorial. Now let NQ be the Nijenhuis tensor of JQ and let us compute w(NQ(w- 1 (A),w- 1(B))) for A, BEg. Since w is constant on the vector fields w- 1 (A) and w- 1 (B), the definition of the exterior derivative implies that
where we also denote by /I'; the curvature function; see 3.1.8. Again by definition JQ(w-1(A)) = w- 1 (iA), and since the bracket on 9 is complex bilinear, we get [A, B] - riA, iB] + iliA, B] + irA, iB] = D, so
w(NQ{w-1(A), w-1(B))) = -4/1';O,2{A + p, B
+ p).
Hence, vanishing of NQ is equivalent to vanishing of /1';0,2' On the other hand, starting with vector fields ~,TJ E X(M) and choosing local lifts t, r" we have already observed that JQ t is a local lift of J~ and similarly for TJ, and brackets of the lifts are lifts of the brackets. Thus, we conclude that for x E M and u E g with p( u) = x, we have N(~, TJ)(x) = Tup' NQ(t, r,)(u). Hence, integrability of J is equivalent w(NQ) having values in p, whenever lifts of vector fields on M are inserted. Taking into account that /I'; vanishes upon insertion of a vertical vector field, this together with the above implies (1). If /1';0,2 = D, then we know that J and JQ are integrable, so p : g -> M is a holomorphic map which, in addition, is a surjective submersion. By the implicit function theorem, p admits local holomorphic sections. On the other hand, consider the principal right action r : g x P -> g. The derivative of r in a point (u, g) is given by T(u,g)r . (~, TJ) = Turg . ~ + Tgru . TJ, where r g : g -> g is the right action by 9 and ru : P -> g is the right action on u. Equivariancy of w reads as w{u· g) (Tu rg .~) = Ad(g)-l(w(u){~)), and since Ad(g-l) : 9 -> 9 is complex linear, we see that Trg is complex linear for the complex structure JQ. Now ru = ru.goAg-l and thus Tgru = Teru.g o TgAg-l. Since w reproduces the generators of fundamental vector fields, we see that w( u· g) oTeru.g is just the inclusion p -> g, which is complex linear, and TgAg-l, is complex linear, too. Thus, the map r is holomorphic, and hence p : g -> M is a holomorphic principal bundle and the proof of (2) is complete. (3): By (2) we know that g is a complex manifold and p : g -> M is a holomorphic principal bundle if and only if /1';0,2 = D. By construction w(u) is complex linear for any u E g, so w is a smooth g-valued (1, D)-form on g. But then w is holomorphic, if and only if 8w = D i.e. if and only if the (1, I)-part of dMJ vanishes; see 1.2.10. But since the bracket on 9 is complex bilinear, this (1,1)part is described by the (1, I)-part of the curvature function /1';, and the result follows. 0
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3.1.18. The equivalence to underlying structures in the complex case. We continue to work in the setting of 3.1.17 and consider a complex parabolic geometry (p : 9 --+ M, w) of type (G, P). Since w is a holomorphic (I,O)-form, we may identify the associated bundle 9 x p (g/p) with the holomorphic tangent bundle T1,0 M of M, and a complex parabolic geometry gives rise to a filtration of the holomorphic tangent bundle by holomorphic subbundles. The constructions of underlying structures done in 3.1.5 and 3.1.15 carries over to the holomorphic setting without changes. There are also obvious holomorphic versions of (abstract) infinitesimal flag structures and P-frame bundles of degree one, and underlying structures of complex parabolic geometries are by construction of that type. Regularity can be defined and characterized in the holomorphic setting completely parallel to the real case. By definition the Lie algebra differential 0 maps complex multilinear maps to complex multilinear maps. For the codifferential 0*, this property follows either by dualization or from the explicit formula in 3.1.11. The Hodge decomposition on the Lie algebra level from 3.1.11 works for the complex multilinear maps case without any changes. Hence, the concept of normality of parabolic geometries works in the holomorphic category without problems. To get the equivalence of categories between complex normal regular parabolic geometries and underlying structures, one has to use slightly different arguments than in the real case. The problem is that in the proofs of this equivalence in the real case, we frequently used that for a vector bundle homomorphism 4> : E --+ F and a smooth section s E r(F) which has values in the image of 4>, one can find a smooth section 8 E r(E) such that s = 4>(8). Otherwise put, we used that any exact sequence of smooth vector bundle homomorphisms splits, and this is not true in the holomorphic category. Thus, we will use a different approach, which is based on Proposition 3.1.17: THEOREM 3.1.18. Let 9 = g-k EB··· EB gk be a complex Ikl-graded semisimple Lie algebra such that none of the simple ideals of 9 is contained in go, let G be a complex Lie group with Lie algebra g, PeG a parabolic subgroup for the given grading and Go c P the Levi subgroup. Then the equivalences of categories from Theorems 3.1.14 and 3.1.16 restrict to equivalences from the category of complex regular normal parabolic geometries to the category of holomorphic regular infinitesimal flag structures, respectively, the category of holomorphic regular normal P-frame bundles of degree one. PROOF. We have already observed that the structures underlying complex parabolic geometries are automatically holomorphic, so we obtain restrictions of functors as claimed. Moreover, any smooth morphism between complex parabolic geometries is automatically holomorphic, since compatibility of a principal bundle map with the Cartan connections implies complex linearity of its tangent map. Now start with a holomorphic regular infinitesimal flag structure, respectively, regular normal P-frame bundle of degree one over a complex manifold M. The procedures from 3.1.14-3.1.16 lead to a regular normal real parabolic geometry (p: 9 --+ M,w). If we can show that this parabolic geometry is actually complex, then this defines a functor in the opposite direction, and the fact that we have obtained an equivalence of categories follows as in the proof of Theorem 3.1.14. In view of Proposition 3.1.17 this can be done by showing that the curvature /'i, of w has complex bilinear values.
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Now this is a local property and locally we can get analogs of 3.1.14-3.1.16 in the holomorphic category: Let us first assume that ({TiM}, Po : E --+ M,O) is a holomorphic regular infinitesimal flag structure of type (G, P). Let U c M be an open subset such that all the filtration components Ti M as well as the bundle E are holomorphically trivial over U. From a simultaneous holomorphic trivialization of the bundles TiU we obtain holomorphic projections 7ri : TU --+ TiU, while a holomorphic trivialization of Elu provides us with a holomorphic principal connection 'Y on pol(U). As in the proof of Theorem 3.1.13, this gives us an extension of 0 to a holomorphic Cart an connection 0 E Ol,O(pOl(U), 9_ EB 90) on the holomorphic principal bundle POl(U) --+ U. The mechanism of extension functors from Cartan geometries of type (POP, Go) to Cartan geometries of type (G, P) works without changes in the holomorphic setting, since it just uses equivariant extension. Hence, we see that the restriction to U of the regular infinitesimal flag structure can be obtained from a complex parabolic geometry on U. If ({TiM}, Pl : E --+ M,O) is a holomorphic regular normal P-frame bundle of degree one, then the underlying regular infinitesimal flag structure ({TiM}, Po : Eo --+ M, fl.) is holomorphic, too. Restricting to an open subset U C M over which all the bundles Ti M, E and Eo are trivial, we can similarly imitate the first part of the proof of Theorem 3.1.16 locally in the holomorphic category. This leads to a complex parabolic geometry on U, such that the underlying P-frame bundle of degree one is isomorphic to the restriction of ({TiM},Pl : E --+ M,O) to u. Now assume that we have given complex parabolic geometry ({TiU}, U x P, w) with holomorphically trivial Cartan bundle, which is what we have obtained in the above constructions. Then we get a holomorphic trivialization of all filtration components of the adjoint tractor bundle AU and thus of all filtration components of the tangent bundle. In particular, we can identify holomorphic differential forms with values in AU with holomorphic functions with values in appropriate vector spaces, and the bundle maps like 8 and 8* are just given by acting on the values of these maps. Taking this into account, we see that the proof of Proposition 3.1.13 works in the holomorphic category in the case of a trivial Cartan bundle. Thus, we conclude that we always obtain a normal complex parabolic geometry (U x P, wu) of type (G, P) over U, which induces the restriction of the regular infinitesimal flag structure, respectively, the regular normal P-frame bundle of degree one that we have started with. From the last part of the proofs of Theorems 3.1.14 respectively 3.1.16 we conclude that there is a (unique) isomorphism ~u : 9 Iu --+ U x P, which induces the identity on the underlying infinitesimal flag structure, respectively, the underlying P-frame bundle of degree one, such that w = ~uwu. This of course implies that the restriction of the curvature function K, of w to p-l(U) is given by K, = K,u 0 ~u, where K,u is the curvature function of Wu. Since Wu is holomorphic, this has complex bilinear values, and hence K, has complex bilinear values, which completes the proof. 0
3.1.19. Abstract adjoint tractor bundles. To finish this section, we discuss an alternative approach to parabolic geometries in which the principal bundle and the Cart an connection is replaced by an induced vector bundle and a linear connection on that bundle. The passage from the Cartan connection to an induced connection was described for general Cartan geometries in 1.5.7, where general bundles associated to actions of the "big" group G were considered. Here we specialize
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to associated vector bundles. The advantage of the tractor approach is that knowing the normal tractor connection on a tractor bundle one immediately gets access to the fundamental derivative on that tractor bundle and all its sub quotients; see 1.5.8. The starting point for the tractor description of parabolic geometries is to introduce an abstract version of the adjoint tractor bundle and its induced linear connection, which is rather straightforward: Let 9 = g-k EB '" EB gk be a Ikl-graded semisimple Lie algebra. Let M be a smooth manifold of the same dimension as gf'p, and consider a bundle of filtered Lie algebras p : A ~ M modelled on 9 = g-k :::> g-k+l :::> ••• :::> gk. By definition, this means that A ~ M is a vector bundle, whose rank equals the dimension of g, which is endowed with a filtration A = A-k :::> A-k+l :::> ••. :::> Ak by smooth subbundles and with a tensorial bracket { , } : A2 A ~ A, which makes each fiber into a Lie algebra, such that there are local charts with values in 9 that are compatible with the filtration and the bracket. For sections 81,82 E r(A) one can use the pointwise bracket to obtain a smooth section {8}, 82} of A, and this makes the space r(A) into a filtered Lie algebra. DEFINITION 3.1.19. Let p : A ~ M be a bundle of filtered Lie algebras modelled on (g, {gi}) as above. (1) A linear connection V' on A is called nondegenerate if for any point x E M and any tangent vector ~ E TxM there is an index i with Iii::; k and a smooth (local) section 8 E r(Ai) such that V'~8(X) ¢ A~. (2) A nondegenerate linear connection V' on A is called a tractor connection if it is compatible with the bracket on A, i.e. if for each vector field ~ E X(M) and sections 81,82 E r(A) one has V'e({8b82}) = {V'~8b82} + {81, V'~82}' (3) The bundle A is called an (ab8tract) adjoint tractor bundle on M if it admits a tractor connection.
Let G be a Lie group with Lie algebra g, PeG a parabolic subgroup for the given grading, Go c P the Levi subgroup, and (p : 9 ~ M, w) a parabolic geometry of type (G, P). Since the adjoint action of P on 9 is by filtration preserving Lie algebra automorphisms, the associated bundle AM = 9 x p 9 is a bundle of filtered Lie algebras modelled on g. From 1.5.7 we know that the Cartan connection w induces a linear connection V' on the bundle AM ~ M. Compatibility of V' with the bracket { , } was proved in Proposition 1.5.7. To prove nondegeneracy of V', we use the relation to the fundamental derivative D. Given sections 81, 82 E r(AM) and denoting by II : AM ~ TM the natural projection, we have V'n(sl)82 = DSl 82+{8b 82} by Theorem 1.5.8. If 82 E r(AO M), then by naturality of the fundamental derivative also Dsl82 E r(AO M). Now suppose that for a point x E M and all 82 E r(AOM) we have V'n(s1l82(X) E A~M. Then {81 (x), 82 (x)} E A~M for all 82 (x) E A~ M. Since the normalizer of 'p in 9 is 'p, this implies 81(X) E A~M and hence II(81)(x) = 0, and nondegeneracy follows. Hence, V' is a tractor connection on AM in the sense of the above definition, which therefore provides an abstract version of the adjoint tractor bundle and its canonical linear connection. 3.1.20. Adapted frame bundles. In this setting, there is a natural choice of a group G with Lie algebra g. We have noted in 2.2.2 that the automorphism group G = Aut(g) has Lie algebra g. Since Aut(g) is a subgroup of GL(g), the adjoint action is given by conjugation, so for ¢ E Aut(g) and A E g, we get
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Ad(cf>)(ad(A)) = cf> 0 ad(A) 0 cf>-1 = ad(cf>(A)). Hence, the adjoint action of G on 9 is given by applying automorphisms. Choosing P to be the maximal parabolic subgroup NG(gi) (see 3.1.3) we see that P = Autf(9), the group of automorphisms of the filtered algebra (g, {gi}). For the Levi subgroup, we obtain Go = Autgr(g), the automorphism group of the graded Lie algebra 9 = g-k$" ·$gk. For this choice of G, we can naturally associate a principal P-bundle to any abstract adjoint tractor bundle A.
n7=-k
PROPOSITION 3.1.20. Let 9 be a Ikl-graded Lie algebra, put G = Aut(g), P = Autf(9) and Go = Autgr(g). Let 11" : A -+ M be a bundle of filtered Lie algebras modelled on g. Then there is a natural principal P-bundle p : y -+ M such that A = Y Xp g. PROOF. For x E M define Yx to be the set of all isomorphisms 'l/J : 9 -+ Ax of filtered Lie algebras, where Ax denotes the fiber of A at x, and define y to be the disjoint union of all Yx as x ranges over M. Then we can naturally view y as a subset of the linear frame bundle GL(g, A) of the vector bundle A, and the restriction of the projection of the frame bundle defines a map p : Y -+ M. By assumption, the bundle A admits local charts with values in 9 which are compatible with the filtration and the bracket, which exactly means that there are local smooth sections of GL(g,A) -+ M that have values in the subset y. On the other hand, GL(g, A) is by definition a principal bundle with structure group GL(g), so the subgroup P = Aut f (g) acts smoothly and freely on G L(g, A) by composition from the right. By construction, this action preserves the subset y, and for 'l/J}' 'l/J2 E Yx, the composition cf> := 'l/J11 0 'l/J2 is an automorphism of the filtered Lie algebra g, and hence ¢ E P. But then 'l/J2 = 'l/J1 0 cf>, so the action of P is transitive on each of the fibers Yx of y. Together with the existence of local smooth y-valued sections, this implies that y is a submanifold of the bundle GL(g,A) and the projection p : y -+ A is a principal P-bundle. Finally, we get a well-defined smooth map y x 9 -+ A given by ('l/J,A) 1---7 'l/J(A), and since Ad(cf»(A) = cf>(A) for cf> E P and A E g, we see that ('l/J, A) and ('ljJocf>, Ad(cf>-1 )(A)) have the same image under this map. Thus, our mapping factors to a bundle map y Xp 9 -+ A covering the identity on M, which by construction is bijective on each fiber and thus an isomorphism of vector bundles. D
To get similar results for other choices of a group G with Lie algebra g, one proceeds as follows: For any Lie group G with Lie algebra g, the adjoint action defines a covering map from G to a subgroup of GL(g) which lies between Aut(g) and Int(g), the connected component of the identity in Aut(g). The elements of this subgroup can usually be characterized as those automorphisms of g, which preserve an additional structure, for example, an orientation. An analog of the above proposition for this subgroup can be obtained by looking at abstract adjoint tractor bundles endowed with that additional structure, and isomorphism compatible with it. To pass further to a nontrivial covering G of that subgroup, one usually needs, in addition, a structure on M which is similar to a spin structure, and the analog of the proposition is obtained via a fibered product construction. Since such versions of the proposition cannot be derived in a uniform way, we require the existence of an appropriate principal bundle as an additional ingredient: DEFINITION 3.1.20. Let 9 = g-k $ ... $ gk be a Ikl-graded semisimple Lie algebra, M a smooth manifold of the same dimension as g/p and p : A -+ M
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a bundle of filtered Lie algebras modelled on g. Let G be a Lie group with Lie algebra g, and let PeG be a parabolic subgroup for the given Ikl-grading with Levi subgroup Go C P. An adapted frame bundle of type (G, P) for A corresponding to the group G is a principal P-bundle g - M such that A = g x p g, the associated bundle with respect to the adjoint action of P on g. 3.1.21. Abstract tractor bundles. We could now directly derive an equivalent description of parabolic geometries via abstract adjoint tractor bundles and normal tractor connections, but in many situations it is easier to pass to simpler tractor bundles. Typically, these are associated to the standard representation of a classical Lie algebra. Hence, we will work in the setting of general abstract tractor bundles and tractor connections, and the most natural way to do this is via (g, P)-modules (rather than G-modules). Recall that a (g, P)-module is a vector space V endowed with representations of the Lie algebra 9 and the Lie group P which are compatible in the sense that the restriction of the g-representation to 13 is the derivative of the P-representation and for A E g, g E P, and v E V we have (Ad(g) . A) . v = g-1 . A . g . v. For any representation of the group G on V, the restriction to P together with the derivative make V into a (g, P)-module. Now assume that 9 is a Ikl-graded Lie algebra, G is a Lie group with Lie algebra g, and PeG is a parabolic subgroup for the given grading with Levi subgroup Go c P. Further, let M be a smooth manifold of the same dimension as g/p, 11" : A - M a bundle of filtered Lie algebras modelled on 9 and p : g - M an adapted frame bundle for A. For a (g, P)-module V we can then consider the associated bundle T := g Xp V. Since A = g Xp g, the representation of 9 on V induces a bundle map. : A ® T - T, which defines an action of the bundle A of Lie algebras, i.e. {81' 82}. t = 81. (82. t) - 82. (81. t) for 81,82 E Ax and t E 'L" and similarly on the level of smooth sections; compare with 1.5.7. Next, we want to define an analog of tractor connections on such bundles. Since T is associated to g, any point u E g with p(u) = x E M gives rise to a linear isomorphism 1! : V - 'L" which is characterized by 1!(v) = [u, v]. In these terms, the correspondence between smooth sections t E r(T) and equivariant functions t E coo(Q, V) reads as t(p(u)) = 1!(t(u)). Now suppose that V T is a linear connection on T, u Egis a point with p(u) = x and ~ E Tug is a tangent vector at the point u. For a smooth section t E r(T) corresponding to the equivariant map t: g - V, consider 1!-1(Vj:"p.et(x)) - (~ . t)(u) E V. If f : M - lR is a smooth = (f 0 p)t, which immediately implies that changing t to ft, the function, then above element of V changes only by multiplication by f(x). Thus, we conclude that this element depends only on t(x) or equivalently on t(u), whence we get a well-defined linear map CI>(~) : V - V which is characterized by
it
1!-l(Vj:"p.et(x)) - (~. t)(u)
= CI>(~)(t(u));
compare with 1.5.8. DEFINITION 3.1.21. (1) The linear connection V T is called a g-connection if and only if for any u E g and any ~ E Tug, there is an element A E 9 such that CI>(~)(v) = A· v for all v E V.
Of course, any structure on V that is invariant under the action of P leads to a corresponding structure on the bundle T, and if it is, in addition, g-invariant,
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287
this structure is preserved by any g-connection on 7. For concrete choices the Lie algebra g, the group G and the representation V it is usually easy to characterize g-connections as those connections which preserve such induced structures. Note further, that in the case of the adjoint tractor bundle itself, this condition is equivalent to compatibility of the connection with the bracket: For 8],82 E r(A) with corresponding functions 81, 82 : Q --> g, the bracket {8], 82} corresponds to the function u f---> [8] (u), 82( u)]. Hitting this function with a tangent vector ~, bi...---....-
linearity of the bracket immediately implies ~. {8],82} = [~. 8],82] + [81'~· 82]. Thus, from the defining equation for (~) above, one immediately concludes that compatibility of a linear connection '\7 A on A with the bracket is equivalent to the fact that (0([8](U), 82(U)]) = [(~)(8] (u)), 82(U)] + [8](U), (~)(S2(U))]. So this is equivalent to (~) being a derivation of 9 for all ~ E TQ and since any derivation of the semisimple Lie algebra 9 is inner, this is equivalent to '\7 A being a g-connection. Let us now assume that the g-action on V is effective, which just means that none of the simple ideals of 9 acts trivially on V. Then the representation identifies 9 with a Lie subalgebra of g((V) and passing to the associated bundles this means that A is naturally a subbundle of the vector bundle L(7,7) of linear endomorphisms of T. Now any linear connection '\7 7 on 7 induces a linear connection on L(7, 7) characterized by ('\7~'l1)(t) = '\7[('l1(t)) - W('\7[t). If we assume that '\7 7 is a gconnection, then this induced connection preserves the subbundle A c L(7,7): By construction, the linear connection '\7 7 is given by '\7~p.~t(x) = Y«~. i(u) - (~)(i(u))).
For 8 E r(A), we have S;-t(u) = s(u) . t(u), and hitting this with a tangent vector ~, bilinearity of the action implies ~. (s;t)(u) = (~ . .§(u)) . i(u) + 8(U) . (~. t(u)). Using this, we immediately compute that '\7~p.~ (8. t)(x) - 8 • '\7~p.~t(x)
= Y«
(~ • s( u) - [(~), s( u)]) . i( u)),
and since (~) is given by the action of an element of g, the right-hand side is given by the action of a section of A on t. Thus, we get an induced connection '\7 A on A, which is compatible with the algebraic bracket since the induced connection on L(7,7) is compatible with the commutator of endomorphisms. DEFINITION 3.1.21. (2) A g-connection '\7 7 on 7 is called a tractor connection if and only if the induced connection '\7 A on A is nondegenerate and thus a tractor connection. (3) The bundle 7 is called the V -tractor bundle corresponding to A (and Q), if it admits a tractor connection (and thus A is an abstract adjoint tractor bundle).
Let us remark that for concrete choices of g, G and V it is usually easy to directly characterize nondegeneracy of g-connections on 7 in terms of natural filtrations available on T. 3.1.22. Tractor description of parabolic geometries. At this stage it is already visible how to obtain a relation between tractor connections on tractor bundles and Cartan connections on the corresponding adapted frame bundles. If 7 = Q x p V for a (g, P)-module V with effective g-action, and '\7 7 is a g-connection on 7, then for any tangent vector ~ E TQ, the action of (O on V is completely determined by '\7 7 , so by effectivity of the g-action, there is a unique element
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288
w(~) E 9 such that wE [21(9,g).
cI>(~)(v) = w(~)
. v. Clearly, this defines a smooth one-form
THEOREM 3.1.22. Let 9 be a Ikl-graded Lie algebra, G a group with Lie algebra g, PeG a parabolic subgroup for the given grading with Levi-subgroup Go c P, M a manifold of the same dimension as g/p, 7r : A - t M a bundle of filtered Lie algebras modelled on 9 and p : 9 - t M an adapted frame bundle for A of type (G, P). Let V be a (g, P) -module which is effective as a 9 -module and T = 9 x p V the corresponding induced bundle. Then we have:
(1) For any g-connection '\IT on T, the corresponding one-form wE [21(9, g) is P-equivariant and it reproduces the generators of fundamental vector fields. Moreover, w is a Carian connection if and only 'if '\IT is a tractor connection. (2) Conversely, any Cartan connection w E [21 (9, g) induces a tractor connection '\IT on T. (3) /f'\lT is a tractor connection, then its curvature R is given by R(~, 'I1)(t) = ",(~,'11) • t, where", E [22(M,A) is the curvature of the corresponding Carian connection wE [21(9, g). PROOF.
(3.4)
(1) From above we know that the one-form w is characterized by '\Ifp.et(x) = ~((~. i)(u) - w(~) . i(u)).
If ~ = (A(U) for some A E p, then the left-hand side vanishes, while equivariancy of i implies that (A' i(u) = (C(Ai)(U) = -A· i(u), and since ~ is injective, this implies W(A) = A. On the other hand, replacing u by U· g and ~ by Trg . ~ in the above equation, the left-hand side remains unchanged. In the right-hand side, we have U· g(v) = ~(g·v), while equivariancy oft reads as i(u· g) = g-1·i(u) and thus implies Tr g • ~. i = g-1 . (~. i), so this term also produces the same value as before. Thus, by injectivity of~ we are left withw(~).i(u) = g.w(Trg.~).g-1.i(u), and since V is a (g, P)-module, the right-hand side coincides with (Ad(g) . w(Tr g · ~)) . i(u). Effectivity of the g-action then implies w(Trg .~) = Ad(g-1 )(w(~)), i.e. equivariancy of w. From above, we also know that the induced connection '\I A on A corresponds to the same one-form w, so to prove the characterization of w being a Cart an connection, we may assume T = A. Let us first assume that '\I A is a tractor connection, and that w(~) = O. If Tp· ~ =I 0, nOlldegeneracy implies that we can find an index i and a smooth section s E r(Ai) such that '\I:j!p.es(x) rt. Ai (and thus, in particular, is nonzero). But since s E r(Ai), the corresponding function s : 9 - t 9 has values in gi, and thus also ~ . s has values in gi. Hence, that class of '\Its(x) in AlAi coincides with the class of _~-1(w(~) . s(u)), and this being nonzero contradicts w(~) = O. Hence, Tp·~ = 0, so ~ is vertical and thus of the form (A, whence w(~) = A so w(~) = 0 implies ~ = O. Thus, the restriction of w to any tangent space is injective and hence a linear isomorphism for dimensional reasons. Conversely, if w is a Cartan connection, then '\ITp.eS(X) E AO for all s E AO implies [w(~), p] C p, thus w(~) E p and Tp· ~ = 0, so '\I A is nondegenerate. (2) If wE [21(M) is a Cartan connection, then we use formula (3.4) from above to define a linear connection '\IT on T. Equivariancy of w immediately implies that the right-hand side of (3.4) is independent of the choice of u with p(u) = x and since w reproduces the generators of fundamental vector fields, the right-hand
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289
side vanishes if ~ is vertical (compare with the computations made in the proof of (1) and with 1.5.8). Thus, the left-hand side really depends only on x and Tp·~. Moreover, starting with a vector field ~ downstairs and choosing a lift ~ E X(O), (3.4) clearly defines a smooth section V~t E r(T), so we get a well-defined operator X(M) x r(T) -+ r(T). Replacing ~ by -f~ for f E COO(M,lR.), we may replace ~ by (fop)~, and (3.4) then immediatelyimpli-;s that Vr~t = fV[t. On the other hand,
it
replacing t by ft, we get = (f op)i and since ~. (f op)i = (Tp·~· f)i+ (f op)~.i equation (3.4) implies V[ ft = ({ . f)t + fV[ t, so VT is a linear connection. By construction, V T is a g~onnection correspo~ding to the one-form w, so from (1) we know that VT is a tractor connection since W is a Cartan connection. (3) Let VT be the tractor connection corresponding to the Cartan connection W E n 1 (g, g), take vector fields { and !1 on M and choose lifts ~,TJ E X(g). For a section t E r(T) with corresponding function i : g -+ V, the function corresponding to V~t is given by u .---. ("I' i)(u) - w(TJ)' i(u). Since [~,TJ] is a lift of [{,!1], the secti~n V~.17Jt corresponds to the function [~, "I]' i - w([~, "I])' i. On the other hand,
V[Vi t (3.5)
c~r~esponds to the function ~ . ("I' i) - w(~) . ("I' i) - ~ . (w(TJ) . i)
+ w(~) . w(TJ) . i.
Bilinearity of the action implies that the third term in (3.5) may be rewritten as (~ . w(TJ)) . i - w(TJ) . (~ . i). Hence, subtracting from (3.5) the same term with ~ and "I exchanged and the function corresponding to V~.!Zl t from above, the terms in which vector fields differentiate corresponding to the function
i
all cancel out and we are left with R(~, TJ)t - -
(~ . w(TJ) - "I . w(~) - w([~, "I])) . i + w(~) . w(TJ) . t - w(TJ) . w(~) . i.
By definition of the exterior derivative, the first terms give dw(~, "I) ·i, while the last ones add up to [w(~), w(TJ)] . t, so by definition of the curvature we get R({, !1)t = K({, !1) • t as required. D Having given an abstract adjoint tractor bundle 1r : A -+ M, an adapted frame bundle p : g -+ M of type (G, P) and a tractor connection V T on the V-tractor bundle T = g Xp V, we get a tractor connection VA on A and a Cartan connection W E n1 (g, g). For the parabolic geometry (p : g -+ M, w) oftype (G, P) on M, we have A = AM. In particular, we get the filtration TM = T-kM :J ... :J T- 1 M coming from the fact that the Cartan connection W gives the identification T M = g Xp (g/p) ~ A/AU. This filtration can be nicely seen directly from the tractor connection VA. By definition, a tangent vector ~ E TxM lies in T~M if and only if for any lift €E Tug, we have w(€) E gi. By the formula for VA, this implies that E r(Ai+i) for all s E r(Ai). Conversely, if E r(Ai) for all s E r(AU), this formula implies that [W(€) , 1'] c gi for any lift € of~. Inserting the grading element E, we see that this implies w(€) E gi, so we see that this gives a characterization of TiM. It remains to characterize regularity and normality of the parabolic geometry corresponding to a tractor connection V T . Note in particular, that having verified these conditions, the corresponding tractor bundle and its normal tractor connection are uniquely determined by the appropriate underlying geometric structure by Theorems 3.1.14 and 3.1.16. For the characterizations, we need one more observation. In
vts
vts
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290
the definition of the codifferential, we used the Killing form B, which is a nondegenerate invariant bilinear form B on g. Of course, this form gives us an identification A ~ A* and thus a trace map A ® A -+ M x R Using this, we formulate. 3.1.22. Let'V T be a tractor connection on the V -tractor bundle V, where V is a (g, P)-module which is effective as a g-module, and let r;, E n2(M,A) be the two form such that R(e,T/)t = r;,(e,T/). t for all e,T/ E X(M) and t E r(T). Then the parabolic geometry (p : 9 -+ M,w) induced by 'V T is regular if and only if r;,(Ti M, Tj M) C Ai+j+l M and it is normal if and only if the trace over the first and third entry of the trilinear map A ® A ® A -+ A given by (Sl,S2,S3) f--t {Sl,r;,(TI(S2),TI(S3))} - !r;,(TI({Sl,S2}),S3) vanishes. Here TI: A-+ T M denotes the canonical projection. PROPOSITION
T
=9
Xp
PROOF. Since r;, is the curvature of w, the characterization of regularity follows immediately from Corollary 3.1.8. For the normality condition, let us analyze the Kostant codifferential on the Lie algebra level. From 3.1.11 we know that for a decomposable element Z /\ W ® A in A2p+ ® 9 and X E 9 we have
8*(Z /\ W ® A)(X) = -B(W, X)[Z, A]
+ B(Z, X)[W, A]- B([Z, Wl, X)A,
and this depends only on the class of X in g/p. Taking dual bases (with respect to B) {Ze} of p+ and {Xe} of g/p we may rewrite [Z,A] as LeB(Z,Xe)[Ze,A] and similarly for [W, A]. But then the first two terms simply add up to 2[Ze, (Z /\ W ® A)(X, Xi)]. Moreover, interpreting Z /\ W ® A as a bilinear map A2g -+ 9 which vanishes if one of its entries lies in p, we may extend the bases to dual bases of 9 and sum over all elements of these bases without changing the result. This expression then makes sense in the same form for arbitrary bilinear maps of that type and the corresponding bundle map applied to r;, is given by taking the trace over the first and last entry of (S1. S2, S3) f--t 2{Sl' r;,(TI(S2) , TI(S3))}. On the other hand, using invariance of B we may rewrite -B([Z, Wl,X) as -!(B(Z, [W,X])B(W, [Z,X])). Rewriting brackets as above, we get -HB(Z, [Ze,X])B(W,Xe)B(W, [Ze, X])B(Z, Xe)) so the last summand from above corresponds to - Le(Z /\ W ®A)([Ze, X], Xe). Extending Xe by elements of p to a basis of 9 and correspondingly extending Ze to the dual basis, we again may sum over all elements without changing the result. This then makes sense for all bilinear maps of the above type and on the bundle level applied to r;, gives the trace over the first and last entry of -r;,(TI({Sl,S2}),S3). Thus, we see that the claimed expression is exactly the extension of 28*r;, to a section of A* ® A which vanishes on (AO)* ® A, and the result follows. 0 3.2. Structure theory and classification In this section, we use the structure theory of semisimple Lie algebras developed in Chapter 2 to give a complete description of the available Ikl-gradings on a semisimple Lie algebra in terms of the Dynkin diagram, respectively, the Satake diagram. We also describe realizations of the homogeneous space G / P using representation theory. Next, we study representations of the subalgebras p = gO coming from such gradings. Finally, we describe the Hasse diagram associated to the pair (g, p), which is a major tool in the study of parabolic geometries. As a first application, we describe the integral homology and cohomology of generalized flag varieties in terms of the Hasse diagram.
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3.2.1. Complex parabolic subalgebras and complex iki-gradings. The basic structural result on complex semisimple Lie algebras is that they are determined up to isomorphism by the associated root system, which in turn is completely determined by the configuration of a subsystem of simple roots. To get these data one has to make two choices, which, however, do not influence the result. Starting from a complex semisimple Lie algebra g, one first has to choose a Cartan subalgebra ~ c g, i.e. a maximal Abelian subalgebra such that the adjoint action ad(H) : 9 --+ 9 is diagonalizable for any element H E ~; see 2.2.2. Cartan subalgebras exist and any two Cartan subalgebras are conjugate by an inner automorphism of g; see Theorem 2.2.2. Having chosen ~, one gets a set of roots, i.e. the finite set Do of linear functionals a E ~*, such that the root space go = {A E 9 : [H, A] = a(H)A VH E ~} is nonzero. The Lie algebra 9 decomposes as 9 = ~ $ ffioEb. go; see 2.2.2. The basic facts about this root decomposition of 9 are, on one hand, that all of the root spaces go are one-dimensional and, on the other hand, that for a, {3 E Do we have [go, g~l = goH if a + {3 E Do and [go, g~] = 0 if a + {3 (j. Do; see 2.2.4. From the fact that the projections onto eigenspaces of an operator are polynomials in the operator, one concludes that any subalgebra of 9 which contains the Cartan subalgebra ~ is (as a vector space) automatically the direct sum of ~ and some root spaces. The subspace ~o c ~ on which all roots are real is a real form of~. Choosing an ordered basis {H1, ... , Hr·} of ~o one defines a real linear functional : ~o --+ IR to be positive, if for some i = 1, ... , r one has (Hj ) = 0 for all j < i and (Hi ) > O. This induces a total ordering on L(~o, 1R) by defining < '¢ if and only if'¢ - is positive. In particular, one then gets the set Do+ of positive roots and Do is the disjoint union of .Do + and {-a: a E Do +}; see 2.2.5 for details. These two choices can be equivalently encoded as the choice of a Borel subalgebra, i.e. a maximal solvable subalgebra, b $ g. In terms of the Cartan subalgebra ~ and the positive system Do + , the associated Borel subalgebra b is given as b = ~$n+, where n+ := ffioEb.+ go is the sum of all positive root spaces. Obviously, n+ is a nilpotent subalgebra of 9 and [b, b] c n+, so b is solvable. On the other hand, any subalgebra of 9 which strictly contains b has to contain at least one negative root space, say g-o for some a E Do+. But then g-o $ [g-o, go] $ go is a subalgebra isomorphic to the simple Lie algebra 5[(2, C); see 2.2.4. Since any subalgebra of a solvable Lie algebra is solvable, we conclude that b really is a maximal solvable Lie subalgebra of g. The Borel subalgebra b is called the standard Borel subalgebra associated to ~ and Do + C Do. The fact that Cartan subalgebras as well as the choice of positive roots are unique up to conjugation can be nicely rephrased as the fact that any two Borel subalgebras of a complex semisimple Lie algebra are conjugate by an inner automorphism of gj see 2.2.2. DEFINITION
3.2.1. Let 9 be a complex semisimple Lie algebra. A parabolic
subalgebra p of 9 is a Lie subalgebra that contains a Borel subalgebra.
Fixing a choice of a Cartan subalgebra ~ for 9 and a system of positive roots, one obtains the corresponding standard Borel subalgebra b as above. Subalgebras of 9 containing this Borel subalgebra are called standard parabolic subalgebras. Since any Borel subalgebra is conjugate to b, any parabolic subalgebra of 9 is conjugate to a standard one. To understand parabolic subalgebras it therefore suffices to deal with the standard parabolic subalgebras.
292
3. PARABOLIC GEOMETRIES
To give a complete description of all standard parabolic subalgebras of a complex semisimple Lie algebra g, we need one more ingredient from the structure theory: A positive root a E ~ + is called a simple root if it cannot be written as the sum of two positive roots. The set of simple roots is denoted by ~ o. Denoting the simple roots by a l , . . . , an one may write any root a E ~ uniquely as a linear combination a = alai + ... + ara r with integral coefficients al, ... , a r E Z, which are either all ~ 0 or all :::; o. Moreover, if a E ~ + is not simple, then there is a simple root ai such that a - ai E~. Using this we can now state PROPOSITION 3.2.1. Let g be a complex semisimple Lie algebra, I) :::; g a Carlan subalgebra, ~ the corresponding set of roots and ~ 0 the set of simple roots for some choice of a positive subsystem. Then standard parabolic subalgebras P :::; g are in bijective correspondence with subsets E c ~ 0 • Explicitly, we associate to P the subset Ep = {a E ~ 0 : g-a ct. p}. Conversely, the standard parabolic subalgebra Pr: corresponding to a subset E is the sum of the standard Borel subalgebra b and all negative root spaces corresponding to roots which can be written as a linear combination of elements of ~o \ E. PROOF. Let E c ~o be an arbitrary subset and consider Pr: C g. Using that for a,j3 E ~ such that a + j3 E ~ we have [ga,g,8] = ga+,8, one easily verifies that Pr: is a subalgebra of g. But then Pr: is evidently a standard parabolic subalgebra. Conversely, let peg be a standard parabolic subalgebra. By definition b C P and we have noted above that since P is a subalgebra containing I), it must be the direct sum of b and some negative root spaces. Let ~ C ~ + be the set of all a such that g-a C p. For a, j3 E ~ such that a + j3 E ~, we have g-a-.B = [g-a, g-.B] C P and hence a + j3 E~. Conversely, for a E ~ and j3 E ~ + such that a - j3 E ~ +, we have g-(a-,8) = [g-a, g,8], and hence a - j3 E ~. Let us write ~ 0 = {al, ... , a r } and suppose that a E ~ is not simple and decomposes as a = al al + ... + ara r . Then there is a simple root ai such that a - ai E ~ and hence in ~ +, so from above we see that both ai and a - ai lie in ~. Inductively, this shows that for each i such that ai =F 0, we must have ai E ~. Hence, we see that ~ is completely determined by E := ~o \ (~n ~O), and that
P = Pr:·
0
Note that the two obvious choices E = 0 and E = ~ 0 lead to the subalgebra g and the standard Borel subalgebra, respectively. Note also that if E C E' C ~ 0 are two subsets, then Pr:' CPr:. The description of standard parabolic subalgebras peg immediately suggest a relation to Ikl-gradings as introduced in 3.1.2. Namely, having given the subset E C ~ 0 = {al, ... , a r } of simple roots and a root a E ~, we define the E-height htr:(a) of a by
For 0 =F i E Z define gi := EBa:htda)=i go and put go := I) EB EBa:htE(a)=O go. Recall from above that we have a total ordering on the set of roots. In particular, there is a maximal root in this ordering, and we define k to be the E-height of this root. Of course, we then have gi = {O} for Iii> k, so the grading has the form g = g-k EB ... EB gk·
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293
THEOREM 3.2.1. Let 9 be a complex semisimple Lie algebra, ~ C £I a Cartan algebra with corresponding roots Ll, Ll + a set of positive roots and Ll c Ll + the set of simple roots. (1) For any standard parabolic subalgebra IJ ::; 9 corresponding to the subset E C Llo, the decomposition 9 = 9-k EEl··· EEl gk according to E-height makes 9 into a Ikl-graded Lie algebra such that IJ = gO = go EEl··· EEl 9k. Moreover, the subalgebra go C £I is reductive and the dimension of its center 3(90) coincides with the number of elements of E. (2) Conversely, for any Ikl-grading 9 = g-k EEl ... EEl gk, the subalgebra gO is parabolic, and choosing a Cartan subalgebra and positive roots in such a way that gO is a standard parabolic subalgebra IJE, the grading is given by the E -height.
°
PROOF. (1) By the properties of the root decomposition, we have [~, go] = go, [£10,9-0] c ~, and [go, £I.e] = 90+.e for roots a and f3 such that a + f3 E Ll. Since obviously htE (a + (3) = htE (a) + htE (f3) in the latter case, it follows immediately that [gi, gj] c gi+j, so we have defined a grading on g. Next, the subalgebra gO = go EEl •.• EEl gk by definition consists of ~, all positive root spaces, and all negative root spaces corresponding to roots with zero E-height, so £10 = IJE. Thus, it remains to verify that the Lie subalgebra 9_ = g-k EEl .•• EEl g-1 is generated by £1-1, Let a C £1_ be the subalgebra generated by g-1. Then [g-1, a] C a and [go, a] C a. 1£ a =F g_, then there has to be a negative root a such that go i a, and we choose a root a of maximal height (not E-height) with this property. Then there is a simple root ai such that a+ai is a root, and by construction this root has a larger height than a, so gO+Oi C a. But then 90 = [£1-0;, gO+Oi]' which is a contradiction since for ai E E, we have g-oi C g-1 while for ai rt E we have g-o, ego. By construction, the subalgebra go decomposes (as a vector space) into the direct sum of ~ and the root spaces go such that htE(a) = O. Now consider the subspace ~' := {H E ~ : ai(H) = 0 Vai E Llo \ E}. For H E ~' and all a with htE(a) = 0, we have a(H) = 0, and thus ~' C 3(90). Since the simple roots form a basis of ~*, the dimension of ~' coincides with the number of elements of E. On the other hand, for ai E Llo we have the canonical element HOi E ~, which generates [goil g-o.] (see 2.2.4). and these elements form a basis of~. Defining ~" to be the span of the HOi for ai E Llo \ E, the fact that ai(Ho;) = 2 implies ~' n ~" = {O}, and thus ~ = ~' EEl ~" by dimensional reasons. To prove that go is reductive, we have to show that any solvable ideal of go is contained in 3(go); see 2.1.2. Now suppose that I is any ideal in go that is not contained in ~/. Since the decomposition go = ~ EEl EB go is the decomposition into eigenspaces for the adjoint action of ~ on go and the projection onto an eigenspace of an operator is a polynomial in the operator, we conclude that I is the direct sum of a subspace of ~ and some root spaces. 1£ I is not contained in ~/, then it has to contain at least one of the root spaces go, since if it contains an element of ~ \ ~/, then there is a root space on which this element acts nontrivially and that root space then is contained in I. But if I contains go, then it also contains [g-o, go] and thus also g-o = [[£1-0, go], g-o]. Hence, I contains the subalgebra g-o EEl [g-o, go] EEl go ~ 5[(2, C) and cannot be solvable. Thus, we see that any solvable ideal I of go is contained in ~' C 3(go), so go is reductive. Moreover, since
294
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3(go) is a solvable ideal in go, we also get 3(go) = ~', which proves the claim about the dimension of the center. Finally, note that by construction the semisimple part g~S = [go, go] of go is given by ~" EB E9 gao (2) Assume that 9 = g-k EB •.• EB gk is a Ikl-grading. By part (1) of Proposition 3.1.2, there is the grading element E E 3(go) whose adjoint action restricts to multiplication by j on the grading component gj. In particular, ad(E) is diagonalizable, so we may extend CE to a maximal abelian subalgebra of 9 consisting of semisimple elements, thus obtaining a Cartan subalgebra that contains E. Since the eigenvalues of ad(E) are {-k, . .. , k} we, in particular, conclude that E lies in the subspace of this Cartan subalgebra on which all roots are real. Taking a basis for this subspace which starts with E, the resulting positive roots have nonnegative values on E. Conjugating by an appropriate inner automorphism, we may therefore assume that E E ~ and Q(E) ;::: 0 for all Q E ~ +. Then [E,~] = 0 which shows that ~ C go and all positive root spaces lie in gO, so gO egis a standard parabolic subalgebra. Since the element E lies in ~, it acts by a scalar on each root space, so any root space is contained in some grading component. For Q E ~ 0 consider the root space g-aj' By construction g-aj E gi for some i S O. In fact, we must have either i = 0 or i = -1 since otherwise the fact that g_ is generated by g-1 contradicts the fact that Qi cannot be written as a nontrivial sum of positive roots. Consequently, all simple root spaces are contained either in go or in gl, and defining E C ~ 0 to be the set of those simple roots whose root spaces lie in gl, we see that gO is the standard parabolic p~ and the grading is given by the E-height. 0 This result immediately gives us additional information on the structure of Iklgraded Lie algebras. In particular, we see that the situation between the subalgebras g- = g-k EB ••• EB g-1 and p+ = g1 EB •.• EB gk is completely symmetric, since changing the sign of the grading just amounts to using the opposite order for the roots, so g_ and p+ are isomorphic as Lie algebras. Moreover, we may conclude that the filtration of 9 is completely determined by the parabolic subalgebra p = gO. In particular, this implies that given a Lie group G with Lie algebra g, the parabolic subgroups corresponding to the grading as defined in 3.1.3 coincide with the parabolic subgroups used in representation theory. COROLLARY 3.2.1. Let 9 = g-k EB ... EB gk be a Ikl-graded semisimple Lie algebra over ][{ = IR or C. Then we have: (1) For i > 0 we have [gi-1,gl] = gi. In particular, the filtration component gi is the ith power of p+ = g1 and p+ ~ g2 ~ ... ~ gk is the lower central series of p+. (2) If for some i < 0, an element X E gi satisfies [X, Z] = 0 for all Z E g1, then X = O. If no simple ideal of 9 is contained in go, this also holds for i = O. (3) The filtration component g1 = p+ is the nilradical of p = gO. (4) For any Lie group G with Lie algebra g, the parabolic subgroups defined in 3.1.3 are exactly the subgroups which lie between the normalizer Na(P) of P in G and its connected component of the identity.
PROOF. As we have noted above, flipping the sign of the grading leads to an isomorphic Ikl-graded Lie algebra ill the complex case. Since the complexification of a reallkl-graded Lie algebra is a complex semisimple Ikl-graded Lie algebra this
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295
also holds in the real case. From this, (1) and (2) follow immediately, since these are just parts (4) and (5) of Proposition 3.1.2 for the flipped grading. (3) By the grading property, g1 is a nilpotent ideal in p, and thus contained in the nilradical n of p. In the complex case, since the Cartan subalgebra I) is contained in p, we conclude that any ideal in p is the direct sum of a subspace of I) and some root spaces. Assume that n contains a root space go C go. Then n also contains [go, g-o] and [[go, g-a], g-o] = g-o· Hence, n contains a subalgebra isomorphic to sl(2, C), which is a contradiction. Hence, n c I) EB g1. But if an element H E I) lies in n, then it must have trivial bracket with all root spaces contained in go, since otherwise one of these root spaces would be contained in n. But, on the other hand, H must also have trivial bracket with any of the root spaces contained in gl, since otherwise we would get nonzero brackets of arbitrary length, which again contradicts n being nilpotent. Hence, H = 0, which completes the proof in the complex case. For the real case we just have to note that the complexification of the nilradical is a nilpotent ideal in the complexification and thus has to be contained in the nilradical of the complexification. (4) Consider the subgroup P := n~=-k Na(gi) c G, which by definition is contained in Na(P). To complete the proof, it suffices to show that P = Na(P). If 9 E Na(P), then Ad(g)(gl) C Ad(g)(p) = P is a nilpotent ideal, and hence contained in the nilradical gl. For i ~ 2, the filtration component gi is just the ith power of gl by part (1), so Ad(g)(gi) C gi for all i ~ O. For i < 0, we know from part (3) of Proposition 3.1.2 that we may characterize gi as the annihilator with respect to the Killing form of g-i+1. Invariance of the Killing form then implies that Ad(g)(gi) C gi for i < 0, and thus 9 E P. 0
3.2.2. Notation. By Theorem 3.2.1, the classification of complex Ikl-graded Lie algebras (up to conjugation by an inner automorphism of g) reduces to the classification of standard parabolic subalgebras of g, which in turn are determined by subsets E C ~ 0 of simple roots. This immediately suggests the following notation. DEFINITION 3.2.2 (Notation for complex parabolic subalgebras and for complex Ikl-gradings). Let 9 be a complex semisimple Lie algebra endowed with a Cartan subalgebra I) C 9 and a set ~ + of positive roots. Then we denote the standard parabolic subalgebra PE C 9 corresponding to E C ~ 0 as well as the corresponding Ikl-grading by ~>height by representing in the Dynkin diagram of 9 the nodes corresponding to elements of E by a cross instead of a dot. From Theorem 3.2.1 we know that for any given Ikl-grading the subalgebra go C 9 is reductive, so it is the direct sum of a semisimple Lie algebra g~S and the center J(go). From the Dynkin diagram representing the Ikl-grading, we immediately get a complete description of the structure of go. PROPOSITION 3.2.2. Let 9 = g-kEB·· ·EBgk be a complex Ikl-graded Lie algebra. Then the dimension of the center of go coincides with the number of crosses in the diagram describing the Ikl-grading, and the Dynkin diagram of the semisimple part g~S is obtained by removing all crossed nodes and all edges connected to crossed nodes. PROOF. By Theorem 3.2.1, the dimension of 3(go) equals the number of elements of E and hence the number of crosses in the Dynkin diagram representing
296
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the Ikl-grading. In the proof of Theorem 3.2.1, we have seen that g~S is the direct sum of the subspace ~" C ~ spanned by the elements HOt; for ai E ~ 0 \ E and the root spaces gOt corresponding to roots a of E-height zero. A root of E-height zero is by definition an integral linear combination of simple roots contained in ~ 0 \ E, and by definition any such root vanishes on ~' C ~. This immediately implies that ~" C g~S is a Cartan subalgebra, and g~S = ~" EB ffihtdOt)=o gOt is the corresponding root decomposition of g~s. Using the induced order on ~", the positive roots of g~S are exactly the positive roots of 9 of E-height zero, and the corresponding simple roots are exactly the elements of ~ 0 \ E. To determine the Dynkin diagram of g~S, we only have to compute the entries aiJ' = 2}0:.;,O:J}'} of the Cartan matrix; see 2.2.5. The inner products in this expression are induced by the Killing form on g~s. Replacing this by the Killing form of g, we obtain the Cartan integers of g, so we have to compare the two Killing forms. By part (1) of Corollary 2.1.5, g~S decomposes into a sum of simple ideals, and this decomposition is compatible with the root decomposition. Suppose first that ai, aj E ~o \ E are such that go:; and gO:j lie in different simple ideals of g~8. Then the two roots are orthogonal with respect to the Killing form of g~8. On the other hand, [go:;> gO:j 1 = 0, so ai + aj cannot be a root of g. Since ai - aj cannot be a root of 9 either, the two roots must also be orthogonal with respect to the Killing form of g; see part (4) of Proposition 2.2.4. On the other hand, on a simple Lie algebra the Killing form is uniquely determined up to scale by invariance; see part (3) of Corollary 2.1.5. Since the restriction of the Killing form of g, is invariant, too, the numbers aij from above coincide with 0 the Cartan integers of g. ,00~,Ol
3.2.3. Complex III-gradings. Let us start by considering the simplest case of III-gradings. These III-gradings are also of interest because of their relation to Hermitian symmetric spaces, which will be discussed in 3.2.7. First, we clarify how to reduce from the semisimple case to the simple case.
LEMMA 3.2.3. Let 9 = g-l EB go EB gl be a Ill-graded semisimple Lie algebra such that no simple ideal is contained in go, then: (i) 9 is the sum of Ill-graded simple Lie algebras g(j). (ii) The dimension of the center 3(9) coincides with the number of simple factors. (iii) The decomposition of the go-module g-l into irreducible components is ' b y g-l = in (j) gwen '\IIj g-l' PROOF. By part (2) of Proposition 3.1.2, any simple ideal of 9 inherits a 111grading or has to be contained in go and the second case is ruled out by assumption, so (i) follows. Writing the decomposition into simple ideals as 9 = ffi j g(j), we get gi = ffi j g~j) for i = -1,0,1 and also 3(go) = ffi j 3(9~»). To complete the proof, it therefore suffices to show that for a simple Ill-graded Lie algebra g, the center 3(go) has dimension 1 and g-l is an irreducible go-module. Representing the highest root as a linear combination of simple roots, any simple root has a nonzero coefficient. Hence, a Ill-grading on a simple Lie algebra can be only obtained if a single simple root is crossed out. Then dim(3(go)) = 1 by Theorem 3.2.1. Since we deal with a Ill-grading, the subalgebra p+ = gl acts
297
3.2. STRUCTURE THEORY AND CLASSIFICATION g~8
9
root
Dynkin diagram
A e, £ > 1
a1
~
A e, £ > 3
ai, 1 < i ~ (£ + 1)/2
Be, £ ~ 2
a1
~
~
Be-1
Ce, £ ~ 3
ae
o---o- ... ~
Ae-1
De, £ ~ 4
a1
De, £ > 5
an
0---0-",
4, crossing a n -l and an differs only by an automorphism of the Dynkin diagram, so there are up to isomorphism exactly two Ill-graded simple Lie algebras of type Dt with f > 4. For the exceptional Lie algebras, the expressions for the highest roots can be found in Table B.2 in Appendix B. 0
3.2.4. Complex contact gradings. These are a special class of complex 121gradings. From the point of view of parabolic geometries, they are interesting since the corresponding geometries have an underlying complex contact structure. They are also of independent interest due to their relation to quaternionic symmetric spaces, which will be discussed in 3.2.7. A contact grading is a 121-grading 9 = 9-2 6) ... 6) 92 such that the dimension of 9±2 is equal to one and such that the bracket [ , 1 : 9-1 x 9-1 -+ 9-2 is nondegenerate. Explicitly, this means that if X E 9-1 is such that [X, Yl = 0 for all Y E 9-1, then X = O. PROPOSITION 3.2.4. Contact gradings can only exist on simple (and not on general semisimple) Lie algebras. On each complex simple Lie algebra of rank larger than one, there is a unique (up to inner automorphism) contact grading. In Dynkin diagram notation, the complete list of complex contact gradings is given by
9 AI', f?:. 2
E
Bt, f> 3
{a2}
{all at}
Dynkin diagram x-----o- ... --o---x ~
...
C 6) Al
~
C e, f ?:. 2
{ad
x-----o- ... ~
D4
{a2}
~
De, e?:. 5
{a2}
E6
{a6}
E7
{a6}
E8
{ad
F4
{a4}
G
G2
{a2}
G=I$=X
~
90 C2 6) Ae-2
. .
kan defines a diffeomorphism K x
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A x N ---+ G. Since the subgroups A and N are by definition connected, they are contained in P, which implies that kanP = kP E G I P. Hence, we see that K acts transitively on GIP and thus GIP ~ KI(K n P). Applying this in the case G = Int(g), the corresponding subgroup K is compact since Int(g) has trivial center. Since the homogeneous spaces for various choices of a connected group with Lie algebra 9 are diffeomorphic, it follows that G I P is compact for any connected group G. 0 This result gives rise to a first realization of a class of generalized flag varieties. The group P = Nc(p) by definition is the stabilizer of the subspace peg under the adjoint action. Hence, we can identify G I P with the orbit of p under the adjoint action of G. This orbit is the variety of all parabolic subalgebras of g, which are conjugate to p via an automorphism from Ad(G). 3.2.7. Remark: Relation of Ikl-gradings to special symmetric spaces. We now have the necessary background to describe the relation of Ill-gradings to Hermitian symmetric spaces and of contact gradings to quaternionic symmetric spaces. Since this is outside of the main line of this book, we only give a brief discussion. We have already discussed the basics about symmetric spaces in 2.3.13. In particular, we have seen that any symmetric space is homogeneous, and if K I L is symmetric, then there is an involutive automorphism u of the Lie algebra t of K, which has the Lie algebra [ of L as its eigenspace with eigenvalue l. The first topic to discuss here are compact Hermitian symmetric spaces. Here, one looks for a compact Riemannian symmetric space K I L which admits a Kinvariant orthogonal complex structure. In terms of the geometry of homogeneous spaces discussed in Section 1.4, it is easy to describe the algebraic data needed for this. In addition to t and an involutive automorphism u with fixed point set [ and appropriate groups K and L C K, we need a complex structure and a Hermitian inner product on t/[, which both are L-invariant. Now suppose that 9 = g-l EB go EB gl is a complex semisimple Ill-graded Lie algebra, G is a simply connected Lie group with Lie algebra g, and P := Nc (p)o c G is the smallest parabolic subgroup for the given grading. Then X := G I P is simply connected, and from Proposition 3.2.6 we know that denoting by KeG the maximal compact subgroup, the inclusion K '---+ G induces a diffeomorphism KIL ---+ GIP, where L:= KnP. Infinitesimally, this means that t/[ ~ g/p, so this carries a L-invariant (and even P-invariant) complex structure. Choosing any Hermitian inner product for this complex structure and averaging over the compact group L, we obtain an L-invariant Hermitian metric on tiL So it remains to realize [ as the (+ 1)-eigenspace of an involutive automorphism of t. Evidently, the map u : 9 ---+ g, which is the identity on go and minus the identity on g-l EB gl defines an involutive automorphism of 9 and it is easy to see that u(t) C t By definition, [= tnp. From the relation of a compact real form with the root decomposition described in 2.3.1, one easily concludes that en p = en go, which is exactly the required result. Hence, we conclude that, viewed as a homogeneous space of K, the generalized flag manifold G I P constructed from a Ill-grading as described above naturally is a simply connected compact Hermitian symmetric space. The main point in the classification of such spaces is to show that this leads to all irreducible simply connected compact Hermitian symmetric spaces; see [Wo64].
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305
Second, we discuss the relation between complex contact gradings and compact quaternionic symmetric spaces, which are also called Wolf spaces. These are symmetric spaces which admit a compatible quaternionic structure; see 4.1.8 for details on almost quaternionic and quaternionic structures. Suppose that K is a Lie group with Lie algebra e, (1 is an involutive automorphism of e with fixed point sub algebra ( c t and L c K is a subgroup corresponding to L Using the theory developed in Section 1.4, a K -invariant almost quaternionic structure on K I L can be described as follows. One needs a three-dimensional, L-invariant subspace Q c L(e/(, til) which admits a basis of the form {I, J, I 0 J} for linear maps I, J : ell ---+ til such that 12 = J2 = -id and J 0 I = -10 J. The elements I and J are not supposed to be L-invariant, however. Now suppose that 9 is a complex simple Lie algebra endowed with a complex contact grading 9 = 9-2EB" 'EB92 as described in 3.2.4. The identity on 9-2EB90EB92 together with minus the identity on 9-1 EB 91 defines an involutive automorphism (1 of 9. Similarly, as above, there is a compact real form e c 9 which is invariant under (1. For the fixed point subalgebra we obtain ( = t n (9-2 EB 90 EB 92) and we can naturally identify til with the t-invariant subspace t n (9-1 EB 9d. Recall from 3.2.4 that [9-2,92] = C . E, where E E 90 is the grading element for our contact grading. Now 9-2 EB C· E EB 92 evidently is an ideal in 9-2 EB 90 EB 92, so its intersection with I is a three-dimensional ideal q C (, which by compactness has to be isomorphic to su(2) ~ sp(l). The adjoint action embeds q into the set of endomorphisms of t n (9-1 EB 91) and one easily concludes that this subspace has the required properties. As before, we can then take as a group K the maximal compact subgroup of the simply connected Lie group with Lie algebra 9, and one shows that [ c e corresponds to a closed subgroup L C K, which we may take to be connected. Now we can take an arbitrary inner product on t n (9-1 EB 91), which is totally real with respect to any of the complex structures in q and average it over the compact group L. Then this gives rise to a quaternion Kahler metric on K I L whose underlying almost quaternionic structure is given by q. In particular, this almost quaternionic structure has to be quaternionic, so we have found a quaternionic symmetric space. As above, the main step in the classification of such spaces is to show that they are all obtained from these examples; see [Wo65].
3.2.8. Projective realization of complex generalized flag varieties. Next, we discuss the realization of the homogeneous spaces G I P as G-orbits in projectivized representations. In the complex case, one not only obtains one such realization, but the projectivization of any finite-dimensional irreducible representation contains a G-orbit which is isomorphic to a generalized flag manifold. If G is a complex Lie group with Lie algebra 9 and PeG is a parabolic subgroup, then G I P is a (by 3.2.6 compact) complex manifold on which G acts holomorphically. Now assume that G is connected and V is a complex irreducible representation of G. Let us fix a Cartan subalgebra ~ :S 9 with corresponding roots ~ and a subset ~ + of positive roots. By Theorem 2.2.11, there is a highest weight vector Vo E V, which is unique up to scale. Thus, we obtain a well-defined point [vol in the complex projectivization PV, i.e. the space of complex lines in V. Let us denote the corresponding weight., which is also called the highest weight of V, by A:~---+C.
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THEOREM 3.2.8. Let 9 be a complex semisimple Lie algebra, G a connected Lie group with Lie algebra g, V a fin'ite-dimensional holomorphic irreducible representation of G with highest weight A : ~ ~ C, and Vo E V a highest weight vector. Let p ~ 9 be the standard parabolic subalgebra corresponding to the set ~ = {a E ~o: (A,a) =f O} of simple roots ofg, and put P:= Nc(p) C G. Then P coincides with the stabilizer of the point [vol in the complex projectivization PV and we get a biholomorphism between G/P and the orbit G· [vol c PV. PROOF. By Proposition 3.2.5, p is the stabilizer subalgebra of [vol. For 9 and X E P and v E V we obtain Ad(g)(X) . v = g. X . g-1 . v, and hence X . g. Vo
= 9 . Ad(g)(X)
. Vo
=
E P
J.l(X)g· Vo,
for some linear functional J.l : p ~ C, where we have used that Ad(g)(X) E P stabilizes the line spanned by Vo. Applying this to X E ~ we see that 9 . Vo is a weight vector. For X in a positive root space, also X . 9 . Vo is a weight vector of different weight, whence J.l(X) = O. This shows that 9 . Vo is a highest weight vector and thus a multiple of Vo. Thus, P is contained in the isotropy subgroup of [vol. On the other hand, any element in an isotropy subgroup must normalize the corresponding isotropy subalgebra, so we conclude that P coincides with the isotropy subgroup of [vol. Now the mapping 9 f-7 g. [vol = [g. vol is a holomorphic submersion from G onto the orbit G . [vol. Since P is the isotropy subgroup of [vo]' this factors to a holomorphic bijection G/P ~ G· [vol, which is a diffeomorphism and thus a biholomorphism by compactness of G / P. D COROLLARY 3.2.8. Let 9 be a complex semisimple Lie algebra, p ~ 9 a complex parabolic subalgebra, G a connected Lie group with Lie algebra g, and P = Nc(p). Then the generalized flag manifold G / P is a compact K iihler manifold and a projective algebraic variety. PROOF. We may assume that p ~ 9 is a standard parabolic, and we denote by ~ the corresponding subset of simple roots. Let 81l be the sum of all fundamental weights corresponding to elements of~, compare with 3.2.16. Since this is a dominant integral weight, there is a finite-dimensional complex irreducible representation V of 9 with highest weight 8P . In view of 3.2.6 we may assume that G is simply connected, so any representation of 9 integrates to a holomorphic representation of G. From the theorem, we get a biholomorphism of G / P with the (closed) orbit of a highest weight vector in the projectivization PV, so G / P is biholomorphic to a closed submanifold of a complex projective space. Since complex projective spaces are Kahler, we see that G / P is Kahler. Further, G / P is an analytic subvariety of projective space, which is algebraic by Chow's theorem (see [GrHa78, 1.3]). D EXAMPLE 3.2.8. Let us start by discussing the An-case, i.e. G = SL(n + 1, q and the maximal parabolic Pk = Nc(p1:) for ~ = {ad for 1 ~ k ~ n. Denoting by V = cn+1 the standard representation and by {V1, ... , Vn+l} the standard basis of V, we have seen in 3.2.5 that the kth fundamental representation is AkV and a highest weight vector is given by V1 A· .. AVk. By the theorem, G / Pk is biholomorphic to the G-orbit of the point [V1 A··· A Vk] E p(AkV) 9:! Cp(ntl)-1. For arbitrary linearly independent vectors WI, ... ,Wk E V there is an element 9 E G such that
3.2. STRUCTURE THEORY AND CLASSIFICATION
307
gVi = 'Wi for all i = 1, ... , k. Thus, we see that the orbit of [VI /\ ... /\ Vk] is exactly the projectivization of the cone of all decomposable elements in AkV. On the other hand, elementary linear algebra shows that Pk also is the stabilizer in G of the subspace C k C C n generated by VI, ... , Vk. Since G acts transitively on the set of all k--dimensional subspaces of C n , we may identify G j Pk with the Grassmannian Grk(C n ) of all k-dimensional complex subspaces of cn. These two realizations of G j Pk give rise to a holomorphic embedding of Grk (C n ) into p(AkV) ~ Cp(nt')-l, whose image is exactly the projectivization of the cone of decomposable elements. This is the well-known PlUcker embedding. Notice that Proposition 3.2.6 in this case states the well-known fact that the group SU(n + 1) acts transitively on Grk(Cn+I), which leads to the identification of the Grassmannian with the homogeneous space SU(n + 1)jS(U(k) x U(n + 1 - k)). Similarly, denoting by P = NG(p~) with I; = {ail''''' ai.} and 1 ::; i l < i2 < ... < ik ::; n, we see that G j P can be realized as an orbit in an appropriate projective space or as the space of flags of nested complex subspaces of dimensions i l , ... , ik. Thus, we obtain a holomorphic embedding of this partial flag manifold into a projective space, which generalizes the Plucker embedding. These examples are the reason why the homogeneous spaces of semisimple groups by parabolic subgroups are called generalized flag manifolds. For the other classical simple groups everything can be done in a very similar way following the treatment of the Lie algebra case in 3.2.5. In any case, one may realize the generalized flag manifolds corresponding to maximal parabolics as the space of decomposable null elements in the projectivization of some exterior power of the standard representation. Alternatively, they may be realized as the space of all isotropic subspaces of some fixed dimension. This leads to a realization of an isotropic Grassmannian or flag manifold generalizing the Plucker embedding. The only exception occurs in the case of D n , i.e. G = SO(2n, C), where there are two orbits of isotropic n-dimensional subspaces and correspondingly two orbits of isotropic decomposable elements in Anc 2n .
R.EMARK 3.2.8. The results of the theorem can be pushed considerably further by using the fact that any complex semisimple Lie group is an algebraic group defined over C and any finite-dimensional irreducible representation is algebraic. This implies that the resulting actions on projective spaces are algebraic, too. One then shows that the orbits of such an action are affine subvarieties, and the closure of an orbit is obtained by adding orbits of lower dimensions. In particular, this implies that all orbits of lowest possible dimension have to be closed. It turns out, however, that the orbit of the line through the highest weight vector is the unique closed orbit in the projectivization of an irreducible representat.ion, and thus t.he unique orbit. of lowest possible dimension. This is proved using t.he Borel fixed point theorem, which st.ates that if V is a representation of a connected solvable algebraic group B and X is a B-invariant projective subvariety of PV, then X contains a fix point of the B-action. For complex groups this is rather easy to prove: Using Lie's theorem (part (2) of Theorem 2.1.1) one shows that for each 1 ::; k ::; dim(V) one finds a k-dimensional B-invariant subspace Vk of V such that Vk :J Vk-I for all k. Element.ary algebraic geometry shows that if one looks at the minimal i such that X n P(Vi) =f. 0, then this intersection is a finite set of points. Since bot.h Vi and X are B-invariant, so
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is the intersection, and since B is connected, all points in the intersection must be fixed by the B-action. This applies to our situation since it turns out that the Borel subgroup of a complex semisimple group is always connected; see 3.2.19. If we have a closed orbit in the projectivization of a finite-dimensional G-representation, then as in the proof of Corollary 3.2.8 one concludes that it is a projective subvariety. Applying the Borel fixed point theorem we obtain a fixed point of the action of the Borel subgroup, which by definition is the line through a highest weight vector.
3.2.9. Parabolic subalgebras in real semisimple Lie algebras. The description of real Ikl-gradings proceeds via complexification. We start be reviewing the main ingredients of the real structure theory. Given a real semisimple Lie algebra g, one first chooses a Cartan involution () on g; see 2.3.2. This is an involutive automorphism such that the bilinear form Bo(X, Y) := -B(X, (}Y) is positive definite, where B denotes the Killing form of g. The decomposition of 9 into eigenspaces for () is called the Cartan decomposition. To avoid confusion with parabolics, we denote the -l-eigenspace by q, so the Cartan decomposition reads as 9 = eE9 q. Now one looks at (}-stable Cartan subalgebras ~ :::; g, i.e. abelian subalgebras such that O(~) = ~ and such that the complexification ~e is a Cartan subalgebra of ge. Then ~ = (~ n e) E9 (~ n q), and ~ is called maximally noncompact if the dimension of n := I) n q is maximal among all (}-stable Cartan subalgebras. Such Cartan subalgebras can be found by first choosing a maximal abelian subspace n c q, then looking at the centralizer m := Je(n) of n in e, choosing a maximal abelian subspace tern and putting I) := t E9 n, see 2.3.7. Having chosen () and I), one can then look at the root system ~ associated to the Cartan subalgebra I)e :::; ge. Let a be the conjugation of ge with respect to the real form g. Then a induces an involutive automorphism a* : ~ -+ ~; see 2.3.8. A positive subsystem ~ + c ~ is called admissible if for a E ~ + we either have a*a = -a or a*a E ~+. DEFINITION 3.2.9. Let 9 be a real semisimple Lie algebra with complexification ge, () a Cartan involution, I) :::; 9 a (}-stable maximally noncompact Cartan subalgebra, ~ the set of roots for the Cartan subalgebra I)e :::; ge, and ~ + c ~ an admissible positive subsystem. A Lie subalgebra P :::; 9 is called a standard parabolic subalgebra with respect to the choices of I) and ~ + if and only if the complexification Pc is a standard parabolic subalgebra of ge with respect to I)e and ~ + .
To formulate the results on real standard parabolics we need a bit more structure theory. First, we need the restricted root decomposition of g. Consider the abelian subspace n = I:J n q. For A E n we by definition have (}(A) = -A, which easily implies that ad(A) : 9 -+ 9 is symmetric for the inner product Bo. Thus, the family {ad( A) : A En} is simultaneously diagollalizable over lR. The corresponding eigenvalues are given by linear functionals >. : n -+ JR, and the nonzero eigenvalues are called the restricted roots. The set of all restricted roots is denoted by ~r. The eigenspaces are called restricted root spaces and they define the restricted root decomposition of g. In 2.3.6 we have seen that the set ~r C n* is an abstract root system, but it is not reduced in general. Still the notions of positive and simple subsystem pose no problems for ~r' In contrast to the root decomposition in the complex case, there is no general result on the dimension of restricted root spaces.
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Second, we briefly describe how to obtain the Satake diagram of g. Having given (), ~ ::; 9 , ~ and the conjugation a as above, we define ~c := {a : a* a = -a} C ~. By 2.3.8, all roots are real on it EB a c ~«> Restricting the roots to ~ C I{~, the map a* becomes complex conjugation, so ~c = {a E ~ : ala = a}. The condition of admissibility of a positive subsystem ~ + c ~ then reads as a* a E ~ + for all a E ~ + \ ~c. Passing to the associated simple system ~ 0 , it turns out that ~~ := ~ 0 n ~c is a simple system for ~c, and for any a E ~ 0 \ ~~ there is a unique a' E ~ 0 \ ~~ such that a* a - a' is a linear combination of compact roots. Mapping a to 0/ defines an involutive automorphism of ~o \ ~~. The Satake diagram of 9 is then obtained by taking the Dynkin diagram of ~ 0 with elements of ~~ indicated by black dots. and elements of ~o\~~ by white dots o. Moreover, for any element a E ~o \ ~~ such that a' i=- a, one connects a and a' by an arrow; see 2.3.8. Finally, let us describe the relation between the Satake diagram and restricted roots. Since all roots are real on itEB a, the restricted roots are exactly the nonzero restrictions of roots to a C ~. Thus, we obtain a surjective restriction map ~ \~c -+ ~r' Since for a E ~, the restrictions to ~ of a and a*a are conjugate, we see that a*ala = ala. For an admissible choice of ~ + c ~, the image in ~r of ~ + under the restriction map is a positive subsystem. This easily implies that the corresponding simple system ~~ for ~r is the quotient of ~ 0 \ ~~ obtained by identifying each simple root a with a'. Having these ingredients at hand, we can now state the classification of real standard parabolics. THEOREM 3.2.9. Let 9 be a real semisimple Lie algebra, () a Cartan involution with associated Cartan decomposition 9 = e EB q, I) = t EB a ega maximally noncompact (}-stable Cartan subalgebra. Put ~ = ~(gC, ~C), let a* be the involutive automorph'ism of ~ induced by the conjugation with respect to 9 C gc and let ~ + C ~ be a positive subsystem such that for a E ~ + we either have a* a E ~ + or a*a = -a. Then we have: (1) Put m = 3p(a) and let neg be the direct sum of all positive restricted root spaces. Then Po := m EB a EB n is a subalgebra of g, and the standard parabolic subalgebras of 9 are exactly the subalgebras containing Po. (2) Let ~o be the set of simple roots and let ~~ C ~r be the corresponding set of simple restricted roots. Then subsets of ~~ are in bijective correspondence with subsets of ~ 0 that are disjoint to ~~ and stable under the involution induced by a*. On the other hand, the set of all subsets of ~ 0 with these two properties is in bijective correspondence with the set of all standard parabolic subalgebras of g. Explicitly, the parabolic subalgebra corresponding to I: C ~ 0 is the sum of Po and the restricted root spaces for those negative restricted roots which can be written as linear combinations of the simple restricted roots which are outside of the image of I: in ~~. PROOF. (1) By definition m and a EB n are sub algebras of 9 and [m, a] = O. On the other hand, [m, a] = 0 immediately implies that ad(m) maps any restricted root space to itself, so in particular, [m, n] C n. Thus, Po = m EB a EB n is a subalgebra of g. From the proof of Proposition 2.3.5 we know that m EB a is the centralizer of a in g, so Po is the direct sum of all nonnegative restricted root spaces. Now consider the complexification (Pok. By construction, I) C m EB a C Po, so I)c c (Pok· We have noted above that a*ala = ala for all a E ~. If a E ~ +, then this restriction is either zero (so a E ~c) or a positive restricted root. If a E ~c,
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then (grc)a C te (see 2.3.8), so [ := tc EEl EBaEt.c (grc)a C te· Since a maps (grc)Q to (grc)cr*Q (see 2.3.8), we get 0'(1) C [ and hence [ is the complexification of [n g. By construction, .any element of [ commutes with any A E a, so (n gem and hence [ C me. One easily verifies that actually [ = me. If a E Ll + is such that a*a E Ll +, then the a-stable subspace (grc)Q EEl (grc)cr*Q intersects 9 in a subspace of a sum of positive restricted root spaces. Conversely, for a positive restricted root .x, the complexification of the corresponding restricted root space has to be contained in EBQ:Qla=>- (grc)Q' Putting the information together, we conclude that
(Pak = ~e EEl
EB ((grc)Q + (grc)cr*a). aEt.+
Thus, (Pak is exactly the sum of the standard Borel subalgebra and its conjugate, which implies the claim. (2) The standard parabolic subalgebras of 9 are exactly the intersections of 9 with a-stable standard parabolic subalgebras p C ge. As we saw in 3.2.1, parabolic subalgebras of ge are in bijective correspondence with subsets of Ll a, so we only have to describe which subsets correspond to a-stable parabolics. If p is a-stable, then for a E Ll such that (grc)Q C p, we also have (gc)cr*a C p. For a E Ll~ we have a*a = -a, which implies that the subset I; C Ll a corresponding to p is disjoint from Ll~. On the other hand, the involutive permutation a f-+ at on Ll a induced by 0'*, is characterized by the fact that 0'* a - at a linear combination of compact roots. Since I; is disjoint from Ll~, we see that htE(a*a) = htE(a t ), so a E I; implies at E I;. Hence, I; is stable under the involutive permutation induced by 0'*. Conversely, if I; is disjoint from Ll~ and stable under the involutive permutation induced by 0'*, then htE(a) = htE(a*a) holds for all a Ella. But this immediately implies the same statement for all a E Ll, and hence the corresponding parabolic is stable under a. The explicit description of the parabolic subalgebra of 9 associated to I; follows as in (1) above. Finally, the bijection between these subsets of Ll a and all subsets of Ll~ follows immediately from the description of Ll~ as a quotient of Ll a \ Ll~. 0 REMARK 3.2.9. In the complex case, the number of standard parabolic subalgebras depends only on the number of simple roots. This equals the (complex) dimension of a Cartan subalgebra and thus the rank of the Lie algebra. For a real semisimple Lie algebra g, the essential quantity is the number of elements of Ll~, which equals the (real) dimension of a~ This dimension is usually referred to as the real rank of g. Note that the real rank is different for different real forms of a given complex semisimple Lie algebra. For a compact real Lie algebra, the real rank is zero, so there are no nontrivial standard parabolics. On the other hand, for a split real form g, there is by definition a maximally noncom pact Cartan subalgebra ~ on which all roots are real, so ~ = a. Hence, the real rank equals the rank for a split real form, and there are as many standard parabolics as in the complex case. As a final example, we see from 2.3.9 that for p 2:: q the real rank of .5u(p, q) equals q.
The relation between Ikl-gradings and standard parabolics is similar to the complex case:
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PROPOSITION 3.2.9. Let 9 be a real semisimple Lie algebra endowed with a Cartan involution {}, a (}-stable maximally noncompact Cartan subalgebra I) and an admissible positive subsystem A +. Then we have: (1) For a standard parabolic subalgebra P ~ 9 corresponding to a subset E c A~, the E-height determines a Ikl-grading of g, such that P = gO. Here, k is the E-height of the maximal restricted root. (2) Given a Ikl-grading of g, there is an automorphism ¢ E Int(g) such that ¢(gO) is a standard parabolic subalgebra of g. Denoting by E c A~ the corresponding subset, the given grading on 9 corresponds to the grading by E-height under ¢. PROOF. (1) is clear. (2) A Ikl-grading of 9 gives rise to a Ikl-grading of the complexification gl(:. The grading element E E 9 also is the grading element for gl(:. Since the map ad(E) is diagonalizable on gl(:, we can find a Cartan subalgebra 6c gl(: containing E. Construct a compact real form of gl(: from 6 as in 2.3.1 and let l' be the corresponding conjugation of gl(:. Since all eigenvalues of ad(E) are real, we get 1'(E) = -E. Denoting by (1 the conjugation of gl(: with respect to g, we obtain (11'(11'(E) = E. Hence, 'lj; := ((11'(11')1/4 (see 2.3.2) also satisfies 'lj;(E) = E. By the proofs of Lemma 2.3.2 and Theorem 2.3.2, 0 := 'lj;1''lj;-1 restricts to a Cartan involution on g, and by construction, O(E) = -E. By part (2) of Theorem 2.3.2, any two Cartan involutions are conjugate by an inner automorphism. Hence, we find ¢1 E Int(g) such that ¢1(E) lies in the (-l)--eigenspace q of {}. Choosing a maximal abelian subspace ii c q containing ¢1(E), we know from Theorem 2.3.6 that there is ¢2 E Int(g) such that ¢2(E) E a = I) n q. By Proposition 2.3.6 any two choices of positive restricted roots are conjugate, so we finally get ¢ E Int(g) such that ¢(E) E a and >"(¢(E)) ;::: 0 for all positive restricted roots >... Since ad(E) is diagonalizable on 9 with real eigenvalues, the same holds for ad(¢(E)) = ¢ 0 ad(E) 0 ¢-1 and obviously ¢ maps gO to the sum of all eigenspaces of ad(¢(E)) corresponding to nonnegative eigenvalues. But this exactly means that ¢(gO) contains the minimal standard parabolic Po and hence is a standard parabolic subalgebra. The relation between the original grading and the E-height is then evident. 0 REMARK 3.2.9. As in the complex case, one may directly obtain information about the subalgebra go directly from the Satake diagram describing the parabolic. The (real) dimension of the center of go again equals the number of crossed nodes in the Satake diagram. Parallel to what we have done in the complex case in Proposition 3.2.2, one may also show that in the real case the Satake diagram of the semisimple part g08 of go is obtained by erasing all crossed nodes as well as all edges and arrows connecting to these nodes from the Satake diagram describing the parabolic subalgebra. Details about this can be found in [Kane93). 3.2.10. Notation for real parabolics and examples. We will use the obvious analog of the notation for complex parabolics in the real case. To describe a standard parabolic subalgebra p ~ g, we consider the Satake diagram of 9 and denote all the simple roots corresponding to elements of the subset E c A 0 by crosses. Since we know that E is disjoint from A e , we can recover the original Satake diagram by replacing all crosses by white dots. In these terms, the classification of real parabolics can be rephrased in terms of replacing white dots in a
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Satake diagram by crosses. The only rule one has to take into account is that two roots joined by an arrow either have to be both crossed or both uncrossed. EXAMPLE 3.2.10. We now go through the most important examples following the discussion of the complex case in 3.2.2-3.2.5.
Real Ill-gradings. To get a complete list of real Ill-gradings, we only have to look at the list of complex Ill-gradings in 3.2.3 and check which of the Ill-gradings are present for the various real forms. Since a complex Ill-grading has only one crossed root, the Satake diagram of a real form has to have a white dot, which is not connected to any other dot by an arrow, at the right place in order to admit a Ill-grading. The quickest way to see the result is to compare the table of complex Ill-gradings in 3.2.3 with the Satake diagrams in Table B.4 in Appendix B. In the An-case, crossing any simple root leads to a Ill-grading, so for a given real form, the available Ill-gradings correspond exactly to the white roots that are not connected to another root by an arrow. For the split real form s[(n, 1R), the real Ill-gradings are in bijective correspondence with complex Ill-gradings of sl(n, C). For su(p, q) with p > q, there are no appropriate roots by example (3) of 2.3.9, so this real simple Lie algebra does not admit any Ill-grading. Similarly, example (4) of 2.3.9 implies that there is a unique Ill-grading on the real simple Lie algebra su(p,p) for p 2: 2. Example (5) of 2.3.9 shows that there are n - 1 Ill-gradings on the real form s[(n,lHI) of sl(2n,C), and by Theorem 2.3.11 the above exhaust the list of noncompact real forms of An. For B n , the unique complex Ill-grading corresponds to the first simple root, and going through the tables of noncompact real forms, one sees that this root is white and not connected to another root by an arrow for any of the noncompact real forms. Thus any of the algebras so(p, q) with p, q =f 0 and p + q odd admits a unique real Ill-grading. In the Cn-case, there is also only one complex Ill-grading which corresponds to the last root in the Dynkin diagram. This time one sees from the tables of noncompact real forms that Ill-gradings are only available on the split real form sp(2n,lR) and on the real form sp(n,n) of sp(4n,C). In both cases, this grading is unique up to isomorphism. Finally, for the Dn case, there are three complex Ill-gradings corresponding to the first and the last two roots. Here, the tables imply that the real Ill-grading corresponding to the first root is available for all the noncompact real forms except so*(2n), while the gradings for the last two roots are available on the split real form so(p,p) and one of them is available on the real form so*(4n) of so(4n, C). For exceptional algebras, a unique Ill-grading is available on the split real forms of E6 and E7 as well as on the noncompact real forms EIV of E6 and EVIl of E 7. Real contact gradings. Similarly, as for Ill-gradings above, we can analyze the contact gradings on real simple Lie algebras, which are by definition 121-gradings such that dim(g±2) = 1 and the bracket g-l x g-1 -> g-2 is nondegenerate. As above, we only have to go through the list of complex contact gradings in 3.2.4 and check to which real form they descend using the Satake diagrams in Table B.4 of Appendix B. The result of this is that except sl(n, IHI), so(n - 1,1), sp(p, q), and the real forms EIV of E6 and FII of F4 , any noncompact noncomplex real simple Lie algebra admits a unique real contact grading.
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Real parabolics as stabilizers of lines and flags. The descriptions of complex parabolics as stabilizers of lines and flags from 3.2.5 have analogs in the real case. Suppose that the inclusion 9 ----) 91(: is chosen in such a way that a O-stable maximally noncom pact Cartan subalgebra of 9 complexifies to the standard Cartan subalgebra of 91(: and the usual positive system is admissible. Then the complexification of a standard parabolic in 9 is the complex standard parabolic in 91(: corresponding to the same set of simple roots, so one may directly carryover the descriptions from 3.2.5 to the real case. In some examples, we have used a different positive system, but the changes caused by this are easy to analyze. We outline this for the real forms of s[(n, q. For the split real form 9 = s[( n, lR) of s[( n, q the real diagonal matrices form a maximally noncompact O-stable Cartan subalgebra, which complexifies to the standard Cartan subalgebra of 91(:. Moreover, the condition of admissibility of a positive subsystem is vacuous in the case of a split form, so we may use the standard positive and simple roots. Hence, we may conclude directly from 3.2.5 that the standard parabolic subalgebra of 9 corresponding to the set {Qil' ... , Qik} of simple roots with 1 ~ i1 < ... < ik < n is the intersection of 9 with the stabilizer of the flag eil C ... C ik c Since 9 itself is the stabilizer in 91(: of the real subspace lRn c we conclude that this parabolic is the stabilizer in 9 of the flag lRil C ... C lRik C lRn. In example (2) of 2.3.4 and example (3) of 2.3.9, we have realized the Lie algebra 9 := su(p, q) for p > q as the subalgebra of 91(: = s[(p+q, q consisting of all
en,
e
matrices of the block form
en.
B C) ( A -A" E D -E" _Co F
,with A E 9[(q, q, C and E arbitrary
complex q x (p - q)-matrices, Band D in u( q) and F E u(p - q). There we have also verified that the tracefree diagonal matrices contained in 9 form a O-stable maximally noncompact Cartan subalgebra whose complexification is the standard Cartan subalgebra of 91(:. However, one has to use a nonstandard set of positive roots, for which the simple roots are given by el - e2,'" ,eq-l - e q , e q - e2q+1, e2q+1 - e2q+2, . .. , e n -1 - en, -e2q + en, -e2q-1 + e2q,. .. , -eq+1 + e q+2. Looking at the Satake diagram, we see that there is a basic set of 121-gradings on 9, which correspond to crossing the ith and the (p + q - i)th nodes in the Satake diagram (which are connected by an arrow). These gradings exist for i = 1, ... , q, and for i = 1 we obtain the real contact grading from the example above. As in 3.2.5, one easily verifies that the standard parabolic subalgebra of 91(: corresponding to the ith simple root is the stabilizer of the subspace generated by the elements VI, ... , Vi of the standard basis, while for the (p + q - i)th simple root one obtains the stabilizer of the subspace generated by VI, ••. , V q , Vq+i+ 1, ... , V n . Thus, we see that the ith basic standard parabolic in 9 is the stabilizer in 9 of the flag formed by these two subspaces. The explicit description of the Hermitian form used for this realization of 9 can be found in 2.3.4. In particular, the first subspace in the flag is isotropic and the second one is the orthocomplement of the first. Since a skew-Hermitian matrix that stabilizes a subspace automatically stabilizes the orthogonal complement, we conclude that the ith basic parabolic in 9 is the stabilizer in 9 of the isotropic subspace e i c Thus, the standard parabolic subalgebras in su(p, q) for p > q are exactly the stabilizers of isotropic flags in p +q , and the existence of parabolics (depending on the signature (p, q)) corresponds exactly to the existence of isotropic subspaces of the appropriate dimensions.
en.
e
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For the real form 9 := su(p,p) of sl(2p, q the situation is completely parallel. The parabolics in 9 are exactly the stabilizers of isotropic flags in C2P • The contact grading corresponds to the stabilizer of a null line, while the unique Ill-grading corresponds to the stabilizer of the maximal isotropic subspace CP c C2P. Finally, let us consider the real form 9 := s((n, JH[) sitting inside gc = sl(2n, q as in example (5) of 2.3.9. Thus, we view 9 as the subalgebra of matrices of the form (_AB ~) such that A, BE g((n, q and the real part of the trace of A vanishes. Again, the diagonal matrices contained in 9 form a (}-stable maximally noncompact Cartan subalgebra whose complexification is the standard Cartan subalgebra of gc. As before, one has to use an ordering of the roots that is different from the usual one, and which leads to the simple roots -el +en+l, el -en+2, -e2+en+2, e2-en+3, ... , en-l - e2n, -en + e2n. These roots are ordered in such a way that one obtains the usual Dynkin diagram. Looking at the Satake diagram, we see only the nodes corresponding to the roots ei - en+i+l may be crossed, and we obtain n - 1 basic parabolics. Each of them gives rise to a Ill-grading on 9. As in 3.2.5, one easily verifies that the standard parabolic of gc corresponding to the simple root ei -en+i+l is the stabilizer of the subspace generated by the elements VI, ... , Vi, Vn+1. ... ,Vn+i of the standard basis of c2n. Thus, the ith basic parabolic in 9 is the stabilizer of this subspace in g. Looking at the description of the isomorphism JH[n - t c2n used to obtain the embedding of sl(n, JH[) into sl(2n, q, we see that this is the quaternionic subspace JH[i C JH[n. Hence, the standard parabolic subalgebras in s(( n, JH[) are exactly the stabilizers of the standard quaternionic flags JH[il C ... C JH[ie C JH[n for 1 ~ il < i2 < ... < it < n. REMARK 3.2.10. Let 9 be a real semisimple Lie algebra and V a finitedimensional complex irreducible representation of g. Then the representation extends to the complexification gc of g, and of course the stabilizers in 9 of the line through a highest weight vector coincides with the intersection of 9 with the stabilizer in gc of that line. According to 3.2.5 we obtain the intersection of 9 with a parabolic subalgebra of gc. This intersection is, however, not a parabolic subalgebra of 9 in general.
3.2.11. Real generalized flag varieties. We switch to the discussion of the homogeneous spaces G / P in the real case, which are again called generalized flag varieties. The considerations on the dependence of the homogeneous space G / P on the choice of the groups G and P (corresponding to a fixed choice of a Ikl-graded semisimple Lie algebra g) carryover to the real case without big changes. The main difference is that even for connected G, the normalizer Nc(p) is not connected in many cases of interest. As in the complex case, this will only replace G/Nc(p) by a (usually finite) covering, so it causes only a minor change. To prove compactness of G / P, we can proceed very similarly as in the complex case. Fixing a Cartan involution {} on 9 and a connected subgroup G with Lie algebra g, there is a unique involutive automorphism e with derivative e. The fixed point group K of e is a closed subgroup of G which is compact if Z(G) is finite. In that case, K is a maximal compact subgroup of G. As in the proof of Proposition 3.2.6, we can next use the global Iwasawa decomposition G S:: K x A x N. Part (1) of Theorem 3.2.9 shows that the Lie subalgebra a EB n is contained in p, so A x N is contained in any parabolic subgroup corresponding to the grading. Hence, K
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acts transitively on G I P, and, provided that Z (G) is finite, compactness follows as in Proposition 3.2.6. This also leads to an interpretation of G I P as the variety of parabolic subalgebras of 9 which are conjugate to P via an element of Ad( G). Now we switch to the interpretation of generalized flag varieties as orbits in projectivized representations. In 3.2.10 we have seen that real parabolic sub algebras can be realized as stabilizers of lines and flags, but the stabilizer of a highest weight space in a general complex irreducible representation of a real semisimple Lie algebra is not a parabolic subalgebra in general. Consequently, we may still realize the corresponding homogeneous space GI P as orbits in certain projectivized representations, but this does not give as much information about orbits in general irreducible representations as in the complex case. Given a real semisimple Lie algebra 9 and a standard parabolic subalgebra P ~ 9, we can consider the complexification Pc, which is a standard parabolic subalgebra in the complex semisimple Lie algebra 9c. Let us denote by I: the set of simple roots of 9c corresponding to this standard parabolic, and consider a dominant integral weight A for 9c, such that for a simple root a we have (A, a) #- 0 if and only if a E I:. For example, we may take the sum of all fundamental weights corresponding to elements of I:. Consider a finite-dimensional representation V of 9c with highest weight A and a highest weight vector Vo E V. From 3.2.5 we know that the stabilizer in 9c of the line through Vo is exactly Pc, which immediately implies that the stabilizer of this line in 9 is p. As in 3.2.8 above, we see that the isotropy subgroup of the line through this point in any connected group G to which the representation integrates coincides with P = Nc(p). Consequently, the G-orbit of the point [vol E P(V) is diffeomorphic to GIP, and we get a realization as a compact submanifold of a complex projective space. Of course, if the representation V is of real type, i.e. if it is the complexification of a real representation W, then we can choose the highest weight vector in W. Then the G-orbit of [vol is contained in the real projectivization of W, so we obtain a realization as a compact submanifold in some real projective space in this case. Let us discuss this for the An-series, where we have looked at the Lie algebraic counterpart in 3.2.10. For the split form s[(n + 1, JR) the parabolics look exactly the same as for s[( n + 1, 0 we know from 3.2.8 that the available parabolic subalgebras corresponds to subsets of the first q simple roots, and with the ith root, always the (n + 1 - i)th root has to be crossed. Looking at the basic parabolics (with two crossed nodes) we see that the procedure above leads to a realization of G I P as the orbit of a highest weight line in the projectivization of the highest weight component in Aicn+l ® An+l-icn+l. In terms of flags, this means that P is the stabilizers of a flag consisting of an idimensional isotropic subspace sitting inside its orthogonal complement. Thus, the orthogonal complement is superfluous, and P is the stabilizer of the i-dimensional isotropic subspace spanned by the first i vectors in the standard basis. Otherwise put, P is the stabilizer of the highest weight line in the projectivization of AiCn+l, and G I P can be realized as the orbit of this point. Similar realizations can be constructed for more general parabolics. Let P be the parabolic subgroup corresponding to I: = {ah, ... , ai€, an+l-il' ... ,an+l-ie}
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with 1 ::; il < ... < if. ::; q. Elementary linear algebra shows that the group G = SU(p, q) acts transitively on all space ofisotropic flags in C p +q • Using this one shows as above that G / P is the manifold of all flags VI C ... C Vi! C C p +q where each Vi is isotropic and has dimension ij. A particularly important example of this situation is the case of the maximal parabolic corresponding to E = {aI, an}. In this case, G / P is the space of all isotropic lines in C p+q, i.e. a quadric in Cpp+q-l. Hence, G / P is a real hypersurface in the complex manifold Cpp+q-l, which indicates the relation of this generalized flag manifold to CR-geometry. Finally, in the case of the real forms s[(m, JHl) of s[(2m, q we can use the group G = SL(m,JHl). As above, one shows that the parabolic subgroups P of G can all be realized as the stabilizers of the standard quaternionic flags JHlil C ... C JHlit C JH[m for 1 ::; il < ... < if. < m. Hence, the homogeneous spaces G / P are the quaternionic flag manifolds. As before, we also get embeddings of flag manifolds into the projectivizations of appropriate complex representations of S L( m, JHl). 3.2.12. Representations of p. From 1.5.5 we know that any representation of P gives rise to a natural bundle on the category of parabolic geometries of type (G, P), so representations of the parabolic subalgebra p are of central interest in the theory. The Lie algebra p is rather complicated, so there is little hope to get a complete picture for general representations of p. Some basic properties are, however, easy to prove. Moreover, there is a complete description of completely reducible representations. There is a standing assumption that we will have to make on representations of p, which comes from the representation theory of 90. Since this subalgebra is only reductive and not semisimple, finite-dimensional representations of 90 are not automatically completely reducible. Indeed, a finite-dimensional representation W of 90 is completely reducible if and only if the center 3(90) acts diagonalizably on W: Since 3(90) acts by a character on any irreducible representation of 90 the necessity of this condition is evident. Conversely, if 3(90) acts diagonalizably, then any eigenspace for this action is invariant under 9gs • Thus, by Theorem 2.1.6 each of these eigenspaces splits into a direct sum of irreducible representations of 9g s and this gives a splitting of W into a direct sum of irreducible 90-modules. Studying representations of p, we will therefore always assume that 3(90) acts diagonalizably. PROPOSITION 3.2.12. Let 9 = 9-k EB··· EB 9k be a JkJ-graded semisimple Lie algebra, p = 90 the corresponding parabolic subalgebra and E E 3(90) the grading element. (1) Any finite-dimensional completely reducible representation W of p is obtained by trivially extending a completely reducible representation of 90 to p. Moreover, E acts by a scalar on each irreducible component of W. (2) Let V be a finite-dimensional representation of p such that 3(90) acts diagonalizably. Then V admits a p-invariant filtration V = VO :J Vl :J ... :J V N :J V N +1 = {O} such that each of the quotients Vi /Vi+l is completely reducible. PROOF. (1) It suffices to consider the case that W is irreducible, and we first assume that W is a complex representation. Then the action of E on W must have at least one eigenvalue. Let So be an eigenvalue with maximal real part and let WSQ be the corresponding eigenspace. Since E E 3(90), the subspace WSQ C W is 90invariant. For wE W SQ and X E 9j C P the equation E·X·w = [E,X)·w+X·E·w implies that X . w is an eigenvector corresponding to the eigenvalue So + j, so
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x .w
= 0 for j > 0 and X . w E Wso for j = O. Thus, Wso c W is p-invariant and by irreducibility Wso = W. This completes the proof in the complex case. For a real irreducible representation W of p which is not complex, the complexification We is irreducible, which immediately implies that E acts by a (real) scalar on Wand the subalgebra p+ = g1 EB ... EB gk acts trivially. (2) Define VN := {v E V: Z·v = 0 VZ E p+}, where as usual p+ = g1 EB···EB gk. For A E P and Z E p+ we have [Z, A] E p+, and since Z·A·v = A·Z·v+[Z, AJ·v we see that VN is a p-invariant subspace of V. Since 3(go) acts diagonalizably on V, the same is true on the invariant subspace VN. In particular, V N is completely reducible as a go-representation. Since p+ acts trivially on VN the decomposition into go-irreducibles is a decomposition into p-irreducibles, so VN is a completely reducible p-module. Next for i = N, N - 1, ... we inductively define
V i - 1 := {v E V: Z· v E Vi
VZ E p+}.
As before, we conclude inductively that each Vi is a p-invariant subspace of V. By construction, p+ acts trivially on the quotient Vi jVi+1. Moreover, 3(go) acts diagonalizably on Vi and Vi+l so it also acts diagonalizably on the quotient. As before, this implies that Vi jVi+l is a completely reducible p-module. Hence, it remains to show that Vi = V for sufficiently small i. Since 3(go) acts diagonalizably on V it suffices to show that any eigenvector for the action of the grading element E lies in some Vi. But if E . v = sv and Z E gj, then E· Z . v = (s + j)Z . v. If e E N is such that s + e is the maximal eigenvalue of E on V of the form s + n with n E N, then by construction v E VN-f. Thus, we conclude that Vi = V for an appropriate i and we can change N in such a way that the largest i with this property is O. 0 We next want to describe complex irreducible representations of p in terms of highest weights. Since such representations extend to the complexification, we may restrict to the case of complex Ikl-graded Lie algebras. Let us first briefly recall the theory for the complex semisimple Lie algebra g. Let ~ ::; 9 be a Cartan subalgebra, ~ the corresponding set of roots, and ~ + a choice of positive subsystem. We will later assume that these choices have been made in such a way that p is a standard parabolic subalgebra, so ~ and all positive root spaces are contained in p. On any finite-dimensional complex representation V of g, the Cartan subalgebra ~ acts diagonalizably. The corresponding eigenvalues A E ~* are called the weights of V and the eigenspaces are called weight spaces. A highest weight vector in V is an element of a weight space which is annihilated by the action of all elements of positive root spaces. Such vectors exist in any finite dimensional complex representation and in irreducible representations they are unique up to scale; see 2.2.11. The highest weight of a finite-dimensional irreducible representation V is the weight of its highest weight vectors. These highest weights are always dominant and algebraically integral, which means that for the inner product induced by the Killing form the expression 2(~::/ is a nonnegative integer for each simple root Q. The theorem of the highest weight (Theorem 2.2.12) says that there is a bijective correspondence between dominant integral weights and isomorphism classes of finite-dimensional complex irreducible representations of g. There is a useful reformulation of the condition of being dominant and integral. Namely, if Q1, ... , Qn are the simple roots of g, one defines the fundamental weights
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,An of 9 by 2((Ai,Oj» = 8i )·. Then any weight can be written as a linear comO!;J ,OJ bination of the fundamental weights. Dominant integral weights are exactly those, for which all coefficients in this expansion are nonnegative integers. The Dynkin diagram notation for weights and representations is then obtained by writing the coefficient of Ai in this expansion over the node of the Dynkin diagram of 9 that corresponds to the simple root ai. Complex irreducible representations of p can be dealt with in a very similar way. By part (1) of the proposition, these coincide with complex irreducible representations of go, which in turn are given by irreducible representations of the semisimple part g~S and linear functionals on the center 3(go). Assuming that p is a standard parabolic, we obtain the corresponding subset ~ = {a E ~ 0 : go C gl} of simple roots. From 3.2.2 we know we can naturally split the Cartan algebra ~ C 9 as ~ = ~' ED ~", with ~' := {H E ~ : a(H) = 0 'Va E ~o \~} and ~" the span of the elements Ho with a E ~o \~. Then ~' = 3(go), while ~" is a Cart an subalgebra for g~s. Hence, complex irreducible representations of go are again in bijective correspondence with a set of functionals on ~, but the dominance and integrality conditions refer only to the restriction to ~". In analogy to the usual notions we define a weight A : ~ ~ C to be p-dominant, respectively, p-algebraically integral if 2(~:~ is real and nonnegative, respectively, an integer for all a E ~ 0 \~. Thus, we obtain )'1, ...
COROLLARY 3.2.12. Let p :::; 9 be a standard parabolic subalgebra in a complex semisimple Lie algebra. Then isomorphism classes of finite-dimensional complex irreducible representations of p are in bijective correspondence with weights A : ~ ~ C which are p-dominant and p-algebraically integral.
The condition of p-dominance and integrality can again be rephrased in terms of fundamental weights as the requirement that the coefficients of all fundamental weights corresponding to simple roots not contained in ~ must be nonnegative integers. In the Dynkin diagram notation this means that the coefficients over all uncrossed nodes are nonnegative integers. As we have noted in the propostion, the grading element acts by a scalar on each irreducible representation. Since this is often needed in applications, let us remark at this point that there is a simple way to compute this scalar. Since the simple roots form a basis for ~*, we can expand any weight in terms of the simple roots. Having determined this expansion, the action of E on a highest weight vector (and hence on all of W) is the sum of the coefficients of the crossed roots. To convert from the expansion in terms of fundamental weights to the one in terms of simple roots, we proceed as follows. Suppose that the fundamental weight Ai can be written as Lk bkiak. Then the defining property of Ai reads as
8. - 2(Ai' aj) _ " b ') -
( aj,aj ) -
L..k h
2(Ok,Oj) (0· 0·) . J,
J
But the last term is just the coefficient ajk of the Cartan matrix of g, so the numbers bki form the ith column of the inverse of the Cartan matrix of g. Hence, the action of E on W can be computed by adding those rows of the inverse of the Cartan matrix which correspond to crossed roots, and pairing the result with the vector of coefficients with respect to the fundamental weights. The inverse Cartan matrices for all complex simple Lie algebras can be found in Table B.4 in Appendix B.
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In the realm of Lie algebra representations, there is no restriction on the coefficients over the crossed nodes. However, if one wants the representation to integrate to at least one parabolic subgroup, then the coefficients over the crossed nodes have to be integers. To see this, note that for a simple root O'.j, we have the corresponding subalgebra SOI.j := g-OI.j EB [g-OI.j' gOl.J EB gOl.j' which is isomorphic to s[(2, C); see 2.2.4. Under this identification, the element HOI.i corresponds to the matrix H = ~d. Since 8L(2,C) is simply connected, the inclusion SOI.j --+ 9 induces a Lie group homomorphism 8L(2, C) --+ G for any group G with Lie algebra g. Now obviously exp(21riH) = id in 8L(2, C), whence exp(21riHOI.j) = e in G and thus also in the parabolic subgroup P. If the p-representation with highest weight A integrates to P, then the action of exp(21riHOI.j) on a highest weight vector is given by multiplication by e21ri ).(H"'j) , which shows that A(HOI.j) E Z and A(HOI.j ) is exactly the coefficient of Aj. Hence, a representation that integrates to some parabolic subgroup has to have integer coefficients over all nodes. Note that in the case of real parabolics there are usually less (or even no) integrality conditions since in this case it may happen that 3(go) integrates to a subgroup isomorphic to ]Ri.
(6
3.2.13. The relation between representations of 9 and p. Let g be a complex simple Lie algebra and let peg be a standard parabolic. For a finitedimensional complex representation V of g, the center 3(go) acts diagonalizably on V since 3(go) C ~. Hence, by part (2) of Proposition 3.2.12, we get a p-invariant filtration V = VO :::> VI :::> ••• :::> V N in which V N is the space VP+ of elements v E V, which are p+ -invariant i.e. which satisfy Z . v = 0 for all Z E p+. Notice that VP+ may also be interpreted cohomologically as HO(p+, V); see 2.1.9. PROPOSITION 3.2.13. Let V be a finite-dimensional complex representation of 9 and let VP+ c V be the subspace of p+ -invariant elements. Then there is a bijective correspondence between p-invariant subspaces of VP+ and g-invariant subspaces of V. In particular, if V is the irreducible representation of 9 with highest weight A, then VP+ is the irreducible p-representation with the same highest weight. PROOF. If V c V is g-invariant, then evidently VP+ = V n VP+ is p-invariant. Conversely, for a subspace W c VP+ we can consider the g-submodule of V generated by W, which is a g-invariant subspace by definition. It suffices to show that these two constructions are inverse to each other. If v E V is a highest weight vector for g, then v E VP+ by definition. On VP+ we have to understand the representation of ggs, since we know that 3(go) acts diagonalizably. This is determined by the highest weight vectors for ggs. But if such a highest weight vector is contained in VP+, then it is by definition a highest weight vector for g. From 2.2.11 we know that any finite-dimensional g-representation is generated by its highest weight vectors. In particular, any g-invariant subspace V c V has to be generated by VP+. If W c VP+ is p-invariant, then W is generated as a p-module by the go-highest weight vectors it contains. Any such vector is a highest weight vector for g, and hence by Theorem 2.2.11 generates an irreducible g-submodule in V. In this submodule, the highest weight vector is unique up to scale. This implies that if W is the g-submodule generated by W, then WP+ = W. Thus, we have established the correspondence and the last statement follows immediately. 0
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We can nicely rephrase these facts in terms of the universal enveloping algebra. Let U(g) be the universal enveloping algebra of g; see 2.1.10. Choosing a basis of 9 that is the union of a basis {Xi} of 9 _ and a basis {Ak} of p, the Poincare-Birkhoff-Witt theorem (see 2.1.10) implies that the monomials of the form X~l ... X~n A{l ... A~ form a linear basis of U (g). In particular, this implies that the multiplication in U(g) defines an isomorphism U(g) ~ U(g_) ®U(p) of vector spaces. Since V is generated by VP+ as a g-module, we get V = U(g)· VP+. Since VP+ c V is a p-submodule, we have U(p) . VP+ C VP+ and hence V = U(g_) . VP+. These ideas naturally lead to the concept of generalized Verma modules. Parallel to the case of ordinary Verma modules, let W be an irreducible representation of the parabolic subalgebra p with highest weight >.. Then we define the generalized Verma module Mp(>.) as U(g) ®U(p) W, compare with 2.2.12. By construction, this is a U(g)-module under left multiplication. The vector space decomposition U(g) ~ U(g_) ®U(p) from above immediately implies that Mp(>.) ~ U(g_) ® Was a vector space and a g_ -module. The generalized Verma module has a universal property. Namely, mapping wE W to 1 ® w defines an inclusion i : W ---+ Mp(>') of p-modules. Suppose that V is any representation of 9 and ¢ : W ---+ V is a homomorphism of p-modules. Then consider the map U(g) ® W ---+ V defined by A ® w t--t A· ¢(w). Obviously, this is a g-homomorphism and since ¢ is a U(p)-homomorphism, it factors to a g-homomorphism ¢ : Mp(>') ---+ V such that ¢ 0 i = ¢, and since Mp(>') is visibly generated by i(W) as a U(g)-module, ¢ is uniquely detennined by this property. Taking V to be the irreducible g-representation of highest weight>. and ¢ : W ---+ V to be the inclusion of VP+ , the homomorphism ¢ must be surjective by irreducibility, so V can be naturally realized as a quotient of Mp(>') for any parabolic subalgebra p of g. For later use, we also want to clarify the relation between ordinary and generalized Verma modules. Of course, mapping 1 ®1 E Mb(>') to 1 ® Wo E Mp(>'), where Wo E W is a highest weight vector induces a homomorphism Mb(>') ---+ Mp(>') of U(g)-modules. Moreover, from 2.2.12 we know that, as a vector space, Mb(>') is isomorphic to U(n_), where n_ is the sum of all negative root spaces. Choosing an appropriate basis for n_ we see that as a vector space U(n_) ~ U(g_) ®U(p_). Here, p_ is the direct sum of all negative root spaces contained in go. Since W is p-irreducible, we have W = U(p_) . wo, and this immediately implies that the homomorphism Mb('\) ---+ Mp('\) is surjective, so any generalized Verma module is a quotient of the Verma module with the same highest weight. 3.2.14. On the Weyl group. Our next aim is to define and analyze the Hasse diagram associated to a parabolic subalgebra p in a complex semisimple Lie algebra g, which encodes an amazing amount of information about a parabolic geometry and its homogeneous model. It is based on the Weyl group of 9 and we first have to prove some facts about the Weyl group. As before, we consider a complex semisimple Lie algebra 9 endowed with a Cartan subalgebra ~ S 9 and an ordering on ~*, and we denote by b., b. +, and b. 0 the corresponding sets of roots, positive roots, and simple roots, respectively. Recall from 2.2.4 that the (real) subspace ~o c ~ on which all roots are real defines a real form of ~, and the Killing form restricts to a positive definite inner product on ~o. The set b. of roots is then a finite subset of the real dual space ~o, and via the duality, we can carryover the Killing form to a positive definite inner product
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( , ) on ~o. For any a E ~ we can consider the root reflection Sa : ~o -+ ~o, defined by Sa (¢) = ¢ - 2 «cP,a}) a,a a, which maps ~ to itself. The Weyl group W = Wg of 9 is then by definition the subgroup of the orthogonal group O(~o) generated by these root reflections. In 2.2.7 we have noted that we may view W as a subgroup of the group of bijections of ~, whence W is finite. Further we noted that W is actually generated by the reflections sai corresponding to simple roots ai. An expression of wE W as a composition of simple root reflections is called reduced if it has the least possible number of factors. This number is called the length £(w) of the element w. On the other hand, the sign sgn(w) of w E W is defined as the determinant of the linear map w : ~o -+ 1)0' By definition, we have sgn(w) = (-l)£(w). For later use, we have to view the Weyl group W not only as a group but also as a directed graph. The vertices of this graph are elements w E W, and we have a directed edge w ~ w' labeled with a E ~ + if and only if £( w') = £( w) + 1 and w' = SaW. To analyze this graph structure, we need one more ingredient, which is closely related to the fact that the Weyl group acts transitively on the set of all simple systems for ~, respectively, the set of all Weyl-chambers. Namely, for w E W define w := {a E ~+ : w- 1 (a) E -~+}, or equivalently w = w(-~+) n ~+. Suppose that a,(3 E w are such that a + (3 E ~ (and hence a + (3 E ~+). Then w- 1 (a + (3) = w- 1 (a) + w- 1 ((3) lies in ~, and since it is the sum of two negative roots, it must be negative, too. Consequently, the set w is saturated, i.e. if a, (3 E w and a + (3 E ~, then a + (3 E w' Similarly, if a, (3 E ~ + \ w are such that a + (3 E ~, then w- 1 (a + (3) must be a positive root. Hence, a + (3 tj w, so the complement ~ + \ w is saturated, too. Further, we define (w) := I:aE., we may consider w(>.). Since we have chosen a total ordering on ~o when choosing the positive subsystem, we may compare any two weights, and we claim that w ::; w' implies w(>.) 2: w' (>.) for any dominant weight >.. To verify this, it suffices to deal with the case that w ~ w' for some a E ~ + . If this is the case, then w' = SaW, so w' (>.) = sa(W(>.)) = w(>.) a. Now since w is orthogonal, we have (w(>.),a) = (>.,w-1(a)), and part (4) of the proposition implies a ~ cP w , whence w-I(a) E ~+. But the condition that>. is dominant by definition means that>. has nonnegative inner product with all simple roots and hence with all positive roots. Consequently, w' (>.) differs from w(>.) by a nonpositive multiple of a positive root, whence w' (>.) ::; w(>.) and our claim follows. Using this we now get
2('(t2r)
COROLLARY 3.2.14. Let g be a complex semisimple Lie algebra with Weyl group W, Wo E W the longest element, >. a dominant integral weight and V the irreducible complex finite-dimensional representation of highest weight >.. Then the highest weight of the dual representation V* is -wo(>.). PROOF. From Theorem 2.2.10 we know that for w E Wand any weight (..L of V also w((..L) is a weight of V. Moreover, the weights of V* are exactly the negatives of the weights of V. In particular, we see that -wo(>') is a weight of V*. Denoting by (..Lo the highest weight of V*, we know again from Theorem 2.2.10 that there are nonnegative integers ni such that -wo(>') = (..Lo - LaiE~O niai. Applying -Wo, we see that -wo((..Lo) = >. - L niwo(ai). But from above we know that for each i, the element wo(ai) is the negative of some simple root, which means -wo((..Lo) = >. + L miai with mi 2: O. Since>. is the highest weight of V, this is only possible if all mj are zero, and hence (..Lo = -wo(>'). 0
3.2.15. The Hasse diagram associated to a complex parabolic. Let g be a complex semisimple Lie algebra and let peg be a standard parabolic
subalgebra corresponding to a Cart an subalgebra ~ ::; g and a positive subsystem ~ +. Let ~ 0 be the set of simple roots and E c ~ 0 the subset determined by p.
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Then we obtain the Ikl-grading of g by ~:-height, (where k is the E-height of the highest root), and in particular, the subalgebras go and p+ = gl of p. The reductive subalgebra go splits as a direct sum of its center 3(go) and its semisimple part gos. The basis for the definition of the Hasse diagram is that the Weyl group of goS can be naturally viewed as a subgroup of the Weyl group W = Wg of g. The Hasse diagram is then a distinguished set of representatives for the corresponding coset space. The Cartan subalgebra I) splits as I)' $1)", where I)' = 3(go) is the common kernel of all elements of tJ..o \ E, while I)" is spanned by the elements Ha for 0: E tJ..o \ E. From 3.2.1 we know that I)" is a Cartan subalgebra for the semisimple part gos. Let 1)0 c I) be the subspace on which all roots are real. By construction, the element Ha for 0: E tJ..o \ E lies in I)" n 1)0, so we may identify this space with I)~. On the other hand, I)~ := I)' n 1)0 is a real form of I)', so we get 1)0 = I)~ $ I)~. By construction, the inner product on 1)0 induced by the Killing form satisfies (Ha, H) = o:(H), so I)~ and I)~ are orthogonal. Passing to the duals we get an orthogonal decomposition of 1)0, and in particular, any simple reflection Saj with O:j E tJ..o \ E acts as the identity on (I)~)*. Defining WI' to be the Weyl group of goB, we see that we may naturally view this as the subgroup of Wg generated by the simple reflections Saj for O:j E tJ.. 0 \ E. The Hasse diagram is a set of distinguished representatives for the set WI' \ Wg of right cosets. To see how such representatives can be obtained, let us decompose tJ..+ = tJ..+(go) U tJ..+(p+) according to the subalgebra containing the corresponding root space. Otherwise put, for 0: E tJ.. + we have 0: E tJ.. + (go) if and only if htE (0:) = O. Since the E-height is additive, both tJ.. + (go) and tJ.. + (p+) are saturated in the sense introduced in 3.2.14. Now assume that 0: E tJ..+(go) and !3 E tJ..+(p+). Then sa(!3) differs from !3 by a multiple of 0:, and thus htE(sa(!3)) = htE(!3). Thus, Sa maps tJ..+(p+) to itself, so the same holds for any element w E WI" In particular, w C tJ..+(go) for any wE WI" Conversely, if wE Wg is such that w C tJ..+(go), then w is saturated. Since tJ.. + \ w and tJ.. +(go) both are saturated, also tJ.. +(go) \ w is saturated. Applying part (2) of Proposition 3.2.14 to goS we find an element in WI' corresponding to this subset, and again by part (2) of Proposition 3.2.14 we conclude that this element coincides with w, whence wE WI" Having characterized WI' as those elements w E W for which w C tJ.. + (go) the following definition is natural. DEFINITION 3.2.15. The Hasse diagram WI' of the standard parabolic subalge-bra p::; g is the subset of Wg consisting of all elements w such that w C tJ..+(p+). We endow WI' with the structure of a directed graph induced from the structure on Wg constructed in 3.2.14. There is a nice alternative characterization of WI': Recall that a weight >.. E 1)0 is g-dominant if (>.., 0:) ~ 0 for all 0: E tJ..o and p-dominant if the same holds for all 0: E tJ.. 0 \ E = tJ.. 0 n tJ.. + (go). Equivalently, one may require these conditions for all elements of tJ.. +, respectively, tJ..+ (go), since they can be written as linear combinations of the corresponding simple elements with nonnegative coefficients. Since any element w E W acts as an orthogonal transformation on 1)0, we get (w(>..),o:) = (>.., w-1(0:)). But this shows that w(>..) is p-dominant for any g-dominant weight>.. if and only if w- 1 (0:) E tJ.. + for any 0: E tJ.. +(go) i.e. if and only if wE WI'. Hence, w E WI' if and only if w(>..) is p-dominant for any g-dominant weight >...
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PROPOSITION 3.2.15. Let w E W be any element. Then there are unique elements wp E Wp and wP E WP such that w = wpwp. Moreover, £(w) = £(wp) + £(w P ). PROOF. Given w E W define q> := q>w n ~+(90). Then q> is saturated since it is the intersection of two saturated subsets of ~ +. Assume that a, f3 E ~ + \ q> are such that a + f3 E~. If one of the elements lies in ~ + (p+), then so does the sum, so in particular, a + f3 tJ. q>. On the other hand, if both a and f3 lie in ~+(90), then they both do not lie in q>w, so a + f3 tJ. q> follows since ~+ \ q>w is saturated. Consequently, also the complement of q> in ~ + is saturated, so there is a unique element wp E W such that q> = q>wp. Since q> c ~ +(90), we conclude that wp E Wp. Now take any element a E ~+(90)' If a E q>w-1, then p -wp(a) E ~ +(90) n q>wp C q>w'
Thus, w-1(-wp(a)) E -~+, and thus w-1(wp(a)) E ~+. If a tJ. q>w-1, then p wp(a) E ~+(90) \ q>wp. Hence, by construction w-1(wp(a)) E ~+. Thus, we see that the element w P := w;lw has the property that its inverse maps any element of ~+(90) to a positive root. By definition wP E WP and we obtain a decomposition of the required form. To prove uniqueness, assume that w = Wl W2 with Wl E Wp and W2 E WP, and assume that a E ~ + (90)' Then w11 (a) is a root of ~-height zero. Since W2 E WP, we conclude that a E q>w, i.e. w21(wl1(a)) E -~+ if and only if w 11 (a) E -~+, i.e. a E q>Wl' Consequently, ~+(90) n q>w = ~+(90) n q>Wll and since Wl E Wp the latter set coincides with q>Wl' By part (2) of Proposition 3.2.14 this implies that Wl coincides with the element wp from above, and thus we also get W2 = wp. To prove the statement on the length, one just has to note that by construction q>w p = ~+(90)nq>w. On the other hand, q>w P C ~+(p+) and thus also wp(q>w p) C ~+(p+). But for a E wp(q>w p) we have w;l(a) E q>w P and thus a E q>w' This shows that Iq>wl ~ Iq>wp 1+ Iwp(q>w p)1· Using part (3) of Proposition 3.2.14, we obtain £(w) ~ £(wp ) + £(wP), and the opposite inequality is obvious. 0 The existence and uniqueness of the decomposition w = wpwP tells us that wP is the unique element in the right coset Wp w that lies in Wp. Thus, WP is a set of distinguished representatives for the right coset space Wp \ W g • The statement about the length then tells us that these representatives are the unique elements of minimal length in each coset. Let us note two simple facts about the Hasse diagram. If w, w' E WP and w ~ w', then a E q>w' whence a E ~+(p+). On the other hand, since both the sets ~+(p+) and ~+(90) are saturated, there is a unique longest element Wb E WP with q> w oP = ~ + (p+). Since Wp is the Weyl group of a semisimple Lie algebra, it contains a unique longest element w~, and W~Wb = wo, the longest element in the Weyl group Wg • 3.2.16. Determining the Hasse diagram. We will describe a two step procedure to determine the Hasse diagram. In the first step, we determine the points of WP and some of the arrows in the Hasse diagram, while in the second step we determine the sets q>w, the remaining arrows, and the labels over all arrows. In many applications of the Hasse diagram, one is mainly interested in (a part of) the orbit
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of a given weight under the action of the elements of WP. For these applications, only the first step of the procedure is needed. First, we observe that it is easy to determine the action of a simple reflection on a weight in the Dynkin diagram notation. By definition, for a simple root 0: E ~ 0 and a weight ..\, we have so:(..\) = ..\ - 2«>.,0:» 0:,0: 0:. Writing..\ as a linear combination of the fundamental weights, the number 2«>.,0:» 0:,0: is the coefficient of the fundamental weight corresponding to 0:. In the Dynkin diagram notation, this is the number over the node representing 0:. Hence, to determine so:(..\) as a linear combination of the fundamental weights, we only have to write a simple root 0: E ~ 0 as a linear combination of fundamental weights. Writing the simple roots as 0:1, •.. ,O:n and the corresponding fundamental weights as WI," . ,Wn , this linear combination is clearly given by 0: = L~-l 2«0:,0:,» Wi' These coefficients form exactly the column .,at ,a: corresponding to 0: in the Cartan matrix of the Lie algebra g. Of course, the coefficient of the fundamental weight corresponding to 0: itself is 2, from which we conclude that applying So: to ..\, the coefficient over the node corresponding to 0: changes sign. Moreover, the coefficient over a node may change only if the inner product of 0: with the corresponding simple root is nontrivial, i.e. if there is an edge between the two nodes in question. Looking at the description of the passage from the Cartan matrix to the Dynkin diagram in 2.2.5, one sees that 2(~~) = -1 if between the nodes corresponding to 0: or j3 there is a single edge or a multiple edge with the arrow pointing towards 0:. On the other hand, if there is a multiple edge with the arrow pointing towards j3, then 2(~~) equals minus the number of edges. Thus, we obtain the following examples for the action of the simple reflection corresponding to the node with coefficient b (with all coefficients not explicitly indicated remaining unchanged by the reflection): t
So:,
C.. ~ .. ·)
so:, ( ...
so:,
= ...
~ .. .
~ ... ) = ... ~ .. .
C.. ~ ... ) = ... ~c
.. .
Since the Weyl group Wg is generated by the simple reflections, we may determine the orbit of any weight under Wg by applying step by step all simple reflections that do not lead "backwards" i.e. we must not apply the same simple reflection twice without another simple reflection in between. If we choose a weight lying in the interior of a Weyl chamber, than the orbit will be in bijective correspondence with Wg , since we know that the action is simply transitive on the set of Weyl chambers. In particular, one may use the lowest form 8 to determine the Weyl group in this way. Choosing a weight adapted to a standard parabolic, one may use the same procedure to determine the points of the subset WP C Wg: PROPOSITION 3.2.16. Let 9 be a complex semisimple Lie algebra and let p :::; 9 be the standard parabolic subalgebra corresponding to a set E of simple roots. Let 8P be the sum of all fundamental weights corresponding to elements of E. Then we have:
328
3. PARABOLIC GEOMETRIES
(1) The map w 1-+ w- 1 (8 P) restricts to a bijection between WP and the orbit of 8P under W g • (2) Suppose that w E WP and a E ~o is a simple root such that a ¢. directly from this diagram. For example, the longest element of Wl> can be written as SOl S02 S03 SOl SC\ + 811, where oX is the highest weight of V. This, for example, implies that the fact whether an irreducible component of C*(p+, V) is contained in the cohomology or not depends only on the highest weight.
3.3.5. Analysis of the weight condition. By Proposition 3.3.4, determining the cohomology H*(p+, V) is equivalent to finding those irreducible components of C*(p+, V), whose highest weights v satisfy IIv + 811 = IloX + 811. To attack this problem, we will start by looking at arbitrary weights of the 90-representation C*(p+, V). The point here is that we get a good enough hand on all these weights to find the ones satisfying the norm condition. In a second step, we will show that all the occurring weights are actually highest weights of irreducible components. Using the Killing form B, we may identify C*(p+, V) with A*9- ® V. Now 9_ is the direct sum of all root spaces 9_0< for 0: E ~+(p+), so the weights of 9_ are exactly the negatives of elements of ~ + (p+), each occurring with multiplicity one. Choosing a basis of 9_ consisting of weight vectors Fo< E 9_0.' and A" is the irreducible representation with highest weight A. Moreover, it follows immediately from the definitions that the Weyl group W of 9 is the product W' x W" of the Weyl groups of the two factors, and the length of (w', w") is the sum of the lengths of the two elements. Hence, from Theorem 3.3.5 we immediately conclude that
Hn(g_, V'
~ V") ~
$
. '+J=n
(Hi(g~, V') ~ Hi (g~, VII)),
as a module over go ~ gb Ef1 g~. In particular, the adjoint representation V = 9 is not irreducible in this case, but we have 9 = g' ~ OC Ef1 OC ~ gil. Consequently, we obtain (3.20) Hn(g_, g)
~ . E9
(Hi (g~, g/)
~ Hi (g~, OC) Ef1 Hi (g~, OC) ~ Hi (g~, gil)).
'+J=n These results can be used to compute the real cohomologies of complex simple Lie algebras, since for a complex simple Lie algebra g, we have glC = 9 Ef1 g. 3.3.7. Zeroth and first cohomologies. The cohomology groups H*(g_, V) will play an important role in many parts of the subsequent developments. As a first application of Theorem 3.3.5 we study the first cohomology groups with coefficients in the adjoint representation. These cohomology groups determine whether a normal parabolic geometry is uniquely determined by the underlying infinitesimal flag structure; see 3.1.14 and 3.1.16. On the way to these results, we will compute HO(g_, V) for any V and Hl(g_,OC). The zeroth cohomology HO(g_, V) can be determined directly for arbitrary V. By definition, this cohomology group is simply the kernel of 8 : V ~ £(g_, V) and thus the go-module V9- of g_-invariant elements in V. Alternatively, we may describe HO(g_, V) as V/im(8*), and 8*(Z Q9 v) = -Z· v for Z E p+ = g: and v E V. Thus, im(8*) = p+ . V, and HO(g_, V) ~ V/(p+ . V), which is a more natural description from the point of view of the p-module structure. In particular, for the trivial representation OC, we obtain HO (g_, OC) = K In the case of the adjoint representation we conclude from 3.3.6 that HO(g_, g) is the direct sum of the zeroth cohomologies of the simple ideals of g, so let us assume that 9 = g-k Ef1 ..• Ef1 gk is simple. Of course, [p+, g] C g-k+l Ef1 .•. EB gk = g-k+l. Moreover, Theorem 3.3.5 together with 3.3.6 tells us that HO(g_, g) = g/([lJ+, g]) is irreducible, which implies that there is no p-submodule of 9 which lies strictly between 9 and [p+, g]. But this immediately implies that [p+,g] = g-k+l, and thus HO(g_,g) ~ g-k ~ g/g-k+l for simple g. The explicit description of the zeroth cohomology in Theorem 3.3.5 tells us that for an irreducible g-representation V with highest weight A the cohomology HO(p+, V) ~ VP+ is the irreducible p-representation of highest weight A, which we have noticed in 3.2.13. To obtain the corresponding statement for HO(9-, V), recall that HO(9-, V) = (HO(p+, V*))* by Corollary 3.3.1. Hence, if V is dual to the irreducible g-representation with highest weight fL (i.e. V has lowest weight -fL), then V9- is dual to the irreducible p-representation with highest weight fl. Let us next consider the cohomology group HI (g_, OC) with trivial coefficients. By definition, 8 : OC -+ £(9-, OC) is the zero map, so HI (9-, OC) coincides with
3.3. KOSTANT'S VERSION OF THE BOTT-BOREL-WElL THEOREM
355
the kernel of 8: L(g_,lK) - t L(A2g_,lK). Since the action is trivial, we simply get 8¢(X, Y) = -¢([X, Y]), which immediately implies H1(g_, lK) = L(g_/[g_, g_],lK). Clearly, [g_, g-l c g-k EB ... EB g-2, and these two spaces coincide since g_ is generated by g-l. Thus, we conclude that H1(g_, lK) ~ (g-l)* ~ gl. Now we are ready to prove the main result on the cohomological obstructions showing up in Theorems 3.1.14 and 3.1.16. Recall from 3.1.14 that the cohomology groups Hn(g_, g) are naturally graded as Hn(g_, g) = ffi t Hn(g_, g)£ by the homogeneous degree of maps. In particular, for the first cohomology this grading is simply induced by the natural grading on p+ 0 g. PROPOSITION 3.3.7. Let 9 = g-k EB ... EB gk be a Ikl-graded semisimple Lie algebra over lK = ~ or C such that none of the simple ideals of 9 is contained in go. Then we have: (1) H 1 (g_,g)£ = {O} for all £ > 1 unless 9 contains a simple ideal isomorphic to 51(2,~) or 51(2, C) with the (unique) grading induced by the Borel subalgebra. (2) H 1 (g_, g)£ = {O} for all £ > 0 unless 9 contains a simple ideal g' such that either g' or gc is isomorphic to >E---O- ... -O--O or >E---O- ... ~. PROOF. We first reduce the problem to the complex simple case. From 3.3.6 we know that for a sum 9 = g' EB gil of ideals, the cohomology HI (g_, g) splits into the direct sum of
HI (g~, g') 181 HO (g~, lK) EB HO (g~, g') 181 H 1 (g~, lK) and the corresponding terms with the roles of g' and gil exchanged. From above we know that HO (g~, g') is concentrated in negative degrees, and HI (g~, lK) is concentrated in homogeneous degree one, so their tensor product sits in nonpositive homogeneity. This also holds for the symmetric term and using that HO(g~, lK) = lK, we conclude that HI (g_, g)£ ~ HI (g~, g')£ EB HI (g~, g")£ for all £ > o. In particular, the first cohomology with coefficients in the adjoint representation of a Ikl-graded semisimple Lie algebra 9 has a nonzero component in some positive homogeneous degree if and only if the same is true for one of the simple ideals of g. On the other hand, the splitting into homogeneous degrees is obviously compatible with complexifications, so it suffices to determine the complex simple Ikl-graded Lie algebras 9 such that H 1 (g_, g)£ :f= {O} for some £ > o. Now the isomorphism H 1 (g_,g) ~ Hl(p+,g)* is compatible with homogeneities, i.e. we may equivalently determine the cases in which Hl(p+, g)£ is nontrivial for some £ < O. We assume that p is the standard parabolic corresponding to a set ~ of simple roots. Now according to Theorem 3.3.5, the irreducible components of HI (p+, g) are in bijective correspondence with the elements of length one in WP, i.e. with the reflections 0"0 corresponding to simple roots a E ~. Moreover, denoting by A the highest weight of the adjoint representation, the highest weight of the irreducible component corresponding to 0"0 is given by 0"0(A+8) -8 = 0"0 (A) -a. By construction, the root space go is contained in gl, from which we immediately conclude that the homogeneous degree of a weight vector of the above weight is given by htdO"o(A)) - 1. Now A simply is the highest root, so O"o(A) is a root, too, and by definition 0"0 (A) differs from A by a multiple of a. If there is more than one simple root in g, then A :f= a, and thus O"o(A) is a positive root and thus htE(O"",(A)) ~ 0 and (1) follows. To prove (2), it remains to determine those 9 and ~ such that there is an element a E ~ with htE(O"o(A)) = o. We have noted above that O"o(A) is a positive root and
356
3. PARABOLIC GEOMETRIES
expanding it as a linear combination of simple roots, the coefficients of all simple roots =f a are the same as for'\. In particular, this implies that htE(a",('\)) > 0 unless E = {a}. Given that E = {a}, the condition that htE(a",('\)) = 0 by definition of the simple reflection is equivalent to the fact that 2«A''''» coincides with "','" the coefficient of a in the expansion of ,\ as a linear combination of simple roots. Now we have to go through the list of complex simple Lie algebras. In the case of An for n ;::: 2, we have simple roots al, ... ,an and the highest root ,\ is given by al + ... + an; see Example (1) of 2.2.6. Consequently, ~~::~:? equals 1 for i = 1 and i = nand 0 otherwise. Since the 111-gradings corresponding to {all and {an} are isomorphic, this is compatible with the statement of (2). Let us next consider the case of Cn with n ;::: 2. As in example (4) of 2.2.6 we have simple roots al, ... , an and ,\ = 2al + ... + 2a n -l + an. From this description and the Dynkin diagram, one immediately concludes that 2( (A''''i» equals 2 for i = 1 and 0 for i =f 1, so we get the second class of exceptions in (2). Since B2 = C2 and this has been dealt with above, we may next consider Bn for n;::: 3. According to example (3) of 2.2.6, the usual Dynkin diagram with simple roots al, ... ,an leads to ,\ = al + 2a2 + ... + 2a n . Assuming n ;::: 3, this implies that 2(0'1.,0:1. (A''''i» equals 1 for i = 2 and 0 for i =f 2, so we get no exception in the case of Bn with n;::: 3. For D n , we may restrict to the case n ;::: 4. Using the simple roots al, ... , an as in example (2) of 2.2.6, the highest root ,\ is given as al + 2a2 + ... + 2a n-2 + an-l + an. Using this, one immediately verifies that 2«A''''i» equals 1 for i = 2 and 0:1. ,0:1. o for i =f 2, so we get no exception in the case of Dn with n ;::: 4. For the exceptional Lie algebras one may read off from Table B.2 in Appendix B that there always is a unique simple root a such that 2«A,"'» =f 0, but the resulting "','" number never coincides with the coefficient of a in the expression of ,\ as a linear combination of simple roots. 0 O:1.,Q't
REMARK 3.3.7. Following the lines of the proof of the above proposition, one may also give a complete list of all simple Ikl-graded Lie algebras such that Hl (g_, g)l = {O} for alIi;::: O. This is of considerable interest from the point of view of parabolic geometries, since it means that the corresponding geometry is essentially determined by the filtration of the tangent bundle only. We will prove this result and study the corresponding geometries in 4.3.1. There are also classification results concerning the second cohomology groups with values in the adjoint representation available. These are important from the point of view of parabolic geometries, since H2(g_, g) determines the possible components of the harmonic curvature, which is a complete obstruction to local flatness; see 3.1.12. The article [Ya93] contains a complete list of the gradings for which there is an irreducible component of H2(g_, g), which is contained in positive homogeneity. If there is no such component, then regular normal parabolic geometries of that type are automatically locally flat (since by Theorem 3.1.12 the lowest nonvanishing component of the Cartan curvature has to be harmonic). Still the results on equivalence to underlying structures can be interesting in these cases; see 4.3.5.
3.3.8. The Bott-Borel-Weil theorem. We next discuss how Kostant's theorem is related to the original Bott-Borel-Weil theorem, mainly following Kostant's original article [Kos61]. This theorem computes the sheaf cohomology groups of a
3.3.
KOSTANT'S VERSION OF THE BOTT-BOREL-WEIL THEOREM
357
complex generalized flag variety GI P with coefficients in the sheaves of local holomorphic sections of certain homogeneous holomorphic vector bundles. As in the smooth case treated in 1.4.3, holomorphic homogeneous vector bundles over GI P are in bijective correspondence with holomorphic representations of the parabolic subgroup P, and in particular, we may look at irreducible representations. From 3.2.12 we know that finite-dimensional complex representations of the Lie algebra p are in bijective correspondence with p-dominant and p-algebraically integral weights )" which may be viewed as weights for g. In the formulation of the theorem it will be more convenient to work with lowest weights rather than highest weights, but symmetry of the weights of a p-representation under the Weyl group W" implies that these are just the negatives of the p--dominant algebraically integral weights. The G-action on a homogeneous holomorphic vector bundle is holomorphic, so it induces an action on the cohomology groups of GI P with values in the sheaf of local holomorphic sections. (This is evident by viewing sheaf cohomology groups as Dolbeault cohomology groups.) Hence, we may analyze the cohomology groups as representations of G. THEOREM 3.3.8 (Bott-Borel-Weil). Let G be a connected complex semisimple Lie group with Lie algebra g, PeG a standard parabolic subgroup with Lie algebra peg. Consider an irreducible representation of P, let -), be its lowest weight, and let DC),) be the sheaf of local holomorphic sections of the corresponding homogeneous holomorphic vector bundle over G I P. Denoting by 8 the sum of all fundamental weights of g, we have: (1) If), + 8 lies in a wall of some Weyl chamber, then the sheaf cohomology H* (G I P, DC),)) is trivial. (2) If), + 8 lies in the interior of some Weyl chamber, then let w E W be the unique element such that w . ), := we)' + 8) - 8 is dominant, and let few) be the length ofw. Then Hk(GIP,O()')) = 0 for k ¥ few) and Hl(w)(GIP,O()')) is an irreducible representation of G with lowest weight -w . ), (and hence dual to the representation with highest weight w . ),). SKETCH OF PROOF. By Proposition 3.2.6, the maximal compact subgroup K c G acts transitively on GIP and GIP ~ KIL where L := K n P. Since G is complex, we know from 2.3.2 that the Lie algebra t of K is a compact real form of g. Let fJ C 9 be the Cartan subalgebra and let fJo C fJ be the subspace on which all roots are real. Then from 2.3.1 we know that for any positive root Q E Ll+ we can choose generators X±a E g±a such that t
= ifJo EB
E9 (~(Xa -
X-a)
+ i~(Xa + X-a)).
aE~+
In particular, 1= t n pC go, and I is a real form of go. Hence, g/go is the complexification of til, and we may view Tc(KI L) as K XL gl go. Since L C Go, we get gl go ~ g- EBp+ as an L-module. The complex structure on the tangent bundle T( K I L) comes from the linear isomorphism til ~ g/p. One verifies that this induces
J(a(Xa - X-a)
+ ib(Xa + X-a)) = -b(Xa -
X-a) - ia(Xa
+ X-a).
This implies that gl go ~ g_ EBp+ is the splitting describing the given complex structure on K I L, with g_ corresponding to the holomorphic part and p+ corresponding to the anti-holomorphic part.
3. PARABOLIC GEOMETRIES
358
Now let V be an irreducible holomorphic representation of P. By restriction, we may view this as a representation of L, and hence we can view the holomorphic vector bundle G x p V also as K x LV. In view of the above discussion and the standard correspondence between sections of associated bundles and equivariant functions on the group, we may view the space nO,q ( G / P, G x p V) of bundle-valued (0, q)-forms as the space COO(K, Mp+ 0 V)L of L-equivariant smooth functions. Via inserting fixed elements of 13+ into such functions, we get the alternative picture of this space as HomdAkp+, COO(K, V)). From above we know that TcK 9:! K x g, so we may view elements of 13+ as sections of the complexified tangent bundle of K, and thus obtain a natural action of 13+ on COO(K, V). We may further identify COO(K, V) with COO(K) 0 V and since V is an irreducible P-module, 13+ acts trivially on V. From the definition, one easily sees that the Dolbeault differential corresponds to
a
where denotes the Lie algebra differential for the representation COO(K) of 13+. By construction, the natural K -action on sections is given by acting only on the COO(K)-factor by (g . I)(g') = I(g-lg'). Putting this together, we see that we may identify the Dolbeault cohomology group Hq (G / P, (G x p V)) with the space (Hq(p+,COO(K)) 0 V)L of L-invariant elements. Here, the action of L on the cohomology groups is obtained from the natural (tensor product) action on Aqp+ 0 COO(K) and L acts on COO(K) via (g. I)(g') = f(g'g). Since G / P is compact, the sheaf cohomology is known to be finite-dimensional, so we may replace COO(K) by the subspace of K-finite vectors, i.e. the subspace of all those functions which are contained in a finite-dimensional K -invariant subspace of COO(K). This subspace is described explicitly by the Peter-Weyl theorem as follows. Consider the space C(K) of complex-valued continuous functions on K. This carries a representation of K x K defined by ((g, h) . I)(g') := f(g-l g'h). Now suppose that W is a finite-dimensional complex representation of K. Since K is compact, there is a K -invariant inner product on W, and for e,.,., E W, we define the matrix coefficient kTJ : K - t C by fe,TJ(g) = (e, g ..,.,). Of course, this is a smooth function, and invariance of the inner product immediately implies that (g, h)· fe,TJ = fg-1.e,h'TJ· In particular, if W is irreducible, then kTJ = 0 implies that either or .,., has to vanish, and we obtain an injection of the K x K -representation W* 181 W into COO(K) and C(K). Consider the set of matrix coefficients of all finite-dimensional representations of K. One immediately checks that a linear combination of matrix coefficients of two representations can be obtained as a matrix coefficient of the direct sum, while the product can be obtained as a matrix coefficient of the tensor product of representations. Hence, this is a sub algebra of COO(K) and of C(K). Provided that K admits a faithful finite-dimensional representation (which can be proved for general compact Lie groups and is obvious for the groups we are concerned with here), this subalgebra separates points and hence is dense in C(K) by the Stone-Weierstrass theorem.
e
The Peter-Weyl theorem now states that the subspace of all matrix coefficients is exactly the subspace of K-finite vectors in C(K) (and thus also in COO(K)), and it can be explicitly described as a K x K-representation as EBw(W* 181 W),
3.3.
KOSTANT'S VERSION OF THE BOTT-BOREL-WEIL THEOREM
359
where W runs through all irreducible representations of K (which are automatically finite-dimensional). See [CSM95] for a nice exposition of the Peter-Weyl theorem. Returning to the description of Dolbeault cohomology, we see that it is given as ffiw (Hq(p+, W* ~ W) ® V)L. By construction, L acts only on the W component and the same is true for p+, since the p+ -action comes from differentiation along left invariant vector fields. On the other hand, the natural K -action only hits the W*-component, and
Hq(G/ P, G
Xp
V) ~
E9 (W* ® (Hq(p+, W) ® V)L),
w with K (and hence G) acting only via the W*-factor. Since V is irreducible and Hq(p+, W) is completely reducible, we conclude that (Hq(p+, W)®V)L is either 0 or C, and the second possibility occurs if and only if Hq (P+, W) contains an irreducible component isomorphic to V*. The Go-irreducible components (which coincide with the L-irreducible components) of Hq(p+, W) are described by Kostant's version of the Bott-Borel-Weil theorem in 3.3.5. In particular, their highest weights always lie in the affine WP-orbit of the highest weight of Wand thus in the interior of some Weyl chamber, and the result follows. 0 REMARK 3.3.8. (1) Let us consider the special case of the Borel subgroup B and a dominant integral weight >.. Then >. + 8 lies in the interior of the dominant Weyl chamber, so part (2) of the theorem applies with w = id. This says that the space of global holomorphic sections of the homogeneous line bundle corresponding to the representation defined by ->. is dual to the representation of highest weight >.. This is commonly referred to as the Borel-Weil theorem, and it gives a uniform construction of all finite-dimensional irreducible representations of complex semisimple Lie algebras. (2) The general version of the Bott-Borel-Weil theorem was proved in [Bott57]. There are several alternative proofs, using sheaf cohomology techniques and the fact that the case of 8L(2, . : ~ . . . . C is simply e). E Z[~*], see 2.2.18 for the notation. Consequently, Theorem 3.3.5 implies that if V is the irreducible g-representation with highest weight >., then the character of the ~-module Hn(b+, V) is given by LWEW:C(w)=n ew ().+d)-c5 or, equivalently, wEW:C(w)=n
We have observed in 2.2.18 that for a finite exact sequence of modules and equivariant maps, the alternating sum of the characters vanishes. For a slight generalization, assume that··· a~l Vi ~ Vi+! ........... is a finite complex of ~-modules with finite-dimensional weight spaces and ~---equivariant differential and denote the cohomology modules by Hi := ker(Oi)/ im(oi-l). Then we claim that
Li( _l)i char(Vi) = Li( _l)i char(Hi ).
3. PARABOLIC GEOMETRIES
360
Indeed, Vi/ker(8i ) ~ im(8i ) and thus char(Vi) = char(ker(8i )) + char(im(8i )). On the other hand, by definition of Hi we get char(Hi) = char(ker(8i ))-char(im(8i _d), and the claim follows immediately by forming alternating sums. Using this we get
n
n
n
n
Now for w E W we have (_l)l!(w) = sgn(w), so we may rewrite the above equation as char(V) sgn(w)e w (6) = sgn(w)eW(A+ 2 and 2A1 - A2 for p = 2, respectively, -Ap + Ap+1 + Ap+q-1 for q > 2 and -Ap + 2Ap+1 for q = 2. Step (C): From the description above, we see that
Viewing elements X E g-l as linear maps IRP ~ IRq, the adjoint action is immediately seen to be given by Ad( Cll C2 ) (X) = C2 X Cl 1 . A moment of thought shows that Ad : Go ~ GL(g-l) is injective if p + q is odd, while for p + q even its kernel is given by {(id, id), (-id, -id)}. A first order structure with structure group Go can be defined on manifolds whose dimension equals dim(g_l) = pq. Given such a structure (Po: go ~ M, ()), we can of course form the associated bundles corresponding to the basic representations IRP and IRq of Go. Hence, we obtain a rank p vector bundle E ~ M and a rank q vector bundle F ~ M. Fixing a Go-invariant isomorphism APIRP IZl A qIRq ~ IR, we obtain a preferred trivialization of APE IZl A qF. The fact that the tangent bundle TM can be identified (using ()) with go xc o g-1 implies that we get an isomorphism If> : E* IZl F ~ T M. Let us conversely assume that on a manifold M of dimension pq we have given vector bundles E and F of rank p, respectively, q, a trivialization ¢ : APE IZl A qF ~ M x IR and an isomorphism If> : E* IZl F ~ T M. Then we consider the fibered product GL(IRP, E) XM GL(IRq, F) of the linear frame bundles of E and F. The fiber of this bundle over x E M consists of pairs ('l/h, 'l/J2) of linear isomorphisms 'l/J1 : IRP ~ Ex and 'l/J2 : IRq ~ Fx. We define a subspace go in this bundle as the set of those pairs for which the second component of ¢o (AP'l/J1IZlAq'l/J2), which is a linear isomorphism APIRP IZl A qIR q ~ IR coincides with the fixed Go-invariant isomorphism. The group Go acts on go by composition from the right, and visibly the action is free and transitive on the fibers of the natural projection Po : go ---+ M. Hence, this becomes a smooth principal Go-bundle. Next, we define () E 0 1 (go, g-l) as follows: For a tangent vector ~ E T(1/Jl,'P2)gO we have Tpo . ~ E TxM, and thus c]>-l(Tpo'~) E L(Ex, Fx). Thus, we may define ()(~) := 'l/J:;1
0 c]>-l(Tpo .~) 0 'l/Jl
E
L(IRP, IRq) = g-l.
This is evidently smooth, its kernel in each point is the vertical subbundle, and by construction it is equivariant in the sense that (rg)*() = Ad(g-l) 0 () for each 9 E Go. Hence, (Po: go ~ M, ()) is a first order Go-structure, and we conclude that such a structure is equivalent to the choice of E, F, ¢ and C]>. Such a structure
376
4. A PANORAMA OF EXAMPLES
is usually referred to as an almost Grassmannian structure of type (p, q) in view of
the homogeneous model being the Grassmannian Grp(~p+q). Of course, for the homogeneous model, the auxiliary bundles E and Fare simply the two tautological bundles over the Grassmannian. The bundle E is the subbundle in Grp(~p+q) x ~p+q whose fiber over a p-dimensional subspace is given by that subspace, while F is simply the quotient of the trivial ~P+Lbundle by the subbundle E. It is a well-known fact that the tangent bundle of the Grassmannian can be canonically identified with the bundle L(E, F) of linear maps.
Step (D): According to Theorem 3.3.5, the irreducible components of H2(gl' g) are in bijective correspondence with elements of length two in the Hasse diagram WP of the parabolic p (or more precisely the complexification pC C sl(p + q, C)). To determine these elements we may use either of the two recipes from 3.2.18, but the recipe for Ill-gradings is simpler. We have to look for two-element subsets in ~+(gl) whose complement in ~+ is saturated, and we have determined all such subsets in Example 3.2.17. Since our grading corresponds to the simple root a p , this simple root has to be contained in any of the subsets. In the picture of matrices, for each root in the subset, any root in ~ + (gl) whose root space lies below or left of the given one, has to be in the subset, too. Hence, the only possible two element subsets are {ap,ap_l + a p} and {ap,ap + a p+1}' These correspond to the Weyl group elements WI := sap_l +ap 0 sap and W2 := sap+ap+l 0 sap' and the sets are recovered as the sets ~Wi associated to the elements Wi as in 3.2.14. From the explicit representatives provided by Theorem 3.3.5 we see that the highest weight of the irreducible component corresponding to W E WP is given by the negative of the sum of the roots contained in ~w plus the image of the highest weight of the adjoint representation under w. From the sets ~Wi we therefore get the contributions -Dp-l - 2ap, respectively, -2ap - a p+1 to the highest weight, and these are easy to analyze: We have noticed above the -ap is the sum of the highest weights of ~P* and ~q. Similarly, -ap-l - a p is the sum of the second highest weight of ~P* with the highest weight of ~q, while -ap - ap+l is the sum of the highest weight of ~P* with the second highest weight of ~q. Thus, we conclude that -ap-l - 2ap is the sum of the highest and the second highest weight of ~P* with twice the highest weight of ~q, so it is the highest weight of A2~p*®S2~q. Likewise, -2a p -a p+1 is the highest weight of S2~P*®A2~q. Notice that g-l ~ ~P* ® ~q implies that the second exterior power A2g_1 decomposes as (A2~p* ® S2~q) EEl (S2~p* ® A2~q), so the two representations naturally sit inside this exterior power. On the other hand, the highest weight A of the adjoint representation of s[(p + q, C) is Al + Ap+q-l = al + ... + ap+q-l. This immediately implies that for p > 2 we have (A,ai) = 0 for i = p - 1,p and thus Wl(A) = A. Similarly, for q > 2 we get W2(A) = A, and A is the highest weight of gl = ~p ® ~q*. In particular, we conclude that for p > 2, the irreducible component corresponding to WI is the highest weight component in (A2~p* ®S2~q) ®gl' (Recall that once we have found an irreducible component in A*g-I ® gi with the right highest weight, then it must be the cohomology component by the multiplicity one result in Theorem 3.3.5.) From 3.3.1 we know that H2(g_l,g) is the dual representation to H2(gI,g*) ~ H2(gI,g). Hence, the irreducible component in H2(g_1,g) corresponding to WI (still for p > 2) is the highest weight component in (A2~p ® S2~q*) ® g-l C A2(g_d* ® g-I.
4.1. STRUCTURES CORRESPONDING TO 111-GRADINGS
377
This highest weight component can be easily described explicitly as follows: According to 9-1 = IRP* ®IRq, we get (A 2]RP ® 8 2]Rq*) ® g-l ~ (A 2]RP ® ]RP*) ® (8 2]Rq* ® ]Rq). Visibly, the first component admits a unique contraction with values in ]RP, while for the second component there is a unique contraction with values in ]Rq*. Hence, from the tensor product we obtain two contractions, one with values in ]RP®(8 2]Rq* ®]Rq) and the other one with values in (A2]Rp ® ]RP*) ® ]Rq*, and the highest weight component is the intersection of the kernels of these two contractions. Similarly, for q > 2 the irreducible component corresponding to W2 is the highest weight component in (8 2]RP ® A2]Rq*) ® 9-1. and the description of this highest weight component is completely parallel to the other one. Let us next consider the case p = 2. Then WI = Sal +02 0 S02' and for the highest weight>. = >'1 + >'q+! of the adjoint representation we get S02 (>.) = >. and WI (>.) = >. - Q:l - Q:2 = Q:3 + ... + Q:q+!' This evidently is the highest weight of the simple component sl(q,]R) c go. Since this is selfdual we conclude that the irreducible component in H 2 (9_1. g) corresponding to WI in the case p = 2 is the highest weight component in (A2]R2 ® 8 2]Rq*) ® sl(q,]R) c A2(9_d* ® go. As before, this highest weight component can be easily described explicitly. The space sl(q,]R) sits as the tracefree part in ]Rq* ®]Rq. The highest weight component then must be contained in A2]R2 ® 8 3 ]Rq* ®]Rq c A2]R2 ® 8 2]Rq* ® IRq* ® ]Rq. On that space there is a unique contraction with values in A2]R2 ® 8 2 ]Rq*, and the highest weight component is exactly the kernel of this contraction. For q = 2 we obtain the parallel description of the irreducible component in H2(g_l> g) corresponding to W2 as the highest weight component in (8 2]RP ® A2]R2*) ® sl(p,]R).
Step (E): Using the correspondence between representations and bundles established in Step (C), we can directly translate the information on H2(g_1. g) obtained in Step (D) above into a description of the harmonic curvature components. This leads to the following table: p = 2, q = 2
two curvatures
p = 2, q> 2
one torsion, one curvature
p> 2, q = 2
one torsion, one curvature
p> 2, q > 2
two torsions
PI E f(A2 E ® 8 2F* ® sl(F)) P2 E f(8 2E ® sl(E) ® A2 F*) T2 E f(8 2E®E* ®A2F* ®F) PI E r(A2 E ® 8 2F* ® sl(F) Tl E f (A 2E ® E* ® 8 2F* ® F) P2 E f(8 2E ® sl(E) ® A2 F*) T1 E r(A 2E ® E* ® 8 2F* ® F) T2 Er(8 2E®E*®A2F*®F)
To interpret the harmonic curvature geometrically using Theorem 4.1.1, we have to clarify the meaning of a principal connection on a Go-structure (Po: go --t M,O). As we have seen in Step (C), we may interpret the bundle go as 8 (G L(]RP ,E) x M GL(]Rq, F»). From this description it follows immediately that a principal connection'Y on this bundle is equivalent to a pair (VE, VF) of linear connections on the
378
4. A PANORAMA OF EXAMPLES
bundles E and F, such that the constant global sections of the bundle APE ® M F determined by the trivialization
2, respectively, p > 2 and q = 2 are completely parallel so we only discuss the first of these. In that case, we must have 71 = 0 and 72 is the basic torsion. Structures for which 72 vanishes identically are called Grassmannian structures of type (2, q) (as opposed to almost Grassmannian structures). If 72 = 0, then the above procedure produces a torsion-free connection on go. Conversely, by Theorem 4.1.1 existence of a torsionfree connection on go implies vanishing of 72. Thus, Grassmannian structures of type (2, q) are exactly those almost Grassmannian structures which are integrable in the sense of G-structures. The second basic obstruction to local flatness for type (2, q) with q > 2 is the curvature PI E r(A2 E ® 8 2F* ® sl(F)). From part (2) of Theorem 4.1.1 we know that we can compute PI as the appropriate component of the curvature of -y. Since the bundle F corresponds to the natural representation of Go on ~q, and \7 F is the linear connection induced by -y, the curvature of \7 F corresponds to the component
4.1. STRUCTURES CORRESPONDING TO 111-GRADINGS
381
in A2T* M 0 L(F, F) of the curvature of 'Y, compare with 1.3.4. Let us denote this curvature by RF, so for s E r(F) and e,,,., E X(M) we have the usual formula RF(e, ".,)(s) =
"\jr'V~ s - "\j~"\j[ s - "\jk,1/]s.
In abstract index notation, R := RF has the form R~,~, cP', where the first two pairs of indices describe the form part. According to the description of the highest weight component in the end of Step (D), we can continue as follows: First, we take the complete symmetrization RtA'~' C') D' in the three lower primed indices. By skew symmetry in the first two pairs of indices, this expression is automatically skew in A and B, so we can compute PI as the tracefree part of this. Expanding the symmetrization, one computes D' _ IRA B D' RA B (A' B' D') - 3 (A' B') D'
1
q+2(XA'B,Dc,
D'
+XC'A,DB'
D'
2R[AB] D' D' (A' B')
+3
+XB'C,DA'
D'
3
) = Q+2 X (A'B,DC')
D'
.
Consequently, we obtain A B D' _ A B D' 1 A B I'D' 2 [AB] I'D' (Pl)A'B'C' - R(A'B'C') - q+2 R (A'B'II'1 DC') - q+2 R l' (A'B' DC')
In the last remaining case p = q = 2, we automatically have 71 = 72 = 0, so the above procedure leads to a principal connection 'Y on go with vanishing torsion. Looking at the corresponding linear connections "\jE and "\jF we can compute the two curvatures PI and P2 from their curvatures as above. Vanishing of one of these two curvatures is usually referred to as semi-flatness. Step (F): The description of the basic tractor bundles is simple in this case. Let T be the standard tractor bundle, i.e. the bundle corresponding to the standard representation of G on jRp+q. By definition, PeG is the stabilizer of IRP c IRp+ q , and the associated bundle to the representation of P on jRP is E. This means that T contains E as a smooth subbundle. On the other hand, the quotient representation of P on jRp+q /jRP is the trivial extension of the representation jRq of the subgroup Go c P. This corresponds to the bundle F, so we get a short exact sequence o -> E -> T -> F -> 0 of natural vector bundles. We will also indicated this short exact sequence (and more general filtrations later on) by writing T = E-f) F. Of course, this sequence does not admit a natural splitting since the representation of P on jRp+q is indecomposable. On the other hand, we can view E eTas a filtration of the vector bundle T, and then the associated graded bundle gr(T) is naturally isomorphic to E ED F. Since any irreducible representation of sl(p + q, jR) is isomorphic to a subrepresent at ion of some tensor power of the standard representation, any tractor bundle corresponding to an irreducible representation can be found in some tensor power of the standard tractor bundle. The filtration E c T gives rise to a filtration of any such tractor bundle, which may, however, be more complicated. As an example, consider the tractor bundles AkT for k = 2, ... ,p + q - 1 which correspond to the other fundamental representations. Then the number of nontrivial subbundles involved in the filtration is the minimum of p, q, and k, and the associated graded bundle has the form gr(AkT) = ffii+j=k Ai E 0 Ai F.
382
4. A PANORAMA OF EXAMPLES
4.1.4. An alternative interpretation in the case p = q = 2. We have observed in example (2) of 2.2.6 that .5[(4, q is isomorphic to .50(6, C), and we can construct an analog of this isomorphism for the real form 9 := .5[(4, lR). Consider the second exterior power A2lR4 of the standard representation. Since A4lR4 is a trivial 9-representation, the wedge product induces a 9-invariant nondegenerate symmetric bilinear form on the six-dimensional space A2lR4. Taking the standard basis e1,.'" e4, we see that the elements e1 /\ ei for i = 2,3,4 span a three-dimensional isotropic subspace. Hence, this bilinear form must have signature (3,3) and we obtain a homomorphism 9 --+ .50(3,3). This must be injective since 9 is simple and thus an isomorphism for dimensional reasons. The parabolic subalgebra p ~ 9 is exactly the stabilizer of the line through e1 /\ e2 (compare with 3.2.10), which is a null vector. Passing to the group level, we obtain a homomorphism from G = SL(4,lR) to SO(3,3), which maps the parabolic subgroup P to the stabilizer of the null line through e1 /\e2' Connectedness of G and the fact that we obtain an isomorphism on the Lie algebra level implies that this homomorphism maps G onto the connected component SOo(3,3) of the identity. The kernel of this homomorphism must be contained in the center ±il of G, so evidently it coincides with this center and the homomorphism is a two-fold covering. It is easy to check that the standard representation of G and its dual realize real forms of the two spin representations of 50(3,3), so our homomorphism actually identifies G with the spin group Spin(3, 3). Hence, we are in the situation of conformal structures of split signature (2,2) on four-dimensional manifolds as discussed (in higher dimensions) in 4.1.2. We can easily show explicitly that Go is the conformal spin group CSpin(2, 2). By definition, 9-1 is the space M2(lR) of real 2 x 2-matrices. The determinant defines a quadratic form on M 2 (lR) and since there clearly are two-dimensional subspaces in M2(lR) which consist entirely of matrices with zero determinant, the inner product inducing the determinant must have split signature (2,2). From 4.1.3 we know that Go ~ S(GL(2, lR) x GL(2, lR)) and given A, B E GL(2, lR) with det(A) det(B) = 1, the adjoint action on X E M2lR is given by BX A -1. Using the relation between the determinants of A and B, we get det(BXA-1) = det(B)2 det(X), so the adjoint action defines a homomorphism Go --+ CSO(2, 2). Evidently, the kernel of this homomorphism is {(il, il), (-il, -iln, and the quotient of G by this subgroup is connected. By dimensional reasons, we obtain a twofold covering Go --+ CSO o(2, 2) of the connected component of the identity of the conformal group. The two obvious two-dimensional representations of Go are easily seen to realize the two spin representations, so Go ~ CSpin(2, 2). In the conformal picture the interpretation of basic curvature components is fairly simple. The principal Go-bundle Po : go --+ M is a two-fold covering of the conformal frame bundle. Taking the Levi-Civita connection of any metric in the conformal class, we obtain a torsion-free connection on go. Then the two basic curvatures P1 and P2 can be computed directly from the curvatures of the induced connections on the two spin bundles using the formulae in 4.1.3. There is an alternative interpretation of these curvatures, which fits better to conformal structures in general dimensions. The curvature of a Levi-Civita connection on M is an element of A2T* M 0 50(T M). As in the identification of Go with CSpin(2, 2) above, one concludes that 50(TM) ~ 5[(E)EB.5l(F), where we denote by E and F the two real spin bundles. On
4.1. STRUCTURES CORRESPONDING TO i1i-GRADINGS
383
the other hand, as we have seen in 4.1.3, the isomorphism T M ~ E*®F gives rise to a decomposition A2T* M = (A2 E®S2 F*)EB(S2 E®A2 F*). One verifies directly that this is exactly the decomposition of two forms into self-dual and anti-self-dual twoforms; see the discussion of conformal structures in 4.1.2. Now so(TM) ~ A2T* M via the metric, so the two decompositions above are just two isomorphic pictures of the same decomposition. Moreover, it is well known that the curvature of a pseudo-Riemannian manifold actually has values S2(A2T* M) c A2T* M®A2T* M. Consequently, the curvature actually has values in
Recall from 1.6.8 that the Weyl curvature is the totally tracefree part of the Riemann curvature. This is also contained in S2(A2T* M) so it splits accordingly, and from the construction it follows that the two components are the curvatures P1 and P2. Hence, the two basic curvatures are exactly the self-dual and the anti-selfdual part of the Weyl curvature, and the two possible semi-flatness conditions are self-duality and anti-self-duality of a conformal structure. 4.1.5. Classical projective structures. Historically, these were among the first examples of parabolic geometries that have been studied. These structures form one of the two basic examples of geometries which are not determined by the underlying infinitesimal flag structure. Therefore, we can only follow the general scheme described in 4.1.1 in the beginning, but will have to deviate from it later on. Step (A): Fix n ~ 2 and consider the Lie algebra 9 = sl(n + 1, lR). The grading 9 = 9-1 EB 90 EB gl is the extremal case p = 1 of the one discussed in Example with X E lRn, 3.1.2 (2). Hence, we view elements of 9 as block matrices (- t~A) Z E lR n* and A E gl(n, lR). The entry X represents g-1, Z represents g1, while the block-diagonal part determined by A is go. The Dynkin diagram describing this grading is simply the An-diagram with the leftmost node crossed. There are two interesting choices for a Lie group G with Lie algebra g. One is the obvious choice G = SL(n + 1, lR). In this case, the good choice for P is to take the connected component of the identity in the subgroup determined by the
!)
grading. Hence, P is the subgroup of all matrices of the form (
detJC)
~),
where
C E GL(n, lR) has positive determinant. This is the stabilizer of the ray spanned by the first basis vector. Since G evidently acts transitively on the set of rays in lRn+1, we conclude that G / P is the space of all such rays and hence diffeomorphic to The subgroup Go C P is characterized by W = 0, so Go ~ GL+(n,lR), the group of invertible n x n-matrices with positive determinant. The second interesting choice of groups is G = PGL(n + 1,lR), the quotient of GL(n + 1, lR) by the closed normal subgroup consisting of all multiples of the identity. Here, we take P to be the maximal parabolic subgroup for the given grading. While G does not act on lRn+1, it does act on the space lRp n of lines through the origin in lRn+1. The subgroup P by definition is the stabilizer of the line generated by the first vector in the standard basis. Hence, G / P ~ lRpn and we exactly recover the situation discussed in 1.1.3. The subgroup Go C P is given by the classes of block diagonal matrices, and since any such class has a unique representative of the form (& ~) we see that Go ~ GL(n, lR) in this case.
sn.
4. A PANORAMA OF EXAMPLES
384
Below, we will treat both choices of groups simultaneously, by working in GL(n + 1,~) and taking into acount that we either work module scalar factors or restrict to matrices of determinant one. In this picture, P consists of the classes of matrices whose first column is a multiple of the first unit vector.
Steps (B) and (C): Consider a block diagonal matrix (0 So) E GL(n+I, ~). Then the adjoint action on 9-1 is given by X f---+ CXc- 1 . Notice that this is unchanged if we replace the matrix by a nonzero multiple. Conversely, if CXc- 1 = X for all X E ~n, then the matrix is a multiple of the identity. Hence, we conclude that for the case G = PGL(n + I,~) the adjoint action induces an isomorphism Go ---t GL(9-1). Likewise, if c = det(C) = 1, then we simply get the standard action of SL(n,~) on ~n. Finally, for>. > 0 we can form the (n + l)st root J.L := >.l/(n+1), and the matrix (1/( acts on 9-1 by multiplication by>.. Hence,
Sd)
for G = SL(n+ 1,~), the adjoint action induces an isomorphism Go ---t GL+(n,~). Hence, determining the basic irreducible representations and the corresponding vector bundles is very easy. The standard representation on 9-1 = ~n corresponds to the tangent bundle, and its dual on 91 corresponds to the cotangent bundle. Any irreducible representation of SL(n,~) can be realized within a tensor product of copies of these representations (see 2.2.13), and correspondingly irreducible bundles can be realized in tensor bundles. On the other hand, there are some natural line bundles. One series of such bundles comes from the one--dimensional representations given by C f---+ Idet(C)ICt for a E ~, so these are density bundles. Finally, for G L( n,~) there is one additional representation corresponding to A f---+ sgn( det( A)). The corresponding bundle with fiber 2:2 is a two-fold covering, usually called the orientation covering. Finally, one may use tensor products of the bundles obtained so far. Tensor products of tensor bundles with a density bundle are often referred to as weighted tensor bundles. At this point, we have to deviate from the standard route. This is due to the fact that HI (9-I,9) has a nontrivial component in homogeneity one. However, we can also see this directly from the developments so far. In the Ill-graded case an infinitesimal flag structure (which is automatically regular) on a manifold M is a reduction of structure group of the frame bundle PM corresponding to Ad : Go ---t GL(9-1). From above, we know that this is either an isomorphism, and hence contains no information at all, or it is the inclusion of the connected component of the identity, and hence we only obtain an orientation on M. In any case, we need more data to describe a Cartan geometry. From Proposition 3.3.7 we know that HI (9- b 9) has trivial components in homogeneities higher than one (since we have assumed n 2: 2). According to Theorem 3.1.16, parabolic geometries of type (G, P) are equivalent to normal P-frame bundles of degree one of type (G, P), so we have to understand these. P-frame bundles of degree one: We have already sketched in 1.1.3 some facts on projective equivalence of connections.
4.1.5. (1) Let M be a smooth manifold of dimension n 2: 2 and let \7 and V be two linear connections on the tangent bundle T M. Then \7 and V are called projectively equivalent if and only if there is a one-form T E n1(M) such that for all vector fields TJ E X(M) we have DEFINITION
e,
VeTJ = \7 eTJ + T(TJ)e
+ T(e)TJ·
4.1.
STRUCTURES CORRESPONDING TO 111-GRADINGS
385
(2) A projective structure on M is a projective equivalence class of linear connections on T M. Note that if Y' and V are projectively equivalent, then their difference is symmetric, so they have the same torsion. In particular, it makes sense to talk about a torsion-free projective structure on M. PROPOSITION 4.1.5. LetG = PGL(n+1,1R) andP C G the subgroup described above. Then there is an equivalence of categories between P-frame bundles of degree one of type G / P and projective structures. Under this equivalence, normal P -frame bundles exactly correspond to torsion-free projective structures. For G = SL(n + 1,1R) and P the subgroup described above, one obtains an analogous statement for oriented projective structures. PROOF. The general definition of of a P-frame bundle of degree one from 3.1.15 simplifies considerably in the Ill-graded case. The only ingredients are a principal P-bundle p : g -+ M and a one-form 9_ 1 E 0 1(9,9/91). In Tg we have two natural subbundles, the vertical subbundle Tog of p : g -+ M and the subbundle T1g C TOg spanned by the fundamental vector fields generated by elements of 91 C p. The defining properties of the frame form can be expressed in terms of (}-1 as: • (rY)*9_ 1 = Ad(g-1) 0 (}-1 for 9 E P, with Ad denoting the action on 9/91 induced by Ad. • For u E g the kernel of (}-1(U) is T~g. • 9_ 1 maps Togo to P/91, and for the fundamental vector field (A generated by A E p, we get (}-1(A) = A + 91 E P/91. As we have observed at the end of 3.1.15, one may pass from a P-frame bundle to an underlying infinitesimal flag structure. One defines go := g/ P+ and observes that projecting the values of 9_ 1 to 9/P, one obtains a one-form that descends to H. E 0 1(go, 9/P)· Now H. is Go--equivariant and strictly horizontal, so this is a first order Go-structure. From the description of Go above, we see that, depending on the choice of G, this gives either the full or the oriented frame bundle. Now take a point Uo E go and a tangent vector ~ E Tuogo. For a point u E g over Uo, choose a lift E Tug of~. Since is unique up to elements of T~g, the 90component of 9_ 1 (e) is indepent of the choice of the lift, so we obtain a well-defined element ,u(~) E 90. For A E 90, the fundamental vector field on g generated by A lifts the one on go, which immediately implies that ,u reproduces the generators of fundamental vector fields. Next, choose a local smooth section a : U -+ g and let Q:.. : U -+ go be the induced section of go. Any point in golu can then be uniquely written as Q:..(x) ·90 for some x E U and 90 E Go. Define,u E 0 1 (golu, 90) by
e
e
'u(Q:..(x) .90) := ,u(x)·yo. By construction, this reproduces the generators of fundamental vector fields. For hE Go, a tangent vector ~ E T.Q:(x).yogo, and a lift E Tu(x).Yog we see that Trh . is a lift of Trh~. Using this we see that
e
((rh)*,u)(Q:..(x). 90)(~) = 'u(Q:..(x)· 90 h)(Trh .~) is the 90-component of
9_ 1 (a(x)· 90 h)(Trh. e) = Ad(h- 1 )(9_ 1(a(x)· 90)(e)).
e
4. A PANORAMA OF EXAMPLES
386
Since Ad(h- 1 ) preserves the grading of g, this equals
Hence, "Yu is Go-equivariant, so it defines a principal connection on golu. Now let us determine how this depends on a. Given a, a general section & of glu has the form &(x) = a(x) . 90(X) exp(Z(x)) for smooth functions 90 : U -+ Go and Z: U -+ g1; compare with Theorem 3.1.3. Let us first assume that Z = 0, i.e. &(x) = a(x) . 90(X). Then by definition
"Ya-(Q:(x))
= "Ya-(x) = "Yu(x)-go(x) = "Yu(g:(x) . 90(X)) = "Yu(Q:(x)).
By equivariancy, this implies "Y& = "Yu, so we may restrict to the case &(x) a(x) . exp(Z(x)), and thus Q: = g:. For ~ E TQ:(x)go and a lift E Tu(x)g we get the lift Trexp(Z(x» . E Ta-(x)g. By construction, "Ya-(g:(x))(~) is the go-component of
€
€
8_ 1(&(x))(Trexp(Z(x)) .~) = Ad(exp( -Z))(8_1(a(x))(~)). This is the sum of the go-component of 8_1(a(x))(~) (which equals "Yu(~)) and the bracket of -Z with the g_1-component of 8_1(a(x))(~), which simply represents the tangent vector~. Now one immediately computes that for Z E g1 and X E g-1, the action on the bracket [-Z,X] E go maps Y E g-1 to XZY + ZXY. (In the first term we evaluate Z on Y and multiply X by the result, and in the second term it is the other way round.) But this means that looking at the linear connection V' on T M associated to "Yu then the connections associated to "Ya- run exactly through the projective equivalence class of V'. Since principal (or linear connections) can be patched together smoothly, we conclude that a P-frame bundle of degree one induces a projective structure. From this construction it is also clear that if we have a morphism of P-frallle bundles, then the base map is compatible (in the obvious sense) with the projective structures. Conversely, assume that we have given a projective equivalence class [V'] of linear connections on a manifold M. Let PM be the (oriented) linear frame bundle of M, and let 8 E n1(PM,g_1) be the soldering form (where we identify IR.n with g-1). Given a point y E PM over x E M, define gy as the values in y of the principal connections associated to the linear connections in the projective class. We can view the disjoint union g of all gy as a subset of the bundle T*P M I3l go of go-valued forms, where we identify gl( n, IR.) with go. We can view PM as a principal bundle with structure group Go, and we want to extend the principal right action to a P-action on g. Given yEP M, a connection form "Y(y) and 9 = 90 exp(Z) E P, we define "Y(y) . 9 to be the connection form at y . 90 defined by ~
1--+
"Y(y . 90)(~) + [Z, 8(~)].
Explicitly, the first summand equals Ad(90 1) (",((y) (Tr9 01 . ~)). Since 8 vanishes on vertical vectors, we immediately conclude that this map reproduces the generators of fundamental vector fields. Hence, we obtain a connection form. Moreover, as above, we conclude that varying Z these forms exactly run through the connections in the projective class. Acting with another element 9b exp(Z') we obtain
4.1. STRUCTURES CORRESPONDING TO 111-GRADINGS
387
Using equivariancy of () and the fact that the adjoint action is by Lie algebra homomorphisms, the second summand can be rewritten as [Ad(go1 )(Z), ()(e)]. Since
go exp(Z)gb exp(Z')
= gogb(gb)-1 exp(Z)gb exp(Z') = gogb exp(Ad((gb)-1 )(Z)+Z'),
we have really defined an action of P on g, which is free and transitive on each fiber. A local smooth section of PM together with the choice of a connection in the projective class gives rise to a local smooth section of g. Hence, g is a smooth principal P-bundle. We can define ()-1 E 0 1(g, g-1 EB go) tautologically by
()-1(-Y(y))(e) = (() EB 'Y(y))(T7l" e), where 7l' : g -7 PM is the projection. One easily verifies directly that this makes g into a P-frame bundle of degree one, and clearly this construction is functorial. Hence, we have established the equivalence, and it remains to relate normality to torsion freeness. If 'Y is the principal connection on PM corresponding to a linear connection V' and () is the soldering form, then the torsion is induced by
d()(e, 17)
+ 'Y(e) (()(17)) -
'Y(17) (()(e));
see 1.3.5. Viewing () as having values in g-1 and'Y as having values in go, this can be rewritten as d()(e, 17) + b(e), ()(17)] + [()(e), 'Y(17)]· In the equivalent picture of P-frame bundles of degree one, () corresponds to the g_1--component of the frame form ()-1, while 'Y corresponds to its go-component. By the compatibility conditions on frame forms from 3.1.15 we can equivalently view 'Y as the go--component of ()o (or any extension 00 of ()o to a form defined on all of Tg). But then the above expression just computes the g_1--component of
d()-1(e, 17) + [Oo(e), ()-1(17)]
+ [()-1(e), 00 (17)],
which by definition computes the torsion of the P-frame bundle. We will see immediately below that 8* is injective on the homogeneous part of degree one, so normality of the P-frame bundle is equivalent to vanishing of the torsion of the connections in the projective class. 0 Step (D): We have already determined the Hasse diagram in 3.2.17. There is only one element of length two, which corresponds to acting first with the second simple reflection and then with the first one. The set w corresponding to this element of length two is {0:1' 0:1 + 0:2}. Notice that the weight vectors with these weights are exactly the highest and the second highest weight vectors of g-1. According to Theorem 3.3.5, the other thing we have to determine is the image of the maximal root 0:1 + ... + O:n under w. For n = 2, this is -0:1, while for n > 2 it is 0:2 + ... +O:n. The latter weight evidently is the highest weight of go. Hence, for n > 2, Theorem 3.3.5 shows that the cohomology H 2 (g_, g) is dual to the highest weight component in A2g_ 1 @ go. In particular, this cohomology is concentrated in degree 2. Since any nonzero map A3g_ 1 -7 9 is homogenous of degree ~ 2, we see that im(8*) C £(A2g_1,g) is contained in homogeneous degree ~ 2. Since there is no cohomology in degree one, we conclude that 8* : £(A2g_1' g) -7 £(g-1' g) is injective on maps homogeneous of degree one, which was the last open point in the proof of the proposition above.
388
4. A PANORAMA OF EXAMPLES
Likewise, in the C88e n = 2, the cohomology H2(g_,g) is dual to the highest weight component in A2g_1 ®g-l. In particular, this cohomology is concentrated in homogeneous degree 3, and the necessary fact on injectivity of 8* follows 88 before. Step (E): Ai; we have noted in the proof of Theorem 4.1.1, the proof remains valid if one starts with a connection 'Y on an infintesimal flag structure (E,9) such that (E x P+, 9$'Y) is a P-frame bundle of degree one. Hence, we may apply the results of Theorem 4.1.1 interpreting "distinguished connection" as "connection from the projective class". Now we can easily compute the Rho tensor in our setting. We use abstract indices and denote by Ri/ l the curvature of a fixed connection V in the projective class, with the first two indices being the form indices. By definition,
To express this in abstract indices, we have to recall that we identify go with L(g_1I g-d via the adjoint action. Now for elements X, Y E g-1 and Z E gl ~ g~1 we get [[X,Z], Y] = Z(X)Y +Z(Y)X. For P we use the abstract index notion fixed by p(e)j = Pijei . Then we get
which in turn immediately implies (8(P»i/l = OfPjl- OjPil- PijO} + Pjio}. To obtain the formula for gro(8*), we observe that in the Ill-graded case the formula for 8* from 3.1.12 simplifies to
Since the bracket between Endo(TM) and T* M is (up to a nonzero factor) just evaluation of the dual map, we see that, up to a factor, 8* is just given by contracting the upper index into one of the form indices, and we use the second form index. Then (8(P»ik kj = -nPij + Pji, and the defining property is that this equals Rij := -Rikkl (which gives the usual sign to the Ricci type contraction). Symmetrizing we obtain -(n - I)P{ij) = R{ij) and alternating we get -(n + I)P[ij] = R[ij]. This implies Pij
=
(n-=!I)R(ij) - {n~I)R[ij]
=
(n i')ln+1) (nRij
+ Rji).
In view of part (1) of Theorem 4.1.1, this gives a complete description of the normal projective Cartan connection, and the description of harmonic curvatures can be deduced directly from part (2) of that theorem. There is never cohomology in homogeneity one, and thus no harmonic curvature in this homogeneity. This reflects the fact that we deal with torsion-free projective cl88ses. If n > 2, the H2(g_l, g) is concentrated in homogeneity 2 and the Weyl curvature R + 8(P) of any connection in the projective class is a complete obstruction to local flatness. For n = 2, cohomology sits in homogeneity 3, Weyl curvature vanishes identically, so R = 8(P) and the complete obstruction to local flatness is the Cotton-York tensor dVP of any connection V in the projective class.
4.1.
STRUCTURES CORRESPONDING TO 111-GRADINGS
389
Step (F): To have the full supply of tractor bundles, we consider the case G = SL(n + 1,JR) of oriented projective structures. By definition, PeG is the stabilizer of the ray generated by the first basis vector in the standard representation V := JRn+1. Hence, this vector spans a one-dimensional invariant subspace VI C V. The associated bundle corresponding to the representation V is the standard tractor bundle T. Associated to VI, we get a line subbundle T1 c T. Now a matrix (det(~)-l ~) in P acts on VI by multiplication by det(C)-1. On the other hand, it acts on g-1 by X 1-+ CX det(C), which has determinant det(C)n+1. Hence, we conclude that, using the convention that 1densities can be integrated, the line subbundle T1 is isomorphic to the bundle of n~1-densities. It is common to denote the latter bundle by £(-1); see [BEG94]. Consequently, in this convention £ (w) denotes the bundle of - n~ 1-densities for each w E JR. From the block decomposition defining the grading of g it is clear, that g-l ~ L(V1, V/V1) as a P-module. Therefore, we see that T/T1®£(1) ~ TM and hence T /Tl = T M (-1) := T M ® £ ( -1). The description of T can be conveniently phrased as a composition series T ~ £(-1)-8 TM(-l). From this composition series it is easy to see the composition series of other tractor bundles. For example, taking into account that an inclusion dualizes to a quotient map, the composition series for the standard cotractor bundle is T* ~ T* M(l)-8 £(1). It turns out (see [BEG94]) that T* can be identified with the first jet prolongation of £(1). The adjoint tractor bundle A = s£(T) can be realized as the tracefree part in T*®T, and the resulting composition series is A ~ T* M -8 (T* M® T M) -8 T M. This of course corresponds to the P-invariant composition series g ~ g1-8 go-8 g-1· For other tensor products of tractor bundles one can proceed similarly. For example, using that S2£( -1) = £( -2) we get S2T ~ £( -2)-8 TM( -2)-8 S2TM( -2), while A2T ~ TM(-2)-8 A2TM(-2). 4.1.6. Projective structures and geodesics. In 4.1.5 above, we have defined a projective structure as an equivalence class of linear connections on the tangent bundle. There is, however, a more geometric interpretation of projective equivalence, that we will briefly discuss here. The basis for this interpretation is the following: PROPOSITION 4.1.6. Consider two linear connections V' and V on the tangent bundle of a smooth manifold M of dimension n ;::: 2 which have the same torsion. Then 'V and V are projectively equivalent if and only if they have the same geodesics up to parametrization. PROOF. The first step is to show that a smooth curve c : I ~ M is a geodesic for V' up to reparametrization if and only if V' c' c' is always proportional to c'. Indeed, by the chain rule one immediately verifies that a reparametrization of a geodesic has this property. Conversely, the same computation shows how the factor of proportionality can be used to set up an ODE whose solution gives a reparametrization to a geodesic. Now suppose that V(rJ = 'V(rJ + T(e)'T] + T('T])e for some T E 01(M). If c: I ~ M is a geodesic for V', then along c we get VC1c' = 2T(c')c'. This implies that 'V and V have the same geodesics up to parametrization.
390
4.
A PANORAMA OF EXAMPLES
Conversely, suppose that \7 and "(7 have the same torsion and the same geodesics up to reparametrization. Consider the difference tensor A(~,7]):= "(7~7] - \7~7].
Since the connections have the same torsion, this tensor is symmetric, and since any tangent vector occurs as the derivative of some geodesic, we conclude that A(~,~) must be a multiple of ~ for each~. Define a: TM -lR. by A(~,~) = a(~)~. Then evidently a is homogeneous of degree one and a(O) = O. Using that A is bilinear and symmetric, we obtain A(~
+ 7], ~ + 7]) = A(~,~) + 2A(~, 7]) + A(7], 7]),
and inserting we get (4.4)
2A(~,
7])
= (a(~
+ 7]) -
a(~))~
+ (a(~ + 7]) -
a(7]))7].
Now expand the equation A(~, t7]) = tA(~, 7]). Assuming that ~ and 7] are linearly independent and t -:f 0, this implies a(~ + t7]) - a(t7]) = a(~ + 7]) - a(7]). Taking the limit t - 0 in the left-hand side, we conclude that a is additive and hence defines a one-form. Then (4.4) reads as 2A(~, 7])
= a(7])~ + a(~)7],
and we have established projective equivalence.
o
This result means that a projective structure can be equivalently described by the family of unparametrized curves defined by the geodesics. It is a bit awkward to formulate the definition precisely in these terms, so instead we will concentrate on the resulting description of morphisms. We will take up the issue of geometries given by families of curves again in the discussion of (generalized) path geometries; see 4.4.3. The relation between projective structures and these path geometries is a nice example of the general concept of correspondence spaces. COROLLARY 4.1.6. Let (M, [\7]) and eM, [V]) be projective structures on smooth manifolds, and let f : M - M be a local diffeomorphism. Then f is a morphism of the projective structures, i.e. j*V is projectively equivalent to \7 if and only if for any geodesic c : I - M of \7 the composition f 0 c is a geodesic of V up to parametrization. PROOF. Of course, f 0 c is a geodesic of V (up to parametrization) if and only if c is a geodesic of j*V (up to parametrization). Hence, the result follows 0 immediately from the proposition above. 4.1.7. Some background on quaternions. Next, we consider the real form sl(n + 1,1HI) of sl(2n + 2,«::). Let us first recall some background on quaternions. The quaternionic multiplication is compatible with the Euclidean norm on 1HI = lR. 4 , so Ipql = Ipllql for all p, q E 1HI. For q E 1HI, one has the conjugate quaternion ij. The basic properties of the conjugation are pq = ijjJ and qij = ijq = Iq121, so q-l = Iq121 ij. A quaternion is called real if ij = q and purely imaginary if ij = -q. The real quaternions are exactly the real multiples of 1, and the purely imaginary ones form a complementary three-dimensional subspace im(lHI) C 1HI. Hence, any q can be split as re(q) + im(q) into real and imaginary part. If q is purely imaginary, then q2 = -qij = -lqI21, so the square roots of -1 form a two-sphere in 1HI. The standard
4.1. STRUCTURES CORRESPONDING TO 111-GRADINGS
391
inner product on ]R4 can be written in quaternionic terms as (p, q) = re(pq). In particular, IH! is the orthgonal direct sum of re(lH!) and im(IH!). An important difference between IH! and the fields ]R and 1, the case n = 1 will be discussed separately below. Step (A): Put 9 := .51(n + l,lHl), the Lie algebra of quaternionic (n + 1) x (n + I)-matrices with vanishing real trace. From 3.2.10 we know that the parabolic subalgebra lJ C 9 corresponding to the second simple root is the stabilizer of the quaternionic line spanned by the first element of the standard basis. Hence, we have a presentation as block matrices of the form (~ ~) with blocks of size 1 and n, where X, E lHl n , a E lHl and A E Mn (lHl) , and re(a) + re(tr(A» = O. The entries a and A span 90, X spans 9-1 ~ lHln and Z spans 91 ~ LIHI(lHln,lHl). Hence, this is the quaterionic analog of the algebraic background used to describe classical projective structures in 4.1.5. As the group G we choose the group PGL(n+l, lHl), the quotient of all invertible quaternionic linear endomorphisms of lHl n+1 by the closed normal subgroup of all real multiples of the identity. Then we define PeG to be the (quotient of the) stabilizer of the quaternionic line spanned by the first basis vector. Hence, the homogeneous space G / P can be identified with the space lHlpn of quaternionic lines in lHln+1, the quaternionic projective space. The subgroup Go C P consists of the classes in P of all block diagonal matrices, i.e. the quotient of
zt
{ (6 g): 0 =1= q E lHl,
2. Under this assumption, we see from the list of Satake diagrams in Appendix B, that this grading exists for two real forms, but we will restrict to the geometry corresponding to the split real form .sp(2n, jR).
Step (A): From 2.2.13 we know that the fundamental representation of sp(2n, jR) corresponding to the last simple root can be realized as a subspace of AnjR2n, namely as the kernel of the natural map to An-2jR2n induced by the symplectic form on jR2n. Hence, we may realize the corresponding parabolic subalgebra peg := .sp(2n, jR) as the stabilizer of the highest weight line in that representation. In the matrix
4.1. STRUCTURES CORRESPONDING TO 111-GRADINGS
399
realization for 9 from example (4) of 2.2.6, the highest weight line is spanned by the wedge product of the first n basis vectors, and we get
with n x n-matrices A, Band C such that Band C are symmetric. Hence, p is the stabilizer of the isotropic n-dimensional subspace spanned by the first n basis vectors. Choosing the group G := Sp(2n, lR) we may therefore use the stabilizer P of that subspace as a parabolic subgroup for the given grading. This means that P consists of all those matrices in G which are block upper triangular with two blocks of size n. From this description it is clear that the homogeneous model G / P is the space of all isotropic n-dimensional subspaces in the symplectic vector space lR2n. Such maximal isotropic subspaces are usually called Lagrangean subspaces, whence the variety of all these subspaces is referred to as the Lagrange-Grassmann manifold. This is also the reason why the corresponding geometries are called almost Lagrangean. The Levi subgroup Go C P is formed by the block diagonal matrices contained in G. This preserves the subspaces spanned by the first, respectively, last n basis vectors, and it is easy to see that the action of Go on either of these subspaces induces an isomorphism Go ~ GL(n,lR). We fix the identification determined by the subspace spanned by the last n basis vectors, which is also in accordance with the matrix presentation for the Lie algebras above. The reason for this choice is that then we obtain 9-1 = S2lRn, and 91 = S2lR n*, which fits with these two spaces corresponding to the tangent respectively cotangent bundle.
Step (B): Since the semisimple part of 90 is .s[(n, lR), the discussion of all Gorepresentations is just a simplified repetition of the Step (B) in 4.1.3. Indeed, we are again dealing with a split real form, so there is no difference between the complexified real and complex representations. The fundamental weights A1>' .. An-1 of 9 lead to the exterior powers AklR n , k = 1, ... , n - 1, as the p+ -irreducible subspaces of the same highest weight. The last fundamental weight gives rise to the one dimensional representation on which the grading element acts by the scalar ~. As we have seen in 3.2.12, the grading element acts on the weight A = alAI + .. ·+anAn by the scalar computed with the help of the last row of the inverse Cartan matrix (determined by the position of the cross in the Satake diagram), i.e. in our case by L~=l ~ai. The grading of the Lie algebra has got irreducible components with highest weights 1
0
1-1
O---O-"'~EB 0
0
0
o
0
2
-2
0---0- .. ·~x
0
2
0
0
0
EB o---o- ... ~
o---o- ... ~
Step (C): As we have seen in Step (A), we get Go ~ GL(n, lR) and 9-1 ~ S2lR n as a representation of Go. From this it follows easily that a first order structure with structure group Go on a manifold M of dimension n(~+l) is given by an auxiliary rank n-vector bundle E --+ M and an isomorphism : S2 E --+ T M. Given the Go-structure, we simply obtain E as the associated bundle with respect to the standard representation. Conversely, given E and an isomorphism : S2 E --+ T M, let go be the full linear frame bundle on E. Together with -1, a point in the fiber of go over x induces a linear isomorphism TxM --+ S2 Ex --+ S2lR n ~ 9-1, which we
400
4. A PANORAMA OF EXAMPLES
can use to define a soldering form. In the homogeneous model, the bundle E is the dual of the tautological bundle.
Steps (D) and (E): There is only one element of length two in the subset WP c W in the Weyl group, namely Sn 0 Sn-l. One easily verfies that the corresponding cohomology component sits in homogeneity -1. Now A2(S2]Rn*) is an irreducible representation of .5l(n, ]R), so our cohomology component is the highest weight subspace in the space A2(s2Rn*) ® S2Rn, which corresponds to A2T* M ® TM. This implies that the harmonic curvature is concentrated in one torsion component and this part of the torsion will be shared by all the distinguished connections 'Y; cf. Theorem 4.1.1. To describe this highest weight bit, one observes that, up to multiples, there is a unique contraction A2 (s2Rn*) ® S2Rn -+ ®3R n* ® Rn. The highest weight bit is simply the kernel of this contraction. This is easily translated to geometry. To use an abstract index notation, we denote the auxiliary bundle E by eA and its dual E* by eA. Then the tangent bundle is isomorphic to S2 E, so it has two symmetric upper indices. Likewise, the cotangent bundle has two symmetric lower indices. Since go -+ M is the full linear frame bundle of E, a principal connection on go is equivalent to a linear connection on E. Via the isomorphism S2E ~ TM, such a connection induces a linear connection on T M. In abstract index notation, the torsion of such a connection then has the form T = TABCD EF with symmetry in each of the pairs AB, CD and EF of indices, and such that TCDABEF = -TABCDEF. Evidently, all contractions of one of the upper indices into one of the lower indices agree up to sign. The basic invariant of an almost Lagrangean structure is the part of the torsion, which is contained in the kernel of this trace. An explicit formula for this tracefree part can be computed along the lines of what we did for almost Grassmannian structures in 4.1.4. Then one may continue to compute one of the preferred connections and determine the corresponding Rho tensor, but we do not go into detail on these issues.
Step (F): Since all representations of Sp(2n,R) live in the tensor products of the first fundamental representation, i.e. the standard representation on R2n, all tractor bundles can be obtained from tensor products of the standard tractor bundle. The standard tractor bundle T = g x p ]R2n comes equipped with a canonical symplectic structure inherited from the skew symmetric bilinear form on ]R2n used to define .5p(2n, ]R). In view of our conventions described in Step (A) it is clear that T contains a natural subbundle Tl ~ E*, and the quotient T ITl is isomorphic to E. This gives rise to natural filtrations on tensor powers of the standard tractor bundles and hence on general tractor bundles. 4.1.12. Almost spinorial structures. From 3.2.3 we know that on complex Lie algebras of type Dm we always have the Ill-grading, which leads to conformal structures as discussed in 4.1.2. For n = 4, this is the unique Ill-grading up to automorphism, but for n ~ 5, there is a second, non-isomorphic Ill-grading. This corresponds to either of the last two simple roots. From the list of Satake diagrams in Table BA in Appendix B we conclude that there is a corresponding real Ill-grading on the real forms of type .5o(n,n) and .50*(4£). We will restrict our discussion to the split real form .5o(n, n) for n ~ 5. The interpretation of the
4.1. STRUCTURES CORRESPONDING TO 111-GRADINGS
401
geometric structures to be discussed below also makes sense for n = 4, but there the result is equivalent to split signature conformal structures in dimension 6. Steps (A) and (B): We put 9 := so(n, n) for n ~ 5. Following the general principles from 3.2.5, we can realize the parabolic subalgebra p c 9 as the stabilizer of a highest weight line in any irreducible representation whose highest weight is a multiple of the last fundamental weight. One possibility would be to take a spin representation, but it is easier to use the representation on self-dual n-forms. In the matrix presentation from example (2) of 2.2.6 (in which the real matrices form the split real form 9 of so(2n, q) it is clear that the resulting highest weight line is spanned by the wedge product of the first n basis vectors. Hence, in terms of block matrices with two blocks of size n each we get p= {
(_~t ~)} c 9 = { (_~t ~)},
where the matrices Band C are skew symmetric. This indicates already that these geometries are very similar to almost Lagrangean structures. As the group G with Lie algebra 9, we choose SOo(n, n), and then we can take the stabilizer of the isotropic n-dimensional subspace spanned by the first n basis vectors as the parabolic subgroup P for the given grading. Hence, these are block upper triangular matrices and the Levi subgroup Go corresponds to block diagonal matrices. The homogeneous model G j P can be realized as the space of all selfdual isotropic n-dimensional subspaces in R(n,n). However, using the spin group rather than G, one may also view GjP as the orbit of the highest weight line in the projectivized spin representation. This orbit consists of the lines spanned by the so-called pure spinors, which also motivates the name "almost spinorial structures" for the corresponding geometries. The action of Go on the space spanned by the last n basis vectors induces an isomorphism Go ~ GL(n, R), which again shows the similarity to the almost Lagrangean case. In view of this similarity, we keep this presentation short. The convention is chosen in such a way that 9-1 ~ A2Rn and gl ~ A2RM as representations of Go. As before, all the basic representations of Go can be obtained from tensor powers of the standard representation and its dual and from one-dimensional representations. Step (C): First order structures with structure group Go C GL(9-1) make sense on manifolds of dimension n(~-l). We have seen above that 9-1 ~ A 2 Rn as a G omodule, which leads to a description analogous to almost Lagrangean structures. Similarly to that case, one proves that these structures are given by an auxiliary vector bundle E ofrank n together with an isomorphism CP: A2E ~ TM. On the homogeneous model, E is again the tautological bundle. Let us briefly describe why these structures specialize to conformal structures for n = 4. The decomposable elements in A2 E define a cone in each tangent space. By the Pliicker relations, an element is decomposable if and only if the wedge product with itself is trivial. But if n = 4, then this wedge product has values in the line bundle A4 E, so any trivialization of this line bundle gives a metric on A2 E having the given cone as its null cone. Alternatively, for n = 4 decomposability is equivalent to non-invertibilty of an element viewed as a skew symmetric matrix. The determinant on antisymmetric matrices allows a square root called the Pfaffian, and this is a quadratic form for n = 4.
402
4. A PANORAMA OF EXAMPLES
Steps (D) and (E): There is only one element oflength two in the subset WP c W, namely Sn 0 Sn-2. The corresponding cohomology component sits in homogeneity -1 (this uses n > 4) and it is the highest weight subspace in A2(A2jRn*) ® A2jRn. This highest weight subspaces can be characterized as the kernel of the unique (up to multiples) nonzero contraction A2(A2jRM) ® A2jRn _
®3jRn* ® jRn.
Geometrically, this means that the full harmonic curvature will be given as the tracefree part of the torsion of any connection on the Go-structure. The connections on T M coming from the Go-structure are exactly those, which are induced from linear connections on E via the isomorphism A2 E ~ T M. Step (F): The standard tractor bundle 7 = 9 xp jR2n comes equipped by the canonical split signature scalar product inherited from the defining one on jR2n. Moreover, it comes with a natural subbundle 7 1 c 7 which is isomorphic to E* and such that 7 /71 ~ E. This gives rise to filtrations on tensor products, and hence on more general tractor bundles. Enlarging G to the spin group Spin(n, n) also the spin representations give rise to tractor bundles, but we do not go into details here. 4.2. Parabolic contact structures These are parabolic geometries for which the underlying geometric structure is a contact structure with some additional structure on the contact subbundle. The most important example is provided by CR-structures, for which the additional structure is a complex structure on the contact subbundle. We start by recalling some background on contact structures. 4.2.1. Contact structures and contact connections. Recall from linear algebra that nondegenerate skew symmetric bilinear forms exist only on vector spaces of even dimension, and there they are uniquely determined up to isomorphism. Looking at an even dimensional smooth manifold M, a smooth family of skew symmetric bilinear forms on the tangent spaces of M is just a two-form T on M. A form T E n2(M) such that Tx : TxM x TxM -jR is nondegenerate for each x EM, is called an almost symplectic form on M, and T is called a symplectic form if, in addition, dT = O. In this case (M, T) is called a symplectic manifold. Note that the nondegeneracy condition can also be expresses as the fact that T /\ ... /\ T (with half the dimension many factors) is a volume form on M. Since there is an obvious symplectic form on jR2n, they exist locally on any manifold of even dimension. The question of global existence of symplectic structures is surprisingly difficult with lots of recent progress, for example, via the Seiberg-Witten equations; see e.g. [Tau94J. The standard example of a symplectic structure is provided by the cotangent bundle T* N of an arbitrary smooth manifold N. This carries a tautological one form a E nl(T* N) defined as follows. Let n : T* N - N be the projection and Tn : TT* N - TN its tangent map. Then for ¢ E T; N and ~ E T¢T* N one defines O'(¢)(O := ¢(Tn . ~). Choosing local coordinates qi on M and using the induced coordinates (qi ,Pi) on T* N one gets 0'= L,i Pidqi and hence dO' = L, dpi/\dqi. This is immediately seen to be nondegenerate and thus defines a symplectic structure. Putting N = jRn, we obtain the (constant) standard linear symplectic structure on
4.2. PARABOLIC CONTACT STRUCTURES
403
JR2n, which usually is written in terms of local coordinates as ((Xl, ... , X2n), (Yl, ... , Y2n))
f-t
L:~=l (XiYn+i - Xn+iYi).
A basic result in symplectic geometry is the Darboux theorem, which states that for any symplectic form 7' on M and around each point x EM, there exist local coordinates (qi, Pi) for M such that 7' = L: dPi 1\ dqi. In particular, symplectic structures do not have any local invariants. Looking for an odd-dimensional analog of this concept, one is led to the notion of a contact form a E ~V(M) on a smooth manifold M of dimension 2n + 1, i.e. a form such that al\(da)n is a volume form on M. Again there are simple examples of such forms on JR2n+1 and hence locally on any manifold of odd dimension. Namely, using coordinates (t, qi,Pi), one defines a = dt+ L:Pidqi, which is immediately seen to be a contact form. There is a contact version of the Darboux theorem which says that given any contact form, one may locally choose coordinates in which it is given by the above expression. So contact forms do not have local invariants either. If a is a contact form on M, then by definition a(x) =f 0 for all x E M, so the pointwise kernels of a form a codimension one subbundle H of the tangent bundle T M, called the contact sub bundle. Again by construction, the restriction of da to H x H is a nondegenerate skew symmetric bilinear form. The quotient bundle Q := T M / H is a real line bundle, which is trivialized by a. Defining T-2 M = T M and T- 1 M := H, we obtain a filtration of the tangent bundle of M with associated graded gr(TM) = Q EB H. The condition on compatibility with the Lie bracket from Definition 3.1.7 is vacuous in this case, so this makes M into a filtered manifold, and we can look at the Levi bracket £, : H x H -. Q induced by this filtration. For sections ~ and rJ of H = ker( a) the definition of the exterior derivative implies that da(~, rJ) = -a([~, rJ]), so we see that, viewing a as a trivialization of Q, we have a 0 £, = -da. Hence, £, is nondegenerate, and the symbol algebra in each point is JR EB JR2n with the bracket JR2n X JR2n -. JR being nondegenerate. This graded Lie algebra is called the real Heisenberg algebra ~2n+l' Now one defines a contact structure on a smooth manifold M of dimension 2n + 1 as a smooth subbundle H c T M of rank 2n such that, putting Q = T M / H, the Levi bracket £, : H x H -. Q is nondegenerate in each point. More elegantly, one can say that a contact manifold is a filtered manifold for which each symbol algebra is a Heisenberg algebra. As we have observed in 3.1.7, the last statement implies that for any contact structure H c TM, the associated graded to the tangent bundle, gr(TM) = Q EB H has a natural frame bundle with structure group Autgr(~2n+d, the group of automorphisms of the graded Lie algebra ~2n+l' The fiber of this bundle over x E M is just the space of isomorphisms ~2n+l -. (Qx EB H x ,[') of graded Lie algebras. Let us first determine the group Autgr(~2n+d. LEMMA 4.2.1. Any automorphism of ~2n+1 is uniquely determined by its restriction JR2n -. JR 2n . Viewed as a subgroup of GL(2n, JR), the group Autgr(~2n+d is generated by Sp(2n, JR), multiples of the identity, and the diagonal matrix lIn,n with the first n entries equal to 1 and the other n equal to -1. The subgroup of those automorphisms, which, in addition, preserve an orientation on JR C ~2n+l is the conformal symplectic group CSp(2n, JR) generated by Sp(2n, JR) and multiples of the identity.
404
4. A PANORAMA OF EXAMPLES
PROOF. As before let [ , ] : JR2n X JR2n -+ JR be the bracket in ~2n+1 given by the standard symplectic form on JR n . Surjectivity of this bracket immediately implies the first statement. For A E Sp(2n, JR) we by definition have [Ax, Ay] = [x, y], so this gives rise to an automorphism. Likewise, for a E JR\O, we get [ax, ay] = a 2 [x, y], so multiples of the identity are in Autgr(~2n+1)' too. Finally, by definition of [ , ], we see that [ITn,nx, ITn,nY] = -[x, y]. Conversely, suppose that A E GL(2n, JR) and a E JR \ 0 are such that [Ax, Ay] = a[x, y]. If a < 0, then replace A by ITn,nA to get an element for which a > O. Dividing then by Va, we obtain an element which preserves [ , ] and thus lies in Sp(2n, JR). The last statement is then obvious. 0
The first statement implies that any isomorphism 1J2n+1 -+ (Hx EB Qx, Cx) is uniquely determined by the component mapping JR2n to Hx. Hence, the natural frame bundle for H EB Q can be viewed as a subbundle of the frame bundle of H. If we choose an orientation on Q (for example the one induced by a contact form), then the structure group of the natural frame bundle is reduced to CSp(2n, JR). We collect some important properties of contact structures in the following: PROPOSITION 4.2.1. Let He TM be a contact structure on a smooth manifold M of dimension 2n + 1 with quotient bundle Q = T M / H. Let p : E -+ M be the natural frame bundle for H EB Q with structure group Autgr (~2n+l)' Let C : A2H -+ Q be the Levi bracket and let AgH c A2H be the kernel of C. (1) Locally, there exists a contact form 0: which has H as its contact sub bundle, and this form is unique up to multiplication by a nowhere vanishing function. In particular, contact structures have no local invariants. There exists a global contact form 0: for H if and only if the quotient bundle Q is orientable and hence trivial. (2) Any principal connection on E is completely determined by the induced linear connection on the vector bundle H. A linear connections on H arises in this way if and only if the induced connection on A2 H preserves the subbundle AgH. (3) If 0: E (21(M) is a contact form with contact subbundle H, then there is a unique vector field r on M such that o:(r) = 1 and irdo: = O. In particular, 0: induces an isomorphism T M ~ H EB JR. (4) Given 0: as in (3), there is a linear connection V on TM such that V preserves the subbundle H, Vo: = 0, Vdo: = 0, and Vr = 0, and such that the restriction to H is induced by a principal connection on E as in (2). PROOF. (1) The bundle Q = T M / H is locally trivial, and a local trivialization of this bundle can be viewed as a local one-form on M whose kernel in each point is the fiber of H. In this picture, do:IA2H = -0: 0 C, which implies that 0: is a contact form. Conversely, a local contact form with contact subbundle H factors to a local trivialization of Q, so we get a bijective correspondence. From this, (1) follows immediately. (2) Since E can be viewed as a subbundle of the frame bundle of H, a principal connection on E is uniquely determined by the induced linear connection on H. On the other hand, by definition, a linear map A E GL(2n, JR) extends to an automorphism of ~2n+1 if and only if the induced map on A2JR2n* preserves the line generated by [ , ]. By duality this is equivalent to the fact that the induced map on A2JR2n preserves the kernel of [ , ]. From this, the description of the induced connections follows immediately.
0:
405
4.2. PARABOLIC CONTACT STRUCTURES
e
(3) Since a is nowhere vanishing, we can locally find a vector field such that a(e) is nowhere vanishing, and mutliplying by an appropriate function we may assume a(e) = 1. Then we can look at the restriction of ieda to H, which defines a section of H*. By nondegeneracy, there is a section 7] of H such that ieda = i1)da, and r := 7] has the required properties. If r has the same properties, then a(r-r) = 0, so r-r E r(H). But then ir_rda = 0 implies r = r by nondegeneracy of da on H, and uniqueness follows. The isomorphism T M ~ H EB ~ is then given a(e)r, a(e))· by ~ (4) The choice of contact form a reduces the structure group of the frame bundle E from part (2) further to Sp(2n, ~). Explicitly, the fiber over x E M of this reduction is given by all linear isomorphisms ¢ : ~2n -> Hx for which da(¢(v), ¢(w)) = [v, w]. This bundle admits a principal connection, and we take the induced linear connection on H and extend it by the trival connection to a connection on H EB~. Via the isomorphism from (3), this gives a linear connection on TM which preserves the subbundle H. By construction of the frame bundle, this linear connection satisfies 'Vda = 0 and by the trival extension we have 'Vr = O. Since H is preserved, this easily implies 'Va = O. 0
e-
e (e -
The connections described in part (2) are called contact connections for the contact structure H. Given a choice a of contact form, the vector field r from (3) is called the Reeb vector field for a. Linear connections as in (4) are called contact connections adapted to the contact form a. There are evident analogs of (2) and (4) for partial connections (see 1.3.7) which leads to the notion of partial contact connections. There is a contact analog of the canonical symplectic structure on a cotangent bundle. Namely, let M be a smooth manifold of dimension n + 1 and let PT* M be the projectivized cotangent bundle. This is the ~pn-bundle over M whose fiber over x is the space of all lines in T; M. If £ is such a line, we define a hyperplane Hi c TtPT* M as the space of those tangent vectors, whose projection to TxM is annihilated by £. This can be viewed as the image of the kernel of the tautological one-form a on T* M under the tangent map of the projection from T* M \ M (the complement of the zero section) to PT* M. Choosing a local section a of this projection, the subbundle H is realized as the kernel of a*a. Now in a point ¢ of T* M, the restriction of da(¢) to ker(a(¢)) is degenerate with null space given by those vertical tangent vectors which are multiples of the foot point ¢. This immediately implies that a*a defines a local contact form for H. By part (1) of the proposition, any contact structure in dimension 2n + 1 is locally isomorphic to PT*M. 4.2.2. Generalities on parabolic contact structures. Recall from 3.2.4 that a contact grading is a 121-grading such that 9-2 is one-dimensional and the Lie bracket 9-1 x 9-1 -> 9-2 is nondegenerate. This exactly means that the Lie algebra 9_ = g-2 EB g-1 is a Heisenberg algebra. In 3.2.4 and 3.2.10 we have seen that contact gradings exist only on simple Lie algebras and obtained a complete classification of both complex and real contact gradings. Given a real contact grading and corresponding groups PeG, the first ingredient for an infinitesimal flag structure of type (G, P) is a filtration T M = T- 2 M :::> T- 1 M, where dim(M) = dim(g_) and H := T- 1 M has corank one. Regularity of the infinitesimal flag structure in particular requires that each symbol algebra of
406
4.
A PANORAMA OF EXAMPLES
this filtration is isomorphic to g_, i.e. that H defines a contact structure on M. The subgroup Go C P acts on g_ by Lie algebra automorphism, so it can be viewed as a subgroup of GL(g-d, and the additional ingredient of a regular infinitesimal flag structure is a reduction of structure group of H to this subgroup. If H1(g_,g) is concentrated in homogeneous degrees:::; 0, then regular normal parabolic geometries are equivalent to regular infinitesimal flag structures and hence to contact structures with an additional reduction of structure group to Ad(G o) C Autgr(g_). There is only one parabolic contact structure for which this condition is not satisfied, namely contact projective structures. These will be discussed separately in 4.2.6 below. Next, let us move towards a description of harmonic curvature components. We will be less detailed here than in the case of 111-gradings. A general approach to the description of harmonic curvature of arbitrary parabolic geometry will be developed using Weyl structures in Chapter 5. First we can prove a general fact on the cohomology for contact gradings, which heavily restricts the possibilities for harmonic curvatures of torsion type. LEMMA 4.2.2. Consider a contact grading 9 = g-2EB" ·EBg-2, and the Kostant Laplacian D on A2g~ @ g. Then ker(D) n (A2g~ @g_) C (A59-1)* @ g_. PROOF. Since the statement of the lemma is invariant under complexification, we may assume that we deal with a complex contact grading. The key point here is that Kostant's version of the Bott-Borel-Weyl theorem asserts that the highest weight of any irreducible component of ker(D) occurs with multiplity one in A*g~ @ g; see part (2) of Theorem 3.3.5. Now as a go-module A2g~ decomposes as g~2 @g~1 EBA2g~1' Via the decomposition A2g_1 = A~9-1 EB 9-2 induced by the bracket, the second summand decomposes further as the sum of (A~9-1)* with g~2 ~ 92. Tensoring with 9 we again obtain a decomposition into three summands. The last of these is (isomorphic to) 92 @ 9. Since the same module is also contained in 9~ @ 9, none of its weights can occur with multiplicity one inside A*9~ @ g. For the first summand, we can apply similar arguments in a more restricted situation. Sillce g-2 is one-dimensional, the representation g~2 @ 9-2 is trivial, so 9~2 @ 9~1 @ 9-2 ~ 91, which also sits in 9 = AOg~ @ 9. Finally, consider the bracket 92 @ 9-1 -+ 91· Recall that the Killing form B induces dualities between 9-1 and 91 and between 9-2 and 92. Now for X, Y E 9-1 and 0 # f3 E 92 we get B([f3, X], Y) = B(f3, [X, Y]). Nondegeneracy of the bracket on g-1 shows that ad(f3) : g-1 -+ gl is injective and thus an isomorphism, so 92 @ g-1 ~ g1. Hence, we conclude that which also sits in 9:" @ 9· Altogether we see that for any weight of A29~ @9- that occurs with multiplicity one in A*9:" @ 9_ the weight space has to be contained in (A~g-I)* @ g_, so the same is true for the 90-submodule generated by this weight space. D Let us analyze the consequences of this proposition for the possible locations of harmonic curvature components. Components of ker(D) contained in homogeneity zero are irrelevant for harmonic curvature. For homogeneity one, there is only
4.2. PARABOLIC CONTACT STRUCTURES
407
one possibility, namely components of (A~g-l)* @ g-l. For homogeneity two, the lemma does not leave any room in torsion types. Therefore, the only possibility is curvatures coming form A2g~1 @ go, and from the proof we again see that these actually have to be contained in (A~g-l)* @ go. To describe the harmonic curvatures in homogeneity one and two, we have to interpret the procedure for constructing a normal Cartan connection from an infinitesimal flag structure similarly as in the proof of Theorem 4.1.1. Suppose that (E,O) is a regular infinitesimal flag structure of some type (G, P) corresponding to a contact grading. The idea of the prolongation procedure in Section 3.1 was to first make some choices in order to construct a Cart an connection won 9 := Ex P+ and then modify this to a normal Cartan connection w. To obtain w, one has to choose a principal connection "Ion E, as well as a projection rr from TM onto the subbundle H. Having made these choices, one constructs from 0 a Cartan connection 0 on E, which then can be trivially extended to a regular Cartan connection won g. Now we can interpret this in terms of linear connections. Since both Hand Q = T M / H are associated bundles to E, the principal connection "I induces linear connections \1 H on Hand \1 Q on Q. By construction, these have the property that C, : H x H ~ Q is parallel, which can be used as a characterization of \1 Q . Since E is a sub bundle of the natural frame bundle of Q EB H, we see from 4.2.1 that \1 H is a contact connection. Conversely, a contact connection \1 H which is compatible with the additional structure induced by E can be used to define "I, and then \1 Q is obtained from \1 C, = O. A choice of a projection rr: TM ~ H induces an isomorphism TM ~ H EB Q via e I---> (rr(e),e + H). Using this isomorphism, the connections \1 H and \1 Q induce a linear connection \1 on TM. Using this, we can now formulate the basic result on the interpretation of harmonic curvature components of parabolic contact structures. THEOREM 4.2.2. Let (E ~ M,O) be an infinitesimal flag structure of type (G, P) corresponding to a contact grading such that Hl(g_, g) is concentrated in homogeneities S; O. Let H C T M be the contact sub bundle, Q = T M / H the quotient bundle, and q : T M ~ Q the natural quotient map. Let f'i,H be the harmonic curvature of a regular normal parabolic geometry of type (G, P) with underlying infinitesimal flag structure (E,O). For a principal connection "I and a projection rr : T M ~ H let \1 H, \1 Q and \1 be the induced linear connections on H, Q, and TM. (1) For any choice of "I and rr, the component (f'i,Hh in homogeneity 1 is represented by the component in ker(D) C A2 H* @ H of the tensor T E r(A 2 H* @ H) defined by
(4.5)
e, fJ E r(H).
The tensor T depends only on \1 H in H -directions. (2) For any choice of "I, there is a unique rr such that component H of the torsion of \1 vanishes. The projection rr is characterized by
for
@
Q~ Q
(4.6)
e
for all E r( H) and all fJ E X( M). In particular, if \1 is compatible with a contact form a, then this is equivalent to rr(fJ) = fJ-a(fJ)r, where r is the Reeb vector field. (3) Suppose that "I and rr are chosen in such a way that the homogeneous component of degree one of the torsion of \1 has values in ker(D) (which in particular
408
4. A PANORAMA OF EXAMPLES
implies that 1r is the projection from {2}}. Then the component ("'H h in homogeneity 2 is represented by the component in ker(D) c A~H* ® (E xGo go) of the curvature R of V' . PROOF. (1) As indicated above, we put 9 = Ex P+, and use "y and 1r to define Denoting by i : E ---+ 9 the inclusion and by K, the curvature of w, the pullback i*K, is given by the torsion and the curvature of V'. Now normalizing W, we see that the homogeneous components of degree one gr 1 (K,) of K, and gr 1 ("') differ by an element in the image of G. Normality of w implies that grl ("') lies in ker(D). As in the proof of Theorem 4.1.1, the harmonic curvature component ("'H h is represented by gr 1 ("'), which coincides with the ker(D)-component of the gr 1 (K,). It follows directly from the definition of torsion that the component H x H ---+ H of the torsion of V' is given by (4.5). The last claim is obvious from the formula. (3) Now if we manage to choose "y and 1r in such a way that grl (K,) actually is a section of ker(D), then the homogeneous component in degree two of K, differs from the one of '" only by elements in the image of G. Hence, the ker(D)-components in homogeneity two coincide and by the lemma we know that ker(D) C L( A2T M, E x Go go). Then the description follows immediately. (2) First look at the right-hand side of (4.6) for fixed "I and variable~. Since q(~) = 0, this is linear over smooth functions, and hence defines (still for fixed "I) a bundle map H ---+ Q. By nondegeneracy, there is a unique section 1r(TJ) such that (4.6) holds. But by the Leibniz rule, the right-hand side is also linear over smooth functions in "I, so we actually get a bundle map 1r : T M ---+ H in this way. Finally, if "I E r(H), then q(TJ) = 0 and q([~, TJD = C(~, "I) and hence 1r(TJ) = "I. Therefore, (4.6) uniquely defines a projection 1r : TM ---+ H. To get the component Q ® H ---+ Q of the torsion, we have to proceed as follows. For a vector field "I consider "I -1r(TJ) (which represents the section q(TJ) of Q), take a section ~ of H and consider
w.
q (V' e( "I -
1r("I))
- V' 1j-7r(1j)~
-
[~, "I - 1r(TJ)]) = q(V' eTJ) - q([~, TJD
+ C(~, 1r(TJ)) ,
where we have used that V' preserves the subbundle H. But by construction of V' we have q(V'eTJ) = V'~q(TJ), so the characterization of 1r follows. In the case that V' is compatible with a contact form a, we can rewrite the characterizing equation as -da(1r(TJ),~)
= ~. a(TJ) -
a([~, TJD.
Since a(~) = 0, the right-hand side equals da(~,TJ), which shows that TJ-1r(TJ) must be a multiple of the Reeb field. But then a(1r(TJ)) = 0 implies that the factor must equal a(TJ). D This result is not sufficient to describe all harmonic curvatures in homogeneity one and two. While it tells us how to choose 1r for given "y, it does not show how to choose "y in order to satisfy the condition in (3). This depends on the concrete choice of structure, and we will indicate it in some cases below. Finally, let us remark that a complete description of the harmonic curvature will be obtained using Weyl structures in Chapter 5. 4.2.3. Lagrangean contact structures. We start the discussion of the individual parabolic contact structures with the An-series. From 3.2.10 we know that in this series the algebras sl(n, JR) and su(p, q) with p, q > 0 admit contact gradings.
4.2. PARABOLIC CONTACT STRUCTURES
409
Lagrangean contact structures correspond to the split real form sl(n, JR). The name of this structures goes back to M. Takeuchi; see [Tak94]. It is derived from the fact that maximal isotropic subspaces in symplectic vector spaces are called Lagrangean subspaces. In contact geometry, maximal isotropic subbundles of the contact bundle are often called Legendrean, so the name Legendrean contact strucutures would also be appropriate. For n ~ 1 consider the Lie algebra 9 := sl(n + 2,JR). The contact grading on this algebra comes from decomposing into blocks of size 1, n, and 1, so
9
~ {(~
¢f)' ",b,p,~ x, E
11.;
WE II.·;Z, Y
E
11.";" + b+t,(A)
~ o}.
The grading components are indicated by
(9~1
9-2
; g-1
:r), go
where we have indicated the splittings 9±1 = g~1 EB g~1 for later use. The following facts are easily seen from this block form. The trace form (and hence any invariant form on g) induces a duality between 9~1 and gf and between g~1 and gf. The splittings of g±1 are invariant under the adjoint action of go which induces a surjection 90 -+ gl(g~1) with one-dimensional kernel. The Lie bracket g-1 x g-1 ---79-2 is trivial on g~1 x g~l as well as on g~1 x g~1 and its restriction to g~1 x g~1 induces an isomorphism g~1 ~ L(g~1,g-2)' In particular, this bracket is nondegenerate, so we really have found a contact grading. Viewing this bracket as a symplectic form on g-l, the subspaces g~l and g~l of g-l are Lagrangean. As a group G with Lie algebra 9 we take PGL(n + 2, JR). We can either realize this group as the quotient of GL(n+ 2, JR) by scalar multiples of the identity or by taking the subgroup of matrices whose determinant has absolute value one and identifying each matrix with its negative. In any case, we will work with representative matrices. For the parabolic subgroup PeG we take the subgroup of matrices which are block upper triangular with blocks of sizes 1, n, and 1. The resulting Levi subgroup Go C P consists of the block diagonal matrices with these block sizes. The homogeneous model G / P is the flag manifold F1,n+l (JR n+2 ) of lines in hyperplanes in JRn+2. Mapping such a flag to its line makes Fl ,n+1 into a fiber bundle over JRpn+1. The fiber of this bundle is the space of all hyperplanes containing a fixed line, which can be identified with hyperlanes in the quotient by that line. Since a hyperplane in a vector space is equivalent to a line in its dual, the fiber is JRpn*. Evidently, the vertical bundle of this fibration corresponds to g~l' Likewise, projecting to the hyperplane shows that F1 ,n+1 is a fiber bundle over JRP(n+1)* with fiber JRp n , and the vertical bundle of this fibration corresponds to g~l' To complete the interpretation of the structure, one shows that the projection to JRpn+l actually identifies F1,n+1 with the projectivized cotangent bundle PT*JRpn+1. The subspace spanned by the two vertical bundles (which are transversal) thereby gets identified with the tautological subbundle, so it defines the canonical contact structure on PT*JRpn+1. By Proposition 3.3.7, HI (g_, g) is concentrated in homogeneous degrees ~ 0, so we only have to understand regular infinitesimal flag structures of type (G, P). Let
410
4. A PANORAMA OF EXAMPLES
us denote elements of g_ as triples (/3, X, Y). Likewise, we denote a block diagonal matrix
(0o g0 8) e
with c,e E IR \ 0, e E GL(n,lR) as (c,e,e). In this language,
the adjoint action is given by (c, e, e) . (/3, X, Y) = (~/3, c- 1 ex, eye- 1 ). Observe that this is unchanged if we replace (c, e, e) by a nonzero multiple. Taking the representative (1, c-1e,~) we see that the second component represents the action on g~l and the last component the one on g-2, and this completely determines the element of Go. In particular, the action (as expected) preserves the bracket and the decomposition of g-l. Conversely, suppose that we take an automorphism of the graded Lie algebra g_, which preserves the decomposition of g-l. If e E GL(g~l) denotes the restriction of this automorphism, then compatibility with the bracket implies that the automorphism must be given by (/3, X, Y) 1-+ (e/3, ex, eye- 1 ) for some nonzero number e. Since this is the action of (1, e, e), we conclude that the adjoint action identifies Go with the subgroup of those autmorphisms of the graded Lie algebra g_, which, in addition, preserve the decomposition g-l = g~l EEl g~l' From the discussion in 4.2.2 we know that an infinitesimal flag structure of type (G, P) on a smooth manifold M of dimension 2n + 1 is given by a contact structure H c T M together with a redcution of structure group corresponding to Go C Autgr(g_). From the description of Go above it is clear that such a reduction is equivalent to a decomposition H = E EEl F of the contact subbundle as a direct sum of two Legendrean subbundles. This means that each of the subbundles has rank n, and the restriction of C to E x E and F x F vanishes identically. Note that this implies that C identifies F with the bundle L(E, Q) of linear maps. A contact structure with an additonal decomposition H = E EEl F into the direct sum of two Legendrean subbundles is called a Lagrangean contact structure. To get an overview of the basic completely reducible natural bundles for these structures we have to look at representations of Go. Now this has a two-dimensional center, so there is a two-parameter family of one--dimensional representations and correspondingly a two-parameter family of natural real line bundles. We do not go into detail of how these are best parametrized, but just observe that Q, An E and AnF ~ AnE* 0 Q are typical examples. The semisimple part of Go is SL(n,lR) with the standard representation g~l corresponding to the bundle E. Hence, all natural bundles can be obtained from tensor bundles of E and natural line bundles. Let us next compute the cohomology group H2(g_,g). This is completely different for n = 1 and n > 1, and we consider the case n = 1 first. If n = 1, then we actually deal with the Borel subalgebra in A 2 , i.e. the Dynkin diagram ~--K There are two elements of length two in the Weyl group, namely the two possible compositions of the two simple reflections. The corresponding sets q)w are {ai, a1 + a2}, respectively, {a2' a1 + a2}' On the other hand, the highest root a1 + a2 is mapped by these two Weyl group elements to -a1, respectively, -a2' By Theorem 3.3.5 (and dualization to get from cohomology of p+ to cohomology of g_), we conclude that the two irreducible components in the cohomology are represented by the one-dimensional representations consisting of maps g-2 x g~l -+ gf, respectively, g-2 x g~l -+ gf. The corresponding harmonic curvatures are represented by Cotton-York type tensors mapping Q0E to E* and Q0F -+ F*. These can be determined more explicity using Weyl structures. For n > 1, the Hasse diagram contains three elements of length two. Denoting by a1, ... , a n +1 the simple roots and by O'i the simple reflection corresponding to
4.2. PARABOLIC CONTACT STRUCTURES
411
ai, these three elements are given as 0'1 00'2, O'n+1 0 O'n, and 0'1 OO'n+1 = O'n+1 00'1' The corresponding sets w evidently are {a1' al + a2}, {an+! , Q n + Qn+!}' and {a1,Qn+1}' The images of the highest root Ql + ... + Qn+1 under these three elements are Q2 + ... + Qn+b Q1 + ... + Qn, and Q2 + ... + Qn, respectively. Again using Theorem 3.3.5 and dualization, we conclude that the irreducible components of ker(D) are the highest weight parts in the sets of maps A2g~1 ---+ g~1' A2g~1 ---+ g~I' and g~1 ® g~1 ---+ go, respectively. The first two components are torsions in homogeneity 1 and the last one is a curvature in homogeneity 2, so we can use Theorem 4.2.2 to analyze the corresponding harmonic curvature components. For the two torsions, the interpretation is simple. Suppose that \1 H is the contact connection induced by a principal connection on the regular infinitesimal flag structure determined by a Lagrangean contact structure H = E EEl F c T M. Then of course \1 H = \1 E EEl \1 F for connections on the subbundles, so, in particular, the subbundles are preserved. Now from 4.2.2 we know that that we have to look at components of
T(~, 1J) := \1r 1J -
\1:{ ~ -
71'([~, 11])
for ~,11 E r( H) and a certain projection 71' from T M onto the subbundle H. To get the first torsion component, we have to take ~,1J E r(E) and project the result to F. But then the covariant derivatives produce sections of E, and since E is Legendrean the bracket [~, 111 is a section of H, so we can leave out 71'. Hence, we end up with mapping ~,11 E r(H) to the F-component of -[~, 111 E r(H). Since ~ and 1J actually have trivial F -components, this is bilinear over smooth functions, and hence defines a tensor TE E r(A2 E* ® F). To understand the highest weight component, recall that F S:! E* ® Q via C. Thus, A2 E* ® F S:! A2 E* ® E* ® Q and the highest weight part in there is the kernel of the alternation map to A3 E* ® Q. Now viewed as a trilinear map on E with values in Q, the torsion TE maps (~, 11, () to C applied to the F-component of [~, 1J1 and (. Replacing the F-component by [~, 1J1 does not change the value of C, so we are left with (~, 11, ()
1---+
C([~, 11], () = q([[~, 11], (]).
But this has trivial alternation by the Jacobi identity. Hence, we obtain PROPOSITION 4.2.3. The two harmonic curvature components in homogeneity one of the regular normal parabolic geometry determined by a Lagrangean contact structure H = E EEl F c T M are represented by the torsions TE E r (A 2 E* 0 F) and TF E r(A 2 F* ® E), induced by projecting the negative of the Lie bracket of two sections of one sub bundle to the other sub bundle. In particular, TE vanishes identically if and only if the sub bundle E c T M is integrable and likewise for TF. Vanishing of both TE and TF is equivalent to torsion freeness of the normal parabolic geometry. PROOF. Apart from the last claim, everything has been proved already above. For the last claim, recall that the lowest nontrivial homogeneous component of the curvature of a regular normal parabolic geometry is harmonic; see Theorem 3.1.12. Vanishing of TE and TF implies that this lowest nonzero component has homogeneity at least two. By Lemma 4.2.2 the harmonic part of homogeneity two cannot produce any torsions. Since the same is true for arbitrary maps of homogeneity at least three, the result follows. 0
4. A PANORAMA OF EXAMPLES
412
To interpret the remaining harmonic curvature component, one has to choose a contact connection \jH and a projection 7r as above, and then modify it in such a way that the homogeneity one component of the torsion is contained in ker(D). Then one looks at the appropriate part of the curvature of the resulting connection. We will give a detailed description of connections adapted to parabolic contact structure in this sense (as well as in stronger senses) in Section 5.2. At this point, we only give a short sketch how such a connection can be constructed. Observe that a contact connection \jH comes from a principal connection on the infintesimal flag strucutre if and only if it is of the form \jE EB \jF. Starting with an arbitrary choice of such a connection, the tensor 7 from above is a section of A2 H* ® H. Decomposing this bundle, we obtain
(A2 E* EB (E ® F)O EB Q* EB A2 F*) ® (E EB F). Here we have denoted by (E ® F)o the kernel of £ : E ® F --t Q and identified a complementary subbundle with Q. Note that by definition of 7, the only part of this that depends on 7r is the part in Q* ® (E EB F). Now take the components in (A2 E* EB (E ® F)i)) ® E, interepret them as a section of H* ® E* ® E and subtract this from \jE. Likewise, take the components in ((E®F)i) EBA2 F*) ®F, view them as a section of H* ® F* ® F and subtract this from \jF. Finally, we can use the component in Q* ® (E EB F) to change the projection 7r. The resulting pair of connnection and projection by construction has the property that the nonzero components of the tensor 7 only lie in A2 E* ® F and in A2 F* ® E, and we know from above that these parts automatically lie in ker(D). We claim that this is already an appropriate connection, i.e. the part of the torsion which maps Q ® H to Q has to vanish automatically. To see this, observe that we are dealing with the lowest homogeneous component of the curvature of a regular Cartan connection, so by the Bianchi identity, it is contained in the kernel of 8; see Theorem 3.1.12. Let us denote the homogeneity one part of the torsion by 'I/J and for sections '11, ( E r(H) expand the equation 0 = 8'I/J(e, '11, (). Using that { , } coincides with C, this gives
e,
0= C(e, 'I/J(ry, ()) - C(ry, 'I/J(e, ()) + C((, 'I/J(e, '11)) - 'I/J(C(e, '11), () + 'I/J(C(e, (), '11) - 'I/J(£(ry, (), e)·
(4.7)
e,
Now assume that '11 E r(E) and ( E r(F). Then in the first two terms, 'I/J already gives zero while in the third term 'I/J has values in F, so this does not contribute either. In the fourth term we get £( '11) = 0 since both are sections of E, so (4.7) reduces to 'I/J(C(e, (),ry) = 'I/J(C(ry,(),e)· But now given (3 E qQ) and '11 E r(E) we can choose E r(E) and (E r(F) such that £(e,() = (3 and £(ry,() = 0, and we get 'I/J((3, '11) = 'I/J(C(e, (), '11) = 'I/J(C(ry, (), e) = o.
e,
e
Thus, 'I/J vanishes on Q ® E and likewise one shows that it vanishes on Q ® F. 4.2.4. Partially integrable almost CR-structures. This is certainly the most important example of a parabolic contact structure, which has often been studied independently. The constructions of canonical Cartan connections by N. Tanaka in [Tan62] and by S.S. Chern and J. Moser in [ChMo76] (for the subclass of integrable CR-structures) are among the best known results of this kind and were a strong motivation for the development of the general theory.
4.2. PARABOLIC CONTACT STRUCTURES
413
We have partly discussed this example in 3.1.7, so we will go through the basics rather quickly. For p + q = n ~ 1 we consider the real form su(p + 1, q + 1) of sl(n + 2, q. We choose the Hermitian form on C n +2 which is given by n
p
((zo, ... , Zn+1), (wo, ... , W n +1)) =
ZOWn+1
+ Zn+1WO + LZjWj j=l
L
ZjWj'
j=p+1
Denoting by IT = ITp,q the n x n-diagonal matrix with the first p entries equal to 1 and the remaining entries equal to -1, we can represent the Lie algebra in block form with blocks of sizes 1, n, and 1, similarly to 4.2.3 as 9=
{(~.
~x
~
_~~*): A E u(n),a. E C,X E cn,~ =- c E 1R, a + tr(A) - a - 0
m
- x*rr
It
-a
,}.
X, Z
The grading components are as for Lagrangean contact structures in 4.2.3 above. Rather than the splitting of 9±1 into two irreducible pieces we have a complex structure on these subspaces. After complexification, the splitting into two components is recovered as the splitting of 9±1 0C into holomorphic and anti-holomorphic parts. This will also be crucial for the interpretation of cohomologies. The bracket 9-1 x 9-1 """"* 9-2 is given by [X, Y] = Y*ITX - X*ITY, so this is twice the imaginary part of the standard Hermitian inner product of signature (p, q). Note that this is compatible with the complex structure in the sense that [iX, iY] = [X, Y]. As a group with Lie algebra 9, we take G = PSU(p + 1, q + 1). The parabolic subgroup P is then the stabilizer of the isotropic complex line generated by the first basis vector. (This automatically stabilizes also its orthocomplement, which is a hyperplane containing the given line.) The subgroup Go again is given by block diagonal matrices, i.e. we have matrices
( 0COO) C 0 o 0 lie
with c E C \ 0 and C
E
U(n)
such that cdet(C)/c = 1. We have to identify matrices which are multiples of each other, which leaves the freedom of multiplying by an (n + 2)nd root of unity. Using a notation similar to 4.2.3, the adjoint action is immediately computed to be given by (c,C)· (ix,X) = (lcl- 2 ix,c- 1 CX), which is complex linear on 9-1 and orientation preserving on 9-2, Notice that there is a p-dimensional subspace in 9-1 on which X f---+ [X, iX] is nonzero with all values of the same sign and a q-dimensional subspace for which the same is true for the opposite sign. Hence, if p i= q, then preserving the bracket and the complex structure on 9-1 implies that the orientation on 9-2 is preserved. For p = q, this is an additional condition. Conversely, assume that A : 9-1 """"* 9-1 is a complex linear isomorphism such that [AX, AY] = A[X, Y] for some A > O. Since the standard Hermitian form of signature (p,q) is obtained as 1/2(i[X,iY] + [X, Y]), we conclude that A has the same compatibility with that Hermitian form. In particular, Idet(A)12 = An. Now choose c E C such that IcI 2 c- n - 2 = det(A). Then we get Idet(A)12 = Icl- 2n = An, and since A > 0 this implies A = Icl- 2 . Hence, cA has the property that [cAX, cAY] = [X, Y] and hence cA E U(n). But then A is realized by the adjoint action of (c, cA) and cdet(cA)/c = cn +2lcl- 2 det(A) = 1 as required. Note that in this procedure c is only unique up to multiplication with an (n + 2)nd root of unity. From the discussion in 4.2.2 we conclude that a regular infinitesimal flag structure (and hence a regular normal parabolic geometry) of type (G, P) on a smooth manifold M of dimension 2n + 1 is equivalent to a contact structure H c T M together with a complex structure J on H such that C(Je, JTJ) = C(e, TJ) for all
4. A PANORAMA OF EXAMPLES
414
~,,,, E f(H). If this last condition is satisfied, then identifying the fiber Qx of Q over x E M with JR, the map £ is the imaginary part of a Hermitian form, and one requires that this form has signature (p, q). If p = q, one in addition has to choose an orientation on Q (which requires Q to be trivial). Since as a complex vector bundle H is canonically oriented, this is equivalent to choosing an orientation on M. For p =f. q, this orientation is automatically chosen by deciding between signature (p, q) and signature (q, p). Let us rephrase this in the language of CR geometry. Given a real smooth manifolds M of dimension 2n + 1, a rank n complex subbundle (H, J) in TM is called an almost CR-structure of hypersurface type. Correspondingly, there is the notion of a (local) CR-diffeomorphism, which requires the tangent map to preserve the CR-subbundle H and the restriction to H being complex linear. The almost CR--structure is called nondegenerate if H defines a contact structure on M. Next, we have the condition that £(J~, J",) = £(~,,,,) for all ~,,,, E H. This condition is not used very often in CR geometry, since it is implied by the integrability condition to be discussed below. One usual terminology (see e.g. [Miz93]) for this condition is partial integrability. Then £ becomes the imaginary part of a Hermitian form and choosing an orientation on Q the signature of this form is called the signature of (M, H, J). Hence, we conclude that regular normal parabolic geometries of type (P SU (p + 1, q + 1), P) are equivalent to oriented nondegenerate partially integrable hypersurface type almost CR-structures of signature (p, q). To understand the terminology "partial integrability" and its relation to the integrability condition, it is best to pass to the complexified setting. Since H is a complex vector bundle, the image H 0C in the complexified tangent bundle T M 0C splits into holomorphic and anti-holomorphic part as H ®C = Hl,o (JJHO,1. Typical sections of HO,l are of the form ~ +iJ~ for ~ E r(H). Applying the complex bilinear extension of £ to two such sections, we obtain
£(~
+ iJ~,,,, + iJ",) =
(£(~,,,,)
- £(J~, J'f/))
+ i(£(J~, 'f/) + £(~, J'f/)).
Evidently, partial integrability is equivalent to vanishing of this expression and hence to the fact that the bracket of two sections of HO,l is a section of H 0 C. Alternatively, this can be phrased as follows. Consider the complex linear extension qc : T M 0 C -4 Q 0 C, and the tensorial map HO,l x HO,l -4 C induced by an imaginary multiple of qd[~, il]) for sections ~, 'f/ E r(HO,l). Partial integrability is equivalent to this form being Hermitian, thus defining (with appropriate normalization) the classical Levi form. The signature of M is then the signature of this form. A partially integrable almost CR-manifold is called integrable or a CR-manifold if the bundle HO,l is involutive. In the real picture, this is expressed by vanishing of the Nijenhuis tensor N : A2 H -4 H, which is induced by (~,,,,)
f-?
[~, ",J- [J~,
J'f/J
+ J([J~, 'f/J + [~, J'f/]).
Note that N is of type (0,2) i.e. conjugate linear in both arguments. The most important examples for CR-structures come from complex analyis. Let (M, J) be a complex manifold of complex dimension n + 1, and let M c M be a smooth real hypersurface. For x E M define Hx := TxM n J(TxM), the maximal complex subspace of TxM C TxM. These subspaces must have complex dimension n and they fit together to define a complex subbundle H C TM. By definiton HO,l = T M 0 C n TO,l M and as the interesection of two involutive subbundles,
4.2. PARABOLIC CONTACT STRUCTURES
415
this is automatically involutive. Generically, the sub bundle H will be maximally nondegenerate, and then (M, H, J) is automatically a CR-structure. Note further that a biholomorphism of M which maps M to itself automatically restricts to a CR-diffeomorphism on M. This picture can be nicely used to understand the homogeneous model G / P. The subgroup P is the stabilizer of a null line and G acts transitively on the space of all such lines. Hence, G / P can be identified with the projectivized null cone, which is a smooth real hypersurface in
e
e.
7l'(e) e e
4. A PANORAMA OF EXAMPLES
422
obvious notation) (Z, W) E 91 and - [[(Z, W),
(ZX2
(~n] ,(~:)]
G:)' G:)
E 9-1, Then one computes that
is given by
+ WY2)(~:) + (ZX1 + WY1)(~:) + (XiY2 -
Y{X2)(~t)'
Taking the traceform on 9 to describe the duality between 9-1 and 91, the expression (ZX 2+ WY2) is simply the pairing between (Z, W) and (~:), so the first two terms are easy to interpret and look exactly as in the projective case. To interpret the last term, we observe that taking its bracket with (~) gives
(ZX
+ WY)2(y1t X 2 - XiY2 ) = (ZX + WY)
Using this we see that changing a to
[(~:), (~:)] .
a, the partial connection changes as
(4.8) for~, TJ E
r(H). Here T is the map Z interpreted as a section of H* and T# : Q ~ H is characterized by £(T#(13), () = T(()13 for 13 E Q and ( E H. For the projection, the interpretation is easier. For (Z, W) E 91 and x E 9-2 we obtain -[(z, W),x] = x(~t). This shows that
= 1T"(~) + 2T#(q(~)), for ~ E X(M), with q: TM ~ TM/H = Q the natural quotient map. 1T&(~)
T
This relation has two nice consequences. On the one hand, consider the tensor from 4.2.2. For the data associated to a this is given by T(~, TJ) = V'~ TJ
- V'~~ -
71''' ([~,
TJ]).
Changing to a, the terms in (4.8) involving T (rather than T#) are symmetric in ~ and TJ and hence do not contribute to the change of T. Hence, the full contribution to the change of T from (4.8) is T#(£(~, TJ) - £(TJ, ~)) = 2T#(£(~, TJ)), which exactly cancels with the contribution from the projection term. Thus, for all choices of sections, we obtain the same torsion tensor T. For the other part of homogeneity one in the torsion, we have to use the induced connection on Q, so we have to compute its change first. This connection is characterized by the fact that £ is parallel, so
V't£(TJ, () = £(V'tTJ, ()
+ £(TJ, V't(),
for ~,TJ, ( E r(H). Expanding the right-hand side, collecting terms and using that £ is parallel for V''', one obtains V'~£(TJ, () + 2T(~)£(TJ, (). In the proof of Theorem 4.2.2, we have seen that the homogeneity one part of the torsion for the choice associated to a is given by V'~q(TJ)
- q([~, 1]]) + £(~, 1T"(TJ)),
for ~ E r(H) and 1] E X(M). Passing to a, the middle term remains unchanged, while the changes caused by the first and last term evidently cancel. Hence, the whole homogeneity one part of the torsion is independent of the choice of a. 4.2.6. Let (M, H) be a contact manifold, and let V' : r(H) x r(H) be a partial contact connection.
DEFINITION
r(H)
~
4.2. PARABOLIC CONTACT STRUCTURES
(1) The contact torsion
T :
A2 H
-+
423
H of V' is the tensor induced by
T(~,11) = V'~11- V'T/~ -11"([~,11]),
where 11" : T M -+ H is the projection associated to V' in part (2) of Theorem 4.2.2. (2) A partial contact connection V on H, is said to be contact projectively equivalent to V' if and only if there is a smooth section Y E r(H*) such that
V~11 = where y# : Q
-+
V' ~11 + Y(~)11 + Y(11)~ + y#(£(~, 11)),
H is characterized by £(Y#(f3), () = Y(()f3 for f3 E Q and ( E H.
The upshot of the above discussion was that a regular P-frame bundle of degree one over M induces a contact structure H on M as well as a contact projective equivalence class of partial contact connections. Parallel to 4.1.5, one establishes the converse. On the other hand, we have seen that contact projectively equivalent partial connections have the same contact torsion, so it make sense to talk about the contact torsion of a projective class. Thus, it remains to discuss the normality condition, which needs some basic information on H2 (9-, 9). There is only one element w of length two in the Hasse diagram, namely acting with the reflection corresponding to the second simple root and then with the one corresponding to the first simple root, which is crossed. Applying this to the highest root 2a1 +-. ·+2an -1 +a n we get 2a2+" ·+2an -1 +an (respectively a2 if n = 2). The corresponding root space lies in 90. It is still a positive root though, so the root spaces corresponding to elements of cI>w must be contained in 91. This shows that the irreducible representation H2(9_,9) sits in A29~1 ® 90. In particular, there is no cohomology in homogeneity one. Using this, one verifies (again parallel to 4.1.5 and using 4.2.2) that a regular P-frame bundle of degree one is normal if and only if it corresponds to a contact projective class with vanishing contact torsion. The harmonic curvature can then be read off as in part (3) of Theorem 4.2.2 using any connection extending a member of the contact projective equivalence class. 4.2.7. Contact projective structures and geodesics. Similarly, as discussed for projective structures in 4.1.6, there is an interpretation of contact projective structures in terms of unparametrized geodesics. We discuss this only briefly, more details can be found in [Fox05a] and [Fox05b]. Suppose first that (M, H) is a contact structure, V' is a contact connection on T M and c : I -+ M is a geodesic for V'. Let a be a contact form for H with Reeb vector field r, and put f(t) := a(c'(t)). Then c'(t) - f(t)r E H for all t, and since V' is a contact connection we get 0= a(V'cl(c' - fr)) = -a(V'c,fr) = -c'· f - fa(V'c,r).
This is a linear first order ODE on f, so if f vanishes in one point, then it vanishes identically. Hence, we conclude that any geodesic for a contact connection that is tangent to H in one point is tangent to H everywhere. We call these geodesics the contact geodesics of V'. Evidently, they depend only on the partial connection underlying V'. Now given a partial contact connection, consider the set of all partial contact connections which have the same contact torsion and the same geodesics up to parametrization. Here the contact torsion is determined with respect to the projection 11" associated to the partial contact connection according to part (2) of
424
4. A PANORAMA OF EXAMPLES
Theorem 4.2.2. Similarly, as in 4.1.6, one shows that this recovers the contact projective equivalence class as defined in 4.2.6 above. The family of contact geodesics can then be viewed as a smooth family of unparametrized curves, with exactly one curve through each point in each direction in the contact subbundle. Among such families there are those, which are the geodesics of a (partial) contact connection with vanishing contact torsion. Suppose that we have contact manifolds (M, H) and (NI, H) endowed with such families of paths and a contact diffeomorphism f : M -+ NI, which is compatible with these families. Then the families determine contact projective equivalence classes of partial contact connections on M and NI. Take a representative of the class on NI, and pull it back to Musing f. The result is a partial contact connection with vanishing contact torsion, which by construction has the distinguished paths as contact geodesics. Thus, it lies in the contact projective equivalence class, and f is a morphism of contact projective structures. Hence, we see that the families of paths provide an equivalent description of projective contact structures. At this point there occurs a subtlety which is not present for classical projective structures. Any linear connection on the tangent bundle of a manifold can be changed into a torsion-free connection. Since the necessary change is given by a skew symmetric tensor, this does not change the set of geodesics. Thus, the family of geodesics of an arbitrary linear connection can always be described by a torsion-free projective structure and hence a regular normal parabolic geometry. This is no longer true in the contact case. The deformation tensor between two partial contact connections is a section of H* 0 csp(H) c H* 0 H* 0 H. Such a change does not affect the contact geodesics, if at the same time it is contained in A2 H*0H, and then it also directly describes the change of torsion. The intersection (H* 0 csp(H)) n (A 2 H* 0 H) turns out to be too small to remove arbitrary torsions. Only those families of contact paths, which can be described by a partial contact connection with vanishing contact torsion, admit an equivalent description as a regular normal parabolic geometry. It turns out, however, that also contact projective structures with nonvanishing contact torsion admit a canonical regular Cartan connection of type (G, P). This was shown by D.J.F. Fox in [Fox05a], where he generalized the normalization condition on the curvature of a Cartan connection. In the case of vanishing contact torsion, the original normalization condition is recovered. Hence, this extends the approach via Cartan geometries to arbitrary families of contact geodesics. 4.2.8. Exotic parabolic contact structures. In this section we briefly indicate what the parabolic contact structures associated to exceptional Lie algebras look like. We also use this to demonstrate that a rough picture of the nature of a parabolic geometry can be obtained with very little effort. To our knowledge, the details have not been worked out yet for any of these geometries. It should, however, be remarked that there are relations between contact gradings on simple Lie algebras and Jordan algebras (see [Kan73]), which should be useful for a more detailed study of these geometries. We will not discuss harmonic curvature components here. Indeed, one shows that in all cases, there is only one irreducible component in H2(g_, g) and that component sits in homogeneity one. Hence, the harmonic curvatures can always be read off as appropriate components of the tensor r associated to any contact connection induced by from a principal connection on the infinitesimal flag structure.
4.2. PARABOLIC CONTACT STRUCTURES
425
The principles along which we discuss the geometries are fairly easy: The list of complex contact gradings can be found in 3.2.4 and from the Satake diagram of each real form one immediately sees whether the contact grading exists on that real form or not. In that table one also finds the type of go. Using this, one computes the dimension, in which the geometry exists as 1/2(dim(g) - dim(go)). For the exceptional algebras, go always has on~imensional center and the Satake diagram of the semisimple part is obtained by removing the crossed node and all edges connecting to it; see [Kane93]. The single crossed node represents the simple root for which the corresponding root space is contained in gl. But this immediately implies that the negative of this root is the highest weight of g-l. From the definition of the Dynkin diagram, one may immediately write out this weight, thus finding the nature of the reduction of structure group of a contact structure which is equivalent to the parabolic contact structure. The parabolic contact structure associated to G 2 • Here there is just one noncompact noncomplex real form, namely the split real form. From the table in 3.2.4 we see that the semisimple part of go is .s((2, ~), so go ~ g((2, ~). Since G2 has dimension 14, the associated parabolic contact structure exists on manifolds 3 -2 of dimension 5. The highest weight of g-l is (X) = ['ljJ(A), X] for all X E 9-. To show this, we have to invoke the cohomological condition. An element f E C 1(g_, g) by definition is a linear map g_ - t 9 and if one requires f to be homogeneous of degree zero, then it has values in g_. The co cycle equation is 0= 8f(X, Y)
= [X, f(Y)]- [Y, f(X)]- f([X, YD,
or equivalently f([X, YD = [f(X), Y] + [X, f(Y)]. Hence, co cycles of homogeneity zero are exactly derivations of the graded algebra g_. Now we compute
ct>([X, YD = 'ljJ([A, 'ljJ-1([X, Y])]) = 'ljJ([[A, 'ljJ-1(X)], 'ljJ-1(y)]
+ ['ljJ-1(X), [A, 'ljJ-1(y)]])
= [ct>(X) , Y]
+ [X, ct>(Y)],
so ct> is indeed a derivation. Vanishing of cohomology says that ct> is given by the adjoint action of an element of go and uniqueness follows from the injectivity of the adjoint action of go on g_. Now we have extended 'ljJ to a linear endomorphism of g, which is evidently invertible, so it remains to check compatibility of'ljJ with the Lie bracket. Compatibility with the bracket of two elements of 9_ holds by definition. For the bracket 90 x 9_ - t g_ we get
['ljJ(A) , 'ljJ(X)] = ct>('ljJ(X)) = 'ljJ([A, Xl).
428
4. A PANORAMA OF EXAMPLES
Using this and the Jacobi identity, we get for A, C E go and X E g_
[[1/1 (A) , 1/1 (C)], 1/I(X)] = [1/I([A, X]), 1/I(C)] + [1/1 (A) , 1/I([C, Xl)] =1/I([[A, C], X]) = [1/I([A, C]), 1/1 (X)] , which implies compatibility with the bracket go x go ---+ go. Thus, we have verified compatibility with all brackets on g Q is an isomorphism. Since the Dynkin diagram Bn has no automorphisms, the group Aut(so(n + 1, n)) coincides with the group of inner automorphisms. This group can be realized as the quotient of any connected Lie group with Lie algebra so(n + 1, n) by its center, and since we are in odd dimensions, the connected component SOo(n + 1, n) has trivial center. The appropriate parabolic subgroup PeG can be identified with the stabilizer of an isotropic subspace of dimension n in JR2n+1; see 3.2.5 and 3.2.11. Using Proposition 4.3.1 we conclude that regular normal parabolic geometries of type (SOo(n+ 1, n), P) are equivalent to distributions of growth vector (n, n(n2+1)). The simplest special case is n = 3, where we obtain growth vector (3,6), so we deal with generic rank three distributions on manifolds of dimension six. In this case, a canonical Cartan connection was also constructed by R. Bryant in his thesis; see [Br79, Br06). From Proposition 4.3.1 we know that the Levi subgroup Go C P coincides with the automorphism group of the graded Lie algebra 9_ = 9-2 EI1 9-1. Since 9_ is generated by 9-1, such an automorphism is determined by its restriction to 9-1. On the other hand, since the bracket induces an isomorphism A29_1 -> 9-2, we see that any invertible linear map on 9-1 extends to an automorphism of 9_. Hence, we conclude that the restriction of the adjoint action identifies Go with GL(9-1) ~ GL(n,JR).
4.3. EXAMPLES OF GENERAL PARABOLIC GEOMETRIES
431
Using this, we can immediately describe the regular infinitesimal flag structure determined by such a generic distribution. The linear frame bundle of the distribution H is a principal bundle Po : go ~ M with structure group GL(n, JR) ~ Go. This carries a canonical soldering form B-1 defined on (TpO)-1(H) c Tg o, and we can view this form as having values in g-1. Explicitly, we can interpret a point U E go as a linear isomorphism u : g-1 ~ H x , where x = po(u). For ~ E Tugo such that TuPo . ~ E Hx c TxM, we then have B-1(~) := u- 1(Tupo . ~). Genericityof the distribution H then implies that given u : g-1 ~ H x , there is a unique map u : g-2 ~ Qx = TxM/Hx such that u([X, Y]) = Cx(u(X),u(Y)). Using this, we define B-2 E 0 1(90, g-2) by B-2 (~) := u- 1 (TuPo' ~ + Hx) for ~ E Tugo. It is an easy exercise to verify that (B-1, B_ 2) makes Po : go ~ M into a regular inifinitesimal flag structure. To understand the basic obstructions to local flatness (Le. local isomorphism to SOo(n+ 1, n)/ P, we can compute the cohomology group H2(g_, g). The recipes from 3.2.18 show that there is just one irreducible component in the cohomology. The nature of the cohomology group depends on n. For n > 3, the cohomology is the highest weight part in g~l 0 g~2 0 g-2. Geometrically, this means that the complete obstruction to local flatness is given by a torsion, which can be interpreted as a bundle map H x Q ~ Q, where H C TM is the distribution and Q = TM/H. We shall see in Chapter 5, how such torsions can be described in terms of compatible connections. For n = 3, the cohomology H 2 (g_,g) is the highest weight component in g~10 g~2 0 go· From above, we know that go = gl(g-l). Hence, for n = 3, the basic obstruction to local flatness is a curvature, which can be interpreted as a bundle map H x Q ~ L(H, H). General tools to describe such curvatures will be developed in Chapter 5, an explicit description of the curvature can be found in [Br06].
Growth vector (2,3,5). This is a classical example whose study goes back to E. Cartan's famous "five variables paper" [CarlO] from 1910. In this fundamental paper, Cartan classified subbundles in the tangent bundle of five-dimensional manifolds, showing that in some cases such subbundles may have local invariants. For the generic type of such distributions, Cart an constructed a canonical Cartan connection related to the exceptional Lie algebra of type G 2 and determined the basic curvature quantity. This paper has motivated many further developments, in particular in the study of differential systems. The complex exceptional Lie algebra of type G 2 has two simple roots al and a2 and we number them in such a way that the highest root becomes 3a1 + 2a2' From Table B.2 in Appendix B, we can see that the positive roots are
If we consider the parabolic subalgebra corresponding to E = {al}, then we obtain a 131-grading with dim(g±l) = dim(g±3) = 2 and dim(g±2) = 1. Since g_ is generated by g-l, the Lie bracket on g_ has to induce isomorphisms A2g_l ~ g-2 and Q-l 0 Q-2 ~ g-3. Conversely, it is clear that, together with the dimensions of the grading components, these two properties characterize the Lie algebra structure of Q_. The split real form of the exceptional Lie algebra of type G2 has an analogous grading and we define G to be the automorphism group of this Lie algebra and
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4. A PANORAMA OF EXAMPLES
PeG as the subgroup of filtration preserving automorphisms. Then the Pinvariant subspace g-1/p C g/p corresponds to a distribution of rank two on the five-dimensional manifold G / P. Moreover, since g-1 generates g_ we see that the growth vector of this distribution is (2,3,5). It turns out that the space G / P is diffeomorphic to (83 X 8 2)/Z2' and the rank two distribution corresponding to g-1/p admits a nice explicit description in this picture; see [Sag06b]. Conversely, consider a manifold M of dimension 5 and a distribution 11. C T M of rank two. Given a local frame {~, 77} for 11., we see that the bracket of any two sections of 11. can be locally written as a linear combination of ~, 77, and [~, 77] with smooth coefficients. Likewise, brackets of three such sections can be written as linear combinations of ~, 77, [~, 77], [~, [~, 77]] and [77, [~, 77]] with smooth coefficients. Hence, we see that (2,3,5) is the quickest possible growth for rank two distributions, so distributions with this growth vector are as non-integrable as possible. Since we have seen above that there exists a distribution with this growth vector such distributions are generic. Given a distribution H with growth vector (2,3,5), brackets of two sections of H together with H span a rank three subbundle of TM. Defining this rank three subbundle to be T-2M and putting T- 1M := H, we obtain a filtration TM = T- 3M ::J T- 2M ::J T- 1M. From the growth vector, we can see that gr -2 (T M) has rank one, while gr_3(TM) has rank two, and that the Levi bracket L has to induce isomorphisms A2T-1M ---+ gr_2(TM) as well as T-1M ®gr_2(TM) ---+ gr_3(TM). We have observed above, that this implies that (gr(TxM), Lx) is isomorphic to g_ for each x E M. Using local frames, we see that the bundle of symbol algebras is locally trivial and modelled on g_. From Proposition 4.3.1 we conclude that regular normal parabolic geometries of type (G, P) are equivalent to rank two distributions on five-dimensional manifolds with (small) growth vector (2,3,5). Similarly as before, we can describe the infinitesimal flag structure determined by a generic rank two distribution H on a manifold M of dimension five explicitly. A moment of thought shows that Go = Autgr(g_) = GL(g_1). Hence, as for growth vector (n, n(~+1)) one concludes that the regular infinitesimal flag structure of type (G, P) describing the distribution is defined on the linear frame bundle of H C T M. Computing the cohomology H2 (g_, g) shows that the basic obstruction to local flatness in this case is a curvature, which can be interpreted as a bundle map H x gr_3(TM) ---+ L(H,H). This basic curvature quantity has already been found by Cartan in [CarlO]. 4.3.3. Quaternionic contact structures. These structures lie on the borderline between generic distributions and general filtrations with prescribed symbol algebra. They have been first introduced (for definite signature) by O. Biquard in his work on conformal infinities of quaternion-Kahler metrics; see [BiOO] and the overview article [Bi02]. It was only realized later, that these structures provide an example of a parabolic geometry. As in 4.1. 7, we will always consider IHln as a right vector space over 1Hl. A bilinear form ( , ) : IHln x IHln ---+ JHI is called quaternionic-Hermitian if (v, wq) = (v, w)q and (w, v) = (v, w), where we use the conjugation on quaternions as discussed in 4.1.7. For nonnegative integers p and q such that p + q = n, one defines the standard quaternionic Hermitian form of signature (p, q) on IHln by ((V1,"" vn), (W1,"" w n )) := 'ihw1
+ ... + Vpwp -
Vp+1 Wp+1 - ... - Vnw n .
4.3. EXAMPLES OF GENERAL PARABOLIC GEOMETRIES
433
It turns out that any nondegenerate quaternionic Hermitian form on an n-dimensional quaternionic vector space is isomorphic to one of these, so such forms are determined by their signature. Given a Hermitian form (, ), we can look at its imaginary part, which has values in the three-dimensional space of purely imaginary quaternions. By definition, this imagnary part can be written as (v,w) I-t !((v,w) - (w,v), so it is skew symmetric. Now one defines the quaternionic Heisenberg algebra of signature (p, q) (with n = p + q) as lHIn EEl im(lHI) endowed with the bracket
[(v,p), (w,q)]
:=
(0, !((v,w) - (w,v))).
Since triple brackets are zero by definition, this satisfies the Jacobi identity, hence making lHInEElim(lHI) into a two-step nilpotent graded Lie algebra. For our purposes, the grading is best written as n = n-2 EEl n-1 with n-2 = im(lHI) and n-1 = lHIn. In particular, dim(n) = 4n + 3. DEFINITION 4.3.3. Let M be a smooth manifold of dimension 4n + 3 for some n ~ 1, and let p, q be nonnegative integers such that p + q = n. A quaternionic contact structure on M of signature (p, q) is a subbundle H c T M of rank 4n such that the bundle of symbol algebras of the filtration H := T- 1 M c TM = T- 2 M is locally trivial and modelled on a quaternionic Heisenberg algebra of signature
(p, q). This is not the basic definition used (for signature (n,O» in [BiOO], but it is noted there as an equivalent definition. We will discuss this further below. To get a connection to parabolic geometries, consider (for fixed p and q) the Lie algebra 9 = sj:J (p + 1, q + 1) of linear maps on lHIv+q+2 which are skew Hermitian with respect to a quaternionic Hermitian form of signature (p + 1, q + 1); see 2.1.7. Let I' C 9 be the stabilizer of an isotropic quaternionic line in lHIv+q+2 . Then 9 is an algebra of type Cv+q+2, the Satake diagram can be found in Table B.4 in Appendix B. Using Theorem 3.2.9 and arguments similar to the ones used in 3.2.10 for the algebras of type SU, respectively, sl(n, 1HI), one shows that the parabolic subalgebras of 9 are exactly the stabilizers of flags of isotropic quaternionic subspaces. In particular, I' is the standard parabolic subalgebra corresponding to the second simple root in the Satake diagram. An explicit realization of 9 = sj:J(p+ 1, q+ 1) can be obtained analogously to the realizations ofso(n+l, 1) in 1.6.3 and ofsu(n+l, 1) in Example 3.1.2 (3). This shows that 9-1 ~ lHIp +q and 9-2 ~ im(lHI) and under this identification the bracket [ , 1: 9-1 x 9-1 -+ 9-2 is the imaginary part of a nondegenerate quaternionic Hermitian form of signature (p, q). This means that 9_ is a quaternionic Heisenberg algebra. Putting G := Aut(9) and P := Autf(9), Proposition 4.3.1 implies that regular normal parabolic geometries are equivalent to quaternionic contact structures of signature (p, q). The subalgebra 90 C 9 turns out to be isomorphic to 1HI EElsj:J(p, q). Since 9_ is generated by 9-1. the group Go ~ Aut gr (9-) can be realized as a subgroup of GL(9-1). The computation is similar as in the complex case which was done in 4.2.4. It turns out that Go is generated by the scalar multiplications by nonzero quaternions and by the elements of Sp(p, q) acting on 9-1 ~ lHIp +q • This shows that 9-1 carries a Go-invariant prequaternionic structure as discussed in 4.1.7. Passing to associated bundles, it follows that, given a quaternionic contact structure H C TM, the subbundle H inherits a natural almost quaternionic structure. Explicitly, this means that there is a canonical rank three subbundle
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4. A PANORAMA OF EXAMPLES
Q c L(H, H), which can be locally spanned by 1, J and 10J, where 12 = J2 = -id and J 0 1 = - 1 0 J. Moreover, the actions of elements of Go leave the real part of the Hermitian form on g-l invariant up to scale, so there is a canonical conformal class of real inner products on H, which are Hermitian with respect to all the complex structures in Q. These data provide Biquard's original definition of a quaternionic contact structure. The homogeneous model of quaternionic contact structures can be equivalently described as the quotient of Sp(p + 1, q + 1) modulo the stabilizer of an isotropic quaternionic line, compare with 3.2.6 and 3.2.5. Hence, this is a real hypersurface N in the quaternionic projective space lHIPp+q+l. A point x EN corresponds to an isotropic quaternionic line £ C lHIP+q+2 • Choosing q E £, we obtain an identification TxN ~ q.l.. / £, where q.l.. denotes the real orthocomplement of q. In particular, the quaternionic orthocomplement of q gives rise to a codimension three subspace in TxN, which is immediately seen to be independent of the choice of q. Passing from q to qa for a E lHI, the quaternionic vector space structure on this subspace changes by conjugation by a. Hence, we can see the ingredients of the quaternionic contact structure coming up. While for n > 1, quaternionic contact structures in dimension 4n + 3 behave uniformly, there are some special features in dimension 7. Consider the imaginary part of a Hermitian form (which is automatically definite in this dimension) as a map A2lR.4 ~ lR. 3. The space of all such maps has dimension 18 and it carries a natural action of the group GL(4,lR.) x GL(3,lR.), which has dimension 25. Now an element in the stabilizer of the bracket can be interpreted as an endomorphism of lR.4 EEl lR. 3 and as such it is by definition an automorphism of the quaternionic Heisenberg algebra. Conversely, such an automorphism defines an element in the stabilizer. Now by Proposition 4.3.1 we know that the automorphism group is Go. In this dimension, go = lHI EEl sp(l), so this has dimension 7. This shows that the orbit of the bracket in L(A 2lR.\lR. 3 ) under GL(4,lR.) x GL(3,JR) has dimension 18 and is therefore open. This shows that any sufficiently small deformation of the bracket of the quaternionic Heisenberg algebra leads to an isomorphic Lie algebra. Hence, the distributions defining quaternionic contact structures in dimension 7 are generic, i.e. stable under small deformations. It turns out (see [Mon02, 7.12]) that the group GL(4, JR) x GL(3, lR.) has only two open orbits on L(A2JR4, JR3), so apart from quaternionic contact structures there is only one other generic type of rank four distributions in dimension seven. These correspond to split quaternionic contact structures to be discussed below. In particular, the genericity of the distributions implies that there are lots of examples. Dimension seven is also special from the point of view of the basic curvature quantities. For any value of n, the cohomology H2 (g_, g) has two irreducible components, one of which is contained in A2g~1 ®go. This component corresponds to a curvature, which can be interpreted as a bundle map A2 H ~ L(H, H). In dimension seven, the second cohomology component is contained in g~l ® g~2 ® g-2. Hence, it gives rise to a torsion, which can be interpreted as a bundle map H x T M / H ~ T M / H, and there are two independent basic curvature quantities in dimension seven. In higher dimensions, the second cohomology component is contained in A2g~1 ® g-2. Hence, it has homogeneity zero, and the corresponding component of the harmonic curvature has to vanish for regular normal parabolic geometries
4.3. EXAMPLES OF GENERAL PARABOLIC GEOMETRIES
435
by definition of regularity. Nontrivial harmonic curvature in this component would change the symbol algebras of the filtration. It turns out that in most respects torsion-free quaternionic contact structures in dimension seven behave similarly to higher-dimensional quaternionic contact structures. Structures with torsion behave differently, for example in twistor theory; see 4.5.5. The different location of the second cohomology also expresses the fact that the seven-dimensional quaternionic Heisenberg algebra is rigid in the algebraic sense (as we have seen above) while higher dimensional quaternionic Heisenberg algebras are non-rigid. As we have noticed already, in general it is not so easy to obtain examples for filtered manifolds with prescribed symbol algebras. In the case of quaternionic contact structures, however, there are general results ensuring the existence of many examples. The work of O. Biquard and C. Le Brun (on quaternion Kahler metrics, see [BiOOJ), even shows that there exist quaternionic contact structures on compact manifolds, which admit an infinite-dimensional family of nontrivial deformations. 4.3.4. Split quaternionic contact structures. The quaternions can be characterized as the unique four-dimensional real algebra, which admits a definite quadratic form N, which is compatible with the multiplication, i.e. satisfies N(pq) = N(p)N(q) for all p, q E IHL If one allows the quadratic form to be indefinite, then there is a second algebra of real dimension four with a multiplicative quadratic form. It turns out that an indefinite multiplicative quadratic form has to be of split signature (2,2). This second algebra is well known. It is simply the algebra M 2 (IR) of real (2 x 2)-matrices with the determinant as the multiplicative quadratic form. (This is the only size of matrices for which the determinant is quadratic.) In this context, we will refer to M 2 (IR) as the algebra of split quaternions and denote it by IHls to emphasize the analogy to the quaternions. One may also realize IHls using a basis {l,i,j,k} for which i 2 = j2 = 1, N(i) = N(j) = -1 and k = ij = -ji, whence k 2 = -1, and N(k) = 1. There is also a notion of conjugation on IHls, with the crucial property being that aa = N(a)l for all a E IHls . Since N is indefinite, this does not imply that any element of IHls is invertible. In the picture of matrices, the conjugation is given by forming the matrix of cofactors, which for (2 x 2)-matrices just amounts to permutation of the entries and sign changes. Having the conjugation at hand, one may consider Hermitian forms on the right IHls-vector space 1Hl~. It turns out that for each n, there is a unique such form up to isomorphism. Using the imaginary part of the standard form to define a bracket 1Hl~ x 1Hl~ ~ im(lHls ), one arrives at the definition of the split quaternionic Heisenberg algebras. DEFINITION 4.3.4. A split quaternionic contact structure on a smooth manifold M of dimension 4n + 3 is a sub bundle H C T M of rank 4n such that the bundle of symbol algebras of the filtration H =: T- 1 M C T- 2 M := T M is locally trivial and modelled on a split quaternionic Heisenberg algebra.
Parallel to the quaternionic case, we can now look at the special unitary algebras of the split quaternions, and the stabilizer of an isotropic split quaternionic line in there. Using matrix realizations similar to the real, complex, and quaternionic cases, dealt with in 1.6.3, 3.1.2, and 4.3.3, respectively, one concludes that one obtains a grading with the negative part forming a split quaternionic Heisenberg algebra. The split quaternionic special unitary algebras admit a more classical description, however.
436
4. A PANORAMA OF EXAMPLES
To see this, we first note that for a (2 x 2)-matrix A E IHIs and II := (~(}), we get II = -llAtll. Now we can view (n x n)-matrices with entries from IHIs as real (2n x 2n)-matrices. In particular, we define .JJ to be the matrix of this type which has ll's on the main diagonal and zeros everywhere else. Then for a matrix A = (Aij) we get A.JJ = (Aijll) and .JJt A = (llAji ). Now A is skew symmetric with respect to .JJ if and only if Aijll = -llAji . This is equivalent to Aij = -Aji and hence to the fact that A is skew Hermitian. But now .JJ evidently defines a symplectic form on jR2n, which shows that the unitary algebras of IHIs can be identified with the symplectic algebras sp(2n, jR). The corresponding parabolic subalgebra is given as the stabilizer of an isotropic plane in jR2n. Having this description at hand, Proposition 4.3.1 shows that split quaternionic contact structures in dimension 4n + 3 are equivalent to regular normal parabolic geometries of type (Sp(2n +4, jR), P), where P is the stabilizer of an isotropic plane in jR2nH. In particular, the homogeneous model of these split quaternionic contact structures is the Grassmannian of isotropic planes in jR2nH. The basic facts about split quaternionic contact structures are closely parallel to quaternionic contact structures. In dimension seven, one may see from the dimensions of the Lie algebras involved that the bracket on the split quaternionic Heisenberg algebra must have open orbit in L( A2jR4 , jR3), so this must be the second type of generic rank four distributions. In particular, there are many examples of seven-dimensional split quaternionic contact structures. In this dimension, there are two basic curvature quantities, one curvature and one torsion, whose form is exactly as in the quaternionic contact case. In higher dimensions there is a single curvature quantity (again of the same form as in the quaternionic contact case) which provides the complete obstruction to local flatness. 4.3.5. Rigid geometries. It can happen that for a semisimple Lie algebra 9 and a parabolic subalgebra p, the second cohomology group H2(g_,g) is concentrated in nonpositive homogeneities. The full information when this happens can be found in the article [Ya93]. If this is the case, then any regular normal parabolic geometry has vanishing harmonic curvature, and hence is locally flat by Theorem 3.1.12. While this means that the theory of regular normal geometries is rather vacuous, it leads to interesting rigidity results for the underlying structures. An interesting example for this phenomenon is provided by octonionic contact structures and split octonionic contact structures. These structures are related to exceptional Lie algebras of type F4 • Consider the complex simple Lie algebra of that type and the parabolic subalgebra corresponding to )( 0----$0 o. By Proposition 3.2.2, the subalgebra go determined by this grading is the sum of a onEMiimensional center and a simple Lie algebra of type B 3. Since F4 has dimension 52 and B3 has dimension 21, we conclude that dim(g_) = 15 in this case. The expression for the highest root in Table B.2 in Appendix B shows that the corresponding grading is a 121-grading. From the list of roots in that table one easily concludes that dim(g-l) = 8 and dim(g-2) = 7. Hence, passing to any real form of this situation, we will arrive at a geometry which exists on manifolds of dimension 15 and involves a subbundle of dimension 8 in the tangent bundle. Using the recipes from 3.2.18 to determine the Hasse diagram for this parabolic, Kostant's version of the Bott-Borel-Weil theorem implies that both Hl (g_, g) and H2(g_, g) are concentrated in negative homogeneous degrees. By Proposition 4.3.1,
4.3. EXAMPLES OF GENERAL PARABOLIC GEOMETRIES
437
the geometries in question are given by a rank 8 subbundle H c T M such that sections of H together with brackets of two such sections span T M and such that the bundle of symbol algebras is locally trivial and modelled on 9-. As we have seen above, Theorem 3.1.12 implies that any such geometry is locally flat and hence locally isomorphic to the homogeneous model. From Table BA in Appendix B, we see that the split real form F I and unique noncompact, nonsplit real form F I I of F4 admit such a grading. The semisimple part of 90 will be .50(3, 4), respectively, .50(7) in the two cases; see Remark 3.2.9. The nilpotent Lie algebra 9_ obtained in these two cases are related to the octonions, respectively, to the split octonions. In our discussion of quaternionic and split quaternionic contact structures, we have met the notion of a composition algebra. This is a finite-dimensional real alternative algebra (Le. any two elements generate an associative subalgebra) endowed with a quadratic form which is compatible with the multiplication. The quaternions and the split quaternionins are the two real composition algebras of dimension four. There are only two real composition algebras of dimension larger than four, and they both have dimension eight. For one of them, the quadratic form is definite, while for the other it has split signature (4,4). These two algebras are called the octonions 0 and the split octonions Os, respectively. On both 0 and Os, there is a natural conjugation map, and (a, b) f-4 ab can be viewed as the standard Hermitian form on 0, respectively, Os. Using the imaginary parts of these, one obtains the octonionic, respectively, the split-octonionic Heisenberg algebra. It turns out that these are exactly the algebras 9_ for the two real forms described above. Due to the non-associativity of the (split) octonions, the notion of (split) octonionic vector spaces does not make sense, so there are no higher-dimensional (split) octonionic Heisenberg algebras. With a bit of trickery, one may, however, define projective planes over 0 and Os. This is done via passing to skew Hermitian 3 x 3-matrices with entries in one of the algebras. Starting from 0, the result is the exceptional Jordan algebra of dimension 27. The projective plane OP2, respectively, Osp2 is then defined as the subspace of these skew-Hermitian 3 x 3-matrices consisting of projections of (octonionic) rank one. It turns out that these spaces are homogenous under the appropriate real form of F4 (which admits an interpretation as automorphisms of the Jordan algebra of skew-Hermitian 3 x 3-matrices, with isotropy group the appropriate parabolic subgroup). More information on these issues (in the case of 0) can be found in [Ba02]. Parallel to the quaternionic case, one defines a (split) octonionic contact strcuture on a smooth manifold M of dimension 15 as a subbundle H C T M such that sections of H together with Lie brackets of two such sections span T M and such that the bundle of symbol algebras is locally trivial and modelled on a (split) octonionic Heisenberg algebra. In view of the above discussion, we obtain the following result, which is stated in [BiOO] with the observation that this follows form Yamaguchi's results in [Ya93] atributed to R. Bryant. PROPOSITION 4.3.5. Any octonionic (respectively split octonionic) contact structure is locally isomorphic to Op2 (respectively OsP2).
4.3.6. Parabolic geometries in dimensions one and two. So far, we have met many different examples of geometric structures in this chapter, each of them in a distinguished dimension (or series of dimensions). Now we will systematically
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4. A PANORAMA OF EXAMPLES
determine all possible parabolic geometries in the lowest dimensions 1 :::; m :::; 5. In fact, we have met most of these geometries already. The dimension of the geometry for a given (g, p) is computed easily using the data in the tables in Appendix B. Indeed, the geometry may exist only in dimension m = dimg_, and the known dimension g (as listed in Table B.l) is given by dimg
= dim go + 2m.
The real dimension of a real form of a complex Lie algebra equals the complex dimension of the complex algebra, so again we only need the information from Table B.1. For the underlying real algebra of a complex one the real dimension is of course twice the complex dimension. The list of real forms can be found in Table B.4. Furthermore, the same table describes the semisimple parts of go once we describe the choice of p by Satake diagrams with crossed nodes. The corresponding rules exhausting all possible choices were discussed in Remark 3.2.9. In particular, the real dimension of the center is equal to the number of crosses while the Satake diagram of the semisimple part of go is obtained by removing in the Satake diagram for g all crossed nodes, all edges connecting to these nodes, and all arrows joining two crossed nodes. Dimension one. Adding any type of node (crossed or uncrossed) to a crossed Satake diagram or changing an uncrossed node into a crossed node clearly increases the corresponding dimension m. Thus, the only possibility to search for dimension one homogeneous spaces is the diagram for A1 with the only node being crossed. The homogeneous model is 8L(2,'R.)jB = 'R.p1, i.e. a circle. Here B denotes the Borel subgroup of G. The other real form 8U(2) of 8L(2, q is compact and hence does not admit any parabolic subgroups. The first cohomology H1 (p, g) has highest weight -4).1 and hence its dual H1(g_, g) is contained in homogeneity two. In analogy with the developments in 3.1.15 and 3.1.16 this implies that normal parabolic geometry of type (G, B) is determined by the underlying P-frame bundle of degree two, but since we are dealing with a Ill-grading here, this is already the full Cartan geometry. So there is no normalization procedure there and the whole geometry is given by the choice of the Cartan connection. (It turns out that the underlying P-frame bundle of degree one is the full second order frame bundle of the one-dimensional manifold M.) On the other hand, since we are in dimension one, there cannot be any curvature. Thus, all Cartan geometries of type G j B are locally isomorphic to the projective line. Since the projective line comes equipped with the space of distinguished projective parametrizations, the Cartan connection on the one-dimensional M carries locally this property over from the homogeneous model. The best way to describe such a geometry is following the description of locally flat geometries in Remark 1.5.2 (3) by an atlas for M with charts that have values in open subsets of 'R.p 1 and chart changes which are restrictions of projective transformations. The Cart an connection can, however, also be described as a linear connection plus a choice of Rho tensor, which turns out to be a fruitful point of view; see [BE90]. Dimension two. To get an overview, we next list all possible crossed Satake diagrams with two nodes together with the dimension of gjp. The types of diagrams we have to consider are A1 x A 1, A 2 , B2 ~ C 2 , and G 2 . For the A-types, we obtain the following five possibilites:
4.3. EXAMPLES OF GENERAL PARABOLIC GEOMETRIES
(g,p) dimg/p
x
x 2
x
x
"'-..A
2
x--o
x--x
x--x
2
3
3
439
"'-..A
For diagrams of types Band G, we get 7 further possibilities, namely (g,p)
x =:=0
x=:=.
o=:=x
x=>=x
X:;::EO
O:3EX
x:::e::x
dim g/p
3
3
3
4
5
5
6
Hence, there are only three possible types of parabolic geometries on two-dimensional manifolds. They all belong to classical objects studied in the literature, however. This is most obvious for the last of these three diagrams, which simply gives rise to projective structures in dimension 2 as discussed in 4.1.5. Recall that this geometry has the curvature obstruction concentrated in homogeneity three, so the only obstruction to a local flatness is a Cotton-York type tensor. Next consider g = .5((2, q, viewed as the real algebra with p = b, the Borel subalgebra (the second case in the first table above). The first crucial observation to be made here is that, as a real Lie algebra, .5((2, q is isomorphic to .50(3,1). This isomorphism is induced via the 4-dimensional invariant subspace A1,1C2 in the real representation AiC2 which has dimension six. The Lorentzian inner product is induced by the wedge product using the fact that AiC 2 ~ lR. This representation integrates to the group G = SL(2, q which thereby, in view of simple connectedness, gets identified with Spin(3, 1). Likewise, the subgroup Go ~ C* can also be viewed as a two-fold covering of CSO(2, JR). Thus, a Cartan connection of type G / B over M will induce a 2-dimensional conformal structure. On the other hand, on any Riemann surface M, the above noticed isomorphisms of groups automatically gives rise to an almost complex structure (which must be integrable by dimensional reasons). So we have encountered the two-dimensional conformal geometry, which we excluded from our considerations in 4.1.2. A Cartan geometry of type G / B, however, is a much stronger structure than that. Clearly, the algebra is Ill-graded and the first cohomology H1(g_, g) is concentrated in homogeneity two. Therefore, exactly as for one-dimensional projective structures, the Cartan connection itself represents the defining ingredient of the structure in question. These geometries, which are modelled on S2, viewed as the projectivized light cone in JR(3,1), are called Mobius structures in the literature. A detailed treament of such structures and links to Einstein-Weyl geometries is presented in [Ca198]. The first entry in our table of A-type diagrams leads to a closely related geometry. Here we get g = .5((2, JR) x .5((2, JR) and p is the product of the Borel subalgebras of the components. Now g is isomorphic to .50(2,2). On the group level, we can realize this by considering the action of G = SL(2, JR) x SL(2, JR) on M2(JR) defined by (A, B)·X := AXB- 1 , which leaves the quadratic form induced by the determinant invariant. This identifies G with Spin(2, 2), with the two standard representations corresponding to the two spinor representations. Likewise, Go = JR* x JR* is realized as a two-fold covering of SOo(I, 1). Note that this is a split--{}uaternionic (see 4.3.4) version of the two-fold covering Sp(1) x Sp(I) ~ SO(4) constructed via quaternions.
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Thus again, every geometry modeled over the product lR.p1 x lR.p1 carries a product structure (which must be integrable by dimension reasons) and we deal with the pseudo-Riemannian conformal structure of signature (1,1) defined by the cone determined by the product structure. The Cartan geometry is then given by the choice of the Cartan connection. Let us briefly analyze how our general procedures of 4.1.1 work in these two special cases. We have started with the choice of appropriate connections on M. In our two cases, there are torsion-free connections compatible with either the almost complex structure or the almost product structure on the manifold, so these have to be the distinguished ones. Clearly, the space of all such connections will be parametrized by one-forms (it is easy to verify this directly, cf. 1.6.4). Next, we can solve the equation 8*(R + 8P) = 0 from Lemma 4.1.1, but the solution is not uniquely determined. In fact, the freedom in the choice of P is given fiberwise by the homogeneity two cohomology component H1(g_lttJh #- O. As we have seen in Theorem 4.1.1, the knowledge of a torsion-free connection "I and the corresponding tensor P determines the normal Cartan connection w, so P cannot be uniquely determined. Fixing one choice of P for a connection "I, we obtain a Cartan connection w and hence a parabolic geometry. Describing the geometry in that way, all the conclusions of Theorem 4.1.1 remain valid. The standard computation shows that the entire cohomology H2 (g-l, g) sits in homogeneity 3. Therefore, the complete obstruction to local isomorhisms to the homogeneous Mobius structure on 8 2 , respectively, to the product of homogeneous projective structures on lR.p1 is given by the Cotton-York tensor d'YP. 4.3.7. The dimensions three through five. Some of the possible types of three-dimensional parabolic geometries are already in tables in 4.3.6. In particular, the last two missing types of A-type diagrams (the two rightmost entries) correspond to Lagrangean contact structures, respectively, CR-structures in the lowest possible dimension three. We have discussed these examples already in 4.2.3 and 4.2.4, respectively. It should be remarked that historically three-dimensional CRstructures were among the early examples of geometries which were studied using Cartan connections; see [Car32). In these low-dimensional cases, the harmonic curvature is concentrated in homogeneity three and so a version of Cotton-York tensor appears as the only curvature obstruction to local flatness. The first two colums in the table of digrams of type Band G in 4.3.6 correspond to conformal pseudo-Riemannian structures of the two signatures (3,0) and (2,1) which are possible in dimension three. We have discussed them in detail in Section 1.6 and in 4.1.2. The complete obstruction to the local flatness is the Cotton-York tensor of any of the Levi-Civita connections of the metrics in the conformal class, and this is the historical source for the name of this type of tensors. The third diagram in that table is better understood by interpreting it as an algebra of type C2 • Then it corresponds to contact projective structures in the lowest possible dimension three. These have been discussed in 4.2.6. This exhausts the three-dimensional geometries which showed up in the tables in 4.3.6, i.e. which correspond to Satake diagrams with two nodes. However, there are also three-dimensional geometries coming from diagrams with three nodes. In the following table, we list all available simple real algebras 9 of rank 3, together with their possible gradings and the dimensions of g/tJ. The choice of the parabolic
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subalgebra is indicated above the colums by the positions of crosses over the Satake diagram: 5[(4, JR) 5[(2, IHI) 5u(1,3) 5u(2,2) 50(4,3) 50(5,2) 50(6,1) 51'(6, JR) 51'(2,1)
xoo oxo oox xxo xox oxx xxx 3 4 3 5 5 5 6 4 5 4 5 6 5 7 6 8 8 8 9 5 7 8 5 8 5 7 6 8 8 8 9 7
In particular, only two (isomorphic) pairs lead to three-dimensional quotients gil', namely the first and third entry in the first row. These are three-dimensional projective structures as discussed in 4.1.5. Apart from the geometries corresponding to simple algebras (which we all have listed), there are also the ones corresponding to semisimple algebras. They are modeled on products of homogeneous spaces of dimensions one and two (two different cases with models CP2 x JRp1 and JRp1 x JRp1 X JRp1). They are never determined by some underlying structure, so the choice of Cartan connection is an essential ingredient of the geometry. While the almost product structures may lead to interesting curvature components, we will not go into detail here. Dimension four. The first example here comes from the table in 4.3.6, namely an algebra of type B2 ~ C 2 with the grading coming from the Borel subalgebra. For the interpretation it is better to view it as C2 , so we are looking at the Borel subalgebra in 51'(4, JR). This is the lowest dimensional example of so called contact path geometries which we will not discuss in this book. Contact path geometries can be viewed as generalizations of contact projective structures with vanishing contact torsion, in a similar spirit as we will exhibit path geometries as a generalization of projective structures in 4.4.3 below. These geometries are j3j-graded, since the highest root of C2 has the form 2(\(1 + (\(2. In dimension four, we get dim(g-d = 2, while the other two components are one-dimensional, so this is the smallest possible j3j-graded algebra. Four-dimensional contact path geometries have been studied classically (although not under this name) due to their relation to the geometry of a single ODE of third order modulo contact equivalence. This geometry has been studied in [Wii1905] and worked out within the Cartan theory in [Ch40] in the context of 3rd order ODEs. An account on contact path geometries in higher dimensions can be found in [Fox05b]. Next, we go through the list of parabolic subalgebras in simple Lie algebras of rank three, such that gil' is 4-dimensional. According to the table above, there are only three examples, all of type A3 with the middle node crossed. These are pseudo-Riemannian conformal structures of the three signatures which are possible in dimension four. We discussed them thoroughly in Section 1.6, and also in 4.1.2, 4.1.4, 4.1.10, and 4.1.9. The two tables above also show that among the simple Lie algebras of rank bigger than three, only the Ai series may lead to further four-dimensional geometries. In fact, the only possibility is the algebra 5[(5, JR) with the grading corresponding
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to either the first or the last node. These are projective structures in dimension four; see 4.1.5. In addition to these real simple algebras with simple complexification, one can also obtain four-dimensional geometries from semismiple Lie algebras or from the underlying real algebras of complex simple Lie algebras. There are examples in this dimension, in which these geometries are determined by an underlying structure. Most notably, there are the cases with homogeneous models JRp2 x JRp2 as a homogeneous space of SL(3, JR) x SL(3, JR), respectively, of CP2 as a homogeneous space of SL(3, q. We will not go into detail of these examples here. One can expect that they behave similarly as the examples of CR-structures of CR-dimension and codimension two which are discussed in 4.3.9 and 4.3.10 below.
Dimension five. From the second table in 4.3.6 we get two geometries in dimension five corresponding to the two maximal parabolic subgebras in a simple Lie algebra 9 of type G 2 • The first of these was discussed in 4.3.2 leading to the geometry of generic rank two distributions on manifolds of dimension five. The study of this example has a long history, it has been among the first applications of Cartan's approach in the early twentieth century; see [CarlO]. The second maximal parabolic subalgebra gives rise to the (unique) parabolic contact structure associated to a simple Lie algebra of type G 2 , which we discussed in 4.2.8. Let us next collect the gradings from the table of rank three Lie algebras above, which lead to five-dimensional geometries. Let us go through the table column by column. In the first column, we find the conformal pseudo-Riemannian structures in the three signatures (5,0), (4,1), and (3,2), which are possible in dimension five. We have discussed them all in section 1.6 and also in 4.1.2. The other 5-dimensional geometry in the first column are five-dimensional projective contact structures as treated in 4.2.6. The next examples arise from the algebras of type A3 in the fifth column. The crosses in the diagram are exactly at the position of the nonzero coefficients in the expession of the highest weight A1 + A3 of the adjoint representation. Thus, all three geometries are parabolic contact structures. In the first row, we find Lagrangean contact structures, while the next two rows correspond to partially integrable almost CR-structures of the two possible non-isomorphic signatures (2,0) and (1,1). The final five-dimensional example in the above table is provided by 5((4, JR) with the grading corresponding to either the first two or the last two roots. These are generalized path geometries in dimension five, which we will discuss in 4.4.3 in connection with correspondence spaces. Now, we come to diagrams with four nodes. Adding a node or cross to an existing Satake diagram with crosses always increases the dimension. Thus, already the table with diagrams with three nodes shows that we may restrict ourselves to the types A and to check the remaining types D, E, and F. But the lowest available dimension with D4 and one cross is already six (the conformal Riemannian 6dimensional structures), while the other option with one cross is of dimension 9. The F diagrams with one cross produce dimensions 15 and 20. The A diagram with four nodes provides examples of dimensions 4 and 6. Thus the only remaining case for us are the five-dimensional projective structures coresponding to the diagram with five nodes and the cross over the first or the last one. Of course, again examples coming from products of homogeneous spaces of lower dimensions are available. With increasing number of nodes and crosses, the
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dimensions expand very quickly. However, there are some more geometries of very special interest in quite low dimensions. We shall see a particularly nice example next.
4.3.8. Codimension two CR-structures on six-dimensional manifolds. It is a well-known phenomenon that submanifolds may inherit geometric structures
from an ambient space. The best known example is the induced Riemannian metric on a submanifold in a Riemannian manifold. In the realm of parabolic geometries, well-known examples are provided by generic hypersurfaces in the projective space lR.lPm +1, which inherit a conformal structure (related to the second fundamental form), and by generic hypersurfaces in en +1 (or in some complex manifold) which inherit a CR-structure; see 4.2.4. There we have also discussed the more general partially integrable almost CR-structures (of hypersurface type). These geometries are obtained on codimension one submanifolds in ambient manifolds with an almost complex structure. The first steps towards the induced CR-structure on a real hypersurface have an analog for submanifolds of higher codimension. For a submanifold M in a manifold M endowed with an almost complex structure j and each point x EM, there is the maximal complex subspace Hx = TxM n j(TxM) c TxM. Assuming that these spaces are of constant complex dimension n (which is automatically satisfied in the case of a hypersurface), they form a smooth subbundle H C TM. The almost complex structure j restricts to an almost complex structure Jon H. Next Q := TMjH is a real vector bundle on M, and by construction, the rank of Q equals the real co dimension k of M in M. Viewing H C T M as a filtration of the tangent bundle, the associated graded then is gr(T M) = Q E9 H. This is the usual setup for the CR geometry in complex analysis. In the language of (abstract) CR geometry, a complex subbundle H C T M of complex rank n on a manifold M of real dimension 2n + k is called an almost CR-structure of CR-dimension nand codimension k. Having H C TM, we get the Levi bracket L : H x H ~ Q. Similarly to the case of hypersurface type CR-structures in 4.2.4, we can next impose nondegeneracy and integrability conditions. In the case of a submanifold in a complex manifold, integrability automatically follows from integrability of the ambient complex structure. In the abstract setting, the partial integrability condition from 4.2.4 continues to make sense. We call an abstract almost CR-structure partially integrable if L(J~, J",) = L(~,,,,) for all~,,,, E H M, i.e. if L is totally real. If this is the case, then for each x E M, there is a Hermitian form on the complex vector space Hx ~ en with values in the complex vector space Qx ® e ~ ek whose imaginary part is Lx. Now suppose that h: en x en ~ ek is an arbitrary vector-valued Hermitian form. Then there is a natural co dimension k submanifold in en + k whose Levi bracket in each point is isomorphic (in the obvious sense) to the imaginary part of h; see [F092]. This is the quadric associated to h. Let us view en+k as en x ek with coordinates z on en and w = u + iv on ek • Then we define the quadric as Qh := {(z, u
+ iv)
: v = h(z, z)} C
en+k.
This is a smooth submanifold, since it can be interpreted as the graph of a smooth function. Now the tangent spaces of Qh are given by T(z,u+ih(z,z)) Qh
= {(w, r
+ 2i re(h(z, w)))
: wEen, r E lR.}.
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Using this, one easily verifies that H(z,u+ih(z,z» = {(w, 2ih(w, z)) : w E (Cn} and that the Levi bracket is induced by a nonzero multiple of the imaginary part of h. For k = 1 and h nondegenerate one easily verifies that the quadric Qh is locally isomorphic to the homogeneous model of partially integrable hypersurface type CR-structures as discussed in 4.2.4. Next, we need an appropriate nondegeneracy condition. Of course, we will require the evident condition that .c(~, 1]) = 0 for all 1] implies ~ = O. This is much to weak, however, since it does not even ensure that .c is onto. We will call the structure nondegenerate at a point x E M if, in addition, for each nonzero '¢ E Q;M the skew symmetric bilinear map .c~ : HxM x HxM -+ IR defined by .c~(~, "') = '¢(.cx(~, "')) is nonzero. Identifying Qx with IRk, the latter condition just means that the components of the vector valued Levi form are linearly independent. An almost CR-structure (H, J) on M is called nondegenerate if it is nondegenerate at each point x EM. From the definition it is evident that nondegeneracy is an open condition. For codimension one, it reduces to the nondegeneracy condition from 4.2.4 and ensures that the Levi bracket is characterized by the signature of the corresponding Hermitian form. Hence, in codimension one, nondegeneracy ensures that the isomorphism type of the Levi bracket is locally constant. This is no more true in higher codimension. Indeed, for general nand k, (Ck-valued Hermitian forms on (Cn admit continuous invariants; see e.g. [GaMi98]. Correspondingly, one may construct continuous families of non-isomorphic intersections of k quadrics; see e.g. [ES94, ES99]. The situation simplifies drastically, if one looks at the case of CR-dimension n = 2 and codimension k = 2 with the basic examples provided by submanifolds of real dimension six in complex manifolds of complex dimension four. Assuming nondegeneracy and partial integrability, there are only three possible types of points, which we shall call hyperbolic, exceptional, and elliptic, according to parts (1), (2), and (3) in the following lemma. LEMMA 4.3.8. Let M be a six-dimensional smooth manifold endowed with a nondegenerate, partially integrable almost CR-structure (H, J) of CR-dimension two and codimension two. Then at each point x E M exactly one of the following possibilities happens: (1) There are two distinct points ['¢1] and ['¢2] in the projectivization P(Q;) such that.c1/J : A2Hx - t IR is degenerate if and only if'¢ E ['¢1] or'¢ E ['¢2]. In this case Hx = Hl1/J 1 l EB Hl1/J21, where each of the null spaces H;1i for .c1/Ji is a complex line. (2) There is one point ['¢o] E P(Q;) such that .c1/J is degenerate only if'¢ E ['¢o]. (3) The forms .c1/J are nondegenerate for all nonzero elements,¢ E Q;. PROOF. Assume first that .c1/Jl and .c1/J2 are degenerate for two linearly independent forms '¢i E Q;. The subsets Hl1/J i l are complex lines since the Levi form is totally real. If ~ E Hl1/J 1 l n Hl1/J21, then C'''(e, "') = 0 for all", E TxM and '¢ E Q;. Thus, ~ = 0 by nondegeneracy of .c, and Hx = Hl1/Jll EB Hl1/J21. If '¢ = a'¢l + b'¢2 is another form for which .c1/J is degenerate and both a, b ::j:. 0, then any two of the lines Hl1/Jl, Hl1/J 1 l, and Hl1/J 21 are complementary. Thus, for each ~ E Hl1/J 1 l, there is a unique 4>(e) E H[1/J21 such that ~ + 4>(~) E Hl1/Jl, and the map 4> defined in this way is a linear isomorphism. According to our choices,
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EXAMPLES OF GENERAL PARABOLIC GEOMETRIES
445
+ 2. The first two components are always represented by maps E q-l X q-2 V q-l X q-2
-+ -+
V q-l,
qQ.
Geometrically, they correspond to a torsion TE : ExT M / H -+ V, respectively, a curvature p : V x TM/H -+ End(V), where H = E ffi V. For n = 2, the last component is represented by maps A2g~1 -+ g~l' so it corresponds to another torsion TV : A 2 V -+ E. However, for n > 2, the component is represented by maps A2g~1 -+ g-2. This has homogeneity zero, and hence cannot correspond to a harmonic curvature component of a regular normal parabolic geometry. At this stage, it is not clear at all how to obtain examples of generalized path geometries. The simplest source of such examples is provided by correspondence spaces of projective structures. THEOREM 4.4.3. Let (N, [V'J) be a classical projective structure on a smooth manifold N of dimension n + 1. Then the projectivized tangent bundle M := PT N carries a canonical generalized path geometry. Explicitly, the bundle H C TP(T N) is the tautological bundle, i.e. ~ E He if and only ifT7r'~ lies in the line f. c T7rCl)N. The sub bundle V cHis the vertical sub bundle of 7r : P(T N) -+ N, and the line bundle E c T M is determined by the horizontal lifts of the connections in the projective class. The curvature p of this generalized path geometry vanishes identically. If n = 2, then also the torsion TV vanishes identically. On the other hand, vanishing of TE is equivalent to (N, [V'J) being locally projectively fiat. PROOF. Let (p: 9 -+ N,w) be the normal parabolic geometry of type (G,P) associated to the projective structure. Similarly, as in the proof of Proposition 4.4.2, one shows the subgroup Q c P is the stabilizer of the line corresponding to q~l in g/p. Since P acts transitively on the projectivization P(g/p), we get P/Q ~ P(g/p). Therefore, the correspondence space CN for Q C P is naturally identified with PT N. Torsion freeness of w then implies that the curvature function has values in p = q~l ffi q. This shows that the parabolic geometry (g -+ CN,w) is regular and it is normal by Proposition 4.4.1. Hence, it is the canonical parabolic geometry associated to a generalized path geometry. The facts that the subbundles V and H = E ffi V of TPT N are the vertical subbundle, respectively, the tautological subbundle are verified exactly as in Proposition 4.4.2. Any connection V' in the projective class gives rise to a horizontal lift of tangent vectors from N to TN; see 1.3.1. By linearity, the horizontal subspaces descend to a horizontal distribution on P(T N), thus defining a general connection on this fiber bundle; see 1.3.2. Passing to a projectively equivalent connection V, the horizontal lift of a vector 'TJ E TxM at the point ~ E TxM changes by a linear combination of ~ and 'TJ. In particular, if'TJ and ~ are collinear, then the only change is by a multiple of~, which is killed by the projection to P(TN). In particular, in a
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point f E P{T N) the horizontal lifts of elements of f are independent of the choice of the connection from the projective class. Hence, one obtains a line subbundle of the tautological bundle, which is complementary to the vertical subbundle. From the description in 4.1.5 we conclude that the horizontal subspaces in TN for the connections in the projective class are just the images of subspaces of the form w;.;-l(P_l). Using this, one immediately verifies that the line subbundle from above exactly corresponds to E. The harmonic curvature components p and TV (for n = 2) take one or two entries from V, so they have to vanish by part (3) of Proposition 1.5.13. But then vanishing of TE is equivalent to vanishing of the harmonic curvature and hence to local projective flatness. 0 4.4.4. Twistor spaces for generalized path geometries. The construction of correspondence space associated to projective structures in Theorem 4.4.3 suggests a way to obtain more examples for generalized path geometries. Suppose that N is an arbitrary smooth manifold, and consider the projectivized tangent bundle M := PT N as above. Let V c H c T M be the vertical subbundle, respecitively, the tautological subbundle. These two bundles exist independently of any choice of a projective structure. Choose a linear connection V on TN. Then by Theorem 4.4.3 we obtain a line subbundle E" C H (which depends only on the projective class of V) that is complementary to V and defines a generalized path geometry on M. Now suppose that E cHis any line subbundle which is complementary to V. Then any section E r(E) can be uniquely written as = 6 +6 with 6 E r(E") and 6 E r(V). By construction for any point x E M we have el(X) = 0 if and only if e(x) = O. Now for a section T/ E r{V) we obtain [6, T/] E r(V) and hence [e, T/](x) E Hx if and only if [6, T/](x) E Hx. This shows that the subbundles E, V c T M satisfy the conditions of Definition 4.4.3, and thus give rise to a generalized path geometry on M = PT N. Such a geometry on PT N is classically called a path geometry on N. Here the word "path" has to be understood as "unparametrized curve" or "immersed 1-dimensional submanifold". A subbundle E C H c TPT N as above defines an integrable distribution. The fact that E is transverse to V implies that the restriction of the projection 7r : PT N - N to an integral submanifold of E is an immersion. Projecting the leaves of the foliation defined by E therefore gives rise to a family of paths in N. For a point f E PTN defined by a line f C TxN, let c C N be the projection of the leaf through f. Since E C H, we see that Txc = f. Hence, we conclude that for each point x E M and each direction f C TxM, there is a unique path in the family which passes through x in direction f. Consider an immersed submanifold c in a smooth manifold N. Then this canonically lifts to an immersed submanifold c in PTN, whose points are the lines given by the tangent spaces of c. A local regular parametrization of c canonically lifts to a regular parametrization of c. Since c is a lift of c, its tangent spaces lie in H and are transverse to V. Now a family of paths in N with exactly one path through each point in each direction therefore lifts to a one-dimensional foliation of PT N . We call the family of paths smooth if this foliation is smooth. By construction, the subbundle E C TPT N defined by such a foliation is contained in H and transverse to V, and we get
e
e
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OBSERVATION 4.4.4. A path geometry on N is equivalent to a smooth family of one-dimensional immersed submanifolds in N with exactly one submanifold through each point in each direction. Such a path geometry gives rise to a generalized path geometry on PT N.
The description of path geometries as generalized path geometries (i.e. via the bundles E and V) has the immediate advantage of generalizing to open subsets. It is not necessary to have one path through each point in each direction. It is sufficient to have pathes through each point in an open set of directions (which may depend on the point) in order to get a generalized path geometry and hence a parabolic geometry. If n =12, this locally exhausts all generalized path geometries: PROPOSITION 4.4.4. Let M be a smooth manifold of dimension 2n+ 1 endowed with a generalized path geometry defined by sub bundles E, VeT M. If n > 2, then the sub bundle V c TM is automatically involutive. In the notation of 4.4-3, this means that there are always local twistor spaces for l' ::J q. If 1/J : U ~ N is a sufficiently small local twistor space of this type, then 1/J canonically lifts to an open embedding'¢; : U ~ PT N such that T'¢; maps V, respectively, E EEl V to the vertical sub bundle, respectively, the tautological sub bundle of PT N ~ N. In particular, M is locally isomorphic to a path geometry on N. PROOF. Let (p: g ~ M, w) be the parabolic geometry of type (G, Q) determined by the given generalized path geometry. Let T E n,2(M, T M) be the torsion of w. By Lemma 1.5.14 we have to prove that T maps V x V to V to conclude integrability of V. By Theorem 3.1.12, the lowest nonzero homogeneous component of the Cartan curvature", of w is harmonic. Looking at the description of the harmonic curvature in 4.4.3, we conclude that, for n > 2, ", is homogeneous of degree ? 2. Since V c T- 1 M, we see that the restriction of T to V x V has to vanish. Now let 1/J : U ~ N be a local leaf space for the foliation determined by the involutive subbundle V C TM. For each x E U, the tangent map Tx1/J by definition induces a linear isomorphism TxM/Vx ~ T..p(x)N. Hence, Ex E TxM is mapped to a line in T.p(x)N, which determines a point ,¢;(x) E PTN. Choosing a local nonvanishing section u of E, we can write'¢; = qoT1/Jou, where q: TN\N ~ PTN is the natural surjection. Hence, '¢; is smooth. From the construction it is evident that T'¢; maps V to the vertical subbundle and E to the tautological subbundle of PT N ~ N. Hence, we can complete the proof by showing that '¢; has invertible tangent maps, since then it locally is an open embedding. To do this, fix a point Xo E U. Since 1/J is a surjective submersion, we can choose local coordinates (uo, ... , u 2n ) around Xo such that 1/J( uO, ... , u 2n ) = (u o, ... , un). We will write Oi for the coordinate vector field a~.. Then by construction the subbundle V is, locally around Xo, spanned by {On+b ... , 02n}. Further, we take a locally nonvanishing smooth section = eiOi of the subbundle E C TM. We may choose the coordinates in such a way that ~(xo) = 00. Evidently, [OJ,e] = ~Oi. The assumption on the Lie bracket between sections of E and V thus implies that that matrix (~(xo)) with i = 1, ... ,n and j = n + 1, ... ,2n is invertible. By construction, we can use (uO, ... , un) as local coordinates around 1/J(xo) on N. Further, Txo1/J(Exo) c T..p(xo)N is the line spanned by 00, so the one-form duO restricts to a nonzero functional on this line. Now consider the set of all lines in tangent spaces with foot point contained in our chart, to which duO has
e
4.4. CORRESPONDENCE SPACES AND TWISTOR SPACES
465
nontrivial restriction. This is an open neighborhood of Txo'lj;(Exo) in PTN, on which we get local coordinates (uo, ... ,un,a\ ... ,an ) which are characterized by duile = ai(£)duoll' (This is an analog of the standard inhomogenous coordinates on ~pn.) We can easily compute the mapping -/p in these coordinates. Namely, by construction T-/p·e = ei{)i, where now the indices run only from 0 to n. But this means that, in our coordinates, -/P(uO, ... ,u2n ) = (uO, ... ,un,eleo, ... ,enleo). In the point xo, we have eo(xo) = 1 and ei(xo) = 0 for i > O. This implies that the partial derivative of i leO in direction of ui in Xo equals .gG(xo). Hence, from above we conclude that Txo -/P has the block form (& ~ ), where IT is the unit matrix 0 of size n + 1 and A is an invertible n x n-matrix. Hence, Txo -/P is invertible.
e
Identifying a generalized path geometry on M locally with a path geometry on PT N, we can next ask when M is locally isomorphic to a correspondence space. Otherwise put, this is the question when a path geometry on PT N comes from a projective structure [V'] on N. In view of our description of the harmonic curvature, this immediately follows from Theorem 4.4.1. COROLLARY 4.4.4. The paths of a path geometry on a manifold N can be realized as the unparametrized geodesics of a linear connection V' on TN if and only if the harmonic curvature P of the canonical Canan connection determined by the path geometry vanishes identically.
4.4.5. Generalized path geometries from Grassmannian structures. In our discussion of generalized path geometries in 4.4.3 we have realized the relevant parabolic subalgebra q C 9 = s[(n+2,~) as the intersection pnp, where p and p are the stabilizers of a line, respectively, a plane in the standard representation ~n+2 of g. Besides the inclusion q ~ p, which gives rise to the correspondence and twistor spaces studied in 4.4.3 and 4.4.4 above, we also have the inclusion q ~ p. Here it is better to choose the group G = 8L(n+2, ~), to take the subgroups P and P to be the stabilizers of the appropriate subspaces, and to put Q := P n P. This causes only minimal changes compared to the discussion in 4.4.3, so regular normal parabolic geometries of type (G, Q) are essentially generalized path geometries. On the other hand, by 4.1.3 normal parabolic geometries of type (G, p) are almost Grassmannian structures on manifolds of dimension 2n. Such a structure on a smooth manifold N comes with two auxiliary bundles E, F -+ N of rank 2 and n together with an isomorphism
9 be the inclusion and let us interpret the values of the curvature functions as bilinear maps on g, respectively, on 9 which vanish if one of their entries is from ji, respectively, from p. Since we view 9 as a subalgebra of g, the result of Proposition 4.5.2 simply reads as fi,(j(u))(X +ji, Y +ji) = ",(u)(X + 1', Y +1') for all X, Y E 9 C g. Since the inclusion of 9 induces a linear isomorphism g/q ---> gfji, this determines fi,(j(u)). By equivariancy of fi" it is determined by its restriction to j (9) C g. Since ()* is also P-equivariant, the same argument shows that to prove normality of W, it suffices to verify that ()*fi,(j(u)) = 0 for all u E g. To compute ()* K,(j (u)), we use Lemma 3.1.11, so we have to choose elements of g which project onto a basis of gfji. Again, since g/q ~ gfji, we choose these elements to be in g. More specifically, we let Xo E 9 be the element with A-block (~ and correspondingly I = 1, while all other blocks are trivial. Next, for i = 1, ... , p + q we denote by Xi and X p +q+i the elements of 9 which have all entries equal to zero except for one entry in the D-block which equals one, namely the element in the ith row of the first column for Xi and the one in the ith row of the second column for X P+ q + i . From the descriptions of the various subalgebras above it is evident that {Xo + ji, ... , X 2 (p+q) + ji} is a basis of g/ji, that Xi E I' for i > p + q, and that {Xo + 1', ... , X p +q + p} is a basis of g/p. Next, we determine the dual bases {Zo, ... , Z2(P+q)} of ji+ and {Zo, ... , Zp+q} of 1'+. Since there is no harm in replacing ()* by a nonzero multiple, we may use the trace-form of g rather than the Killing forms to induce the dualities for both algebras. Consider the matrix which has a 1 in the ith column of the first row of
g)
4. A PANORAMA OF EXAMPLES
488
the B-block and all other entries equal to zero. Then this lies both in p+ and in 13+, it pairs to 1 with Xi under the trace-form and to zero with all other X j . Hence, this element equals Zi and Zi, so in particular, we have Zi = Zi for i = 1, ... ,p+q. The elements Zi for i = p+q+ 1, ... , 2(p+q) can be described similarly, but we will not need them. Hence, it remains to determine Zo and Zoo Since Zo E p+ it may have nonzero entries only in the B-block and the block elIl,l. Since tr(ZoXi ) = 0 for i = 1, ... , 2(P + q), also the B-block must be zero, while tr(ZoXo) = 1 implies that e = 1/2. Likewise, one determines Zo using that it has to lie in 9 c 9 and have nonzero entries only in the first row and in the last colum and above the main diagonal. One obtains that the only nontrivial blocks in Zo are the A-block, which equals (~ 1b4 ), and correspondingly the one for e = 1/4. According to Lemma 3.1.11, we get 2(p+q)
8*K(j(U))(X) = 2 L
2(p+q)
[Zi,K(j(U))(X,Xi)]- L
i=O
K(j(U))([Zi,XJ,Xi )
i=O
for all X E g. As before, it suffices to take X E g. Now we have K(j(U)) (X, Xi) = /'i:(u) (X, Xi) and K(j(U))([Zi, X], Xi) = /'i:(U)(Y,Xi)' where Y E 9 has the property that Y + p = [Zi, X] + p. In both cases, the expression vanishes if i > p + q, since then Xi E p. Hence, in both summands, we only have to sum up to i = p + q. Now for i = I, ... ,p + q, we have Zi = Zi E 13+ and since X E 9 this implies [Zi, X] E p. From the definitions, one immediately concludes that [Zo, Zi] = 0, and hence 0 = tr(X[Zo, Zi]) = - tr([Zo, X]Zi) for all X and i = I, ... ,p + q. But this implies that [Zo, X] is congruent to a multiple of Xo mod 13, and we see that the second summand in the above formula does not contribute at all. Hence, we are left with p+q
8*(K(j(U)))(X) = 2[Zo - Zo, /'i:(u)(X, Xo)]
+ 2 L[Zi, /'i:(u) (X, Xi)]. i=O
On g, we are dealing with a Ill-grading, so [g,p+] c p. Hence, the second summand expresses the full normalization condition for (g, p), and therefore vanishes by normality of w. Finally, for Zo - Zo the only nonzero blocks are the A-block, 4 ), and correspondingly the one for e = 1/4. This matrix has which equals the property that it vanishes on V = ij.l and maps ij to a multiple of el. On the other hand, /'i:(u)(X,Xo) E g, so it vanishes on ij and has values in V. Hence, (Zo - Zo) O/'i:(u)(X, Xo) vanishes identically and /'i:(u) (X, Xo) o(Zo -Zo) vanishes on V. But from Corollary 1.6.8 we know that /'i:(u)(X,Xo) E 13 and its go~omponent is totally tracefree. In particular, it acts trivially on the preferred null line spanned by el, and we also get /'i:(u) (X, Xo) 0 (Zo - Zo) = O. 0
(g -V
4.5.5. Twistor theory for quaternionic contact structures. For all the other analogs of the Fefferman construction that we are aware of, stronger tools are needed in order to settle the question of normality of the induced Cartan connections. In the rest of this chapter, we will therefore only briefly outline several other examples of analogs of the Fefferman construction. The first of these examples has been mainly worked out by O. Biquard; see [BiOO]. This was done only in terms of the underlying structures without any reference to Cartan connections. In the picture of underlying structures, the construction is similar to the twistor
4.5. ANALOGS OF THE FEFFERMAN CONSTRUCTION
489
construction for quaternionic structures as described in 4.4.9 and 4.4.10. In terms of Cartan connections, it is closely parallel to the classical Fefferman construction. We have discussed quaternionic contact structures in 4.3.3. Fixing a signature (p, q) and putting n = p+q, these geometries exist on manifolds of dimension 4n+3. (In Biquard's work, only the definite case q = 0 is considered.) A quaternionic contact structure of signature (p, q) on a manifold M of that dimension is given by a corank three subbundle H c T M, such that the bundle of symbol algebras is locally trivial and modelled on a quaternionic Heisenberg algebra of signature (p,q). This means that one can locally identify H with M x lHIn and TM/H with M x im(lHI) such that the Levi bracket becomes the imaginary part of a quaternionic Hermitian form of signature (p, q). In 4.3.3 we have seen that quaternionic contact structures of signature (p, q) are equivalent to regular normal parabolic geometries of a certain type (G, P) to be discussed in more detail below. We have also seen there that the subbundle H c T M inherits an almost quaternionic structure, i.e. a preferred rank three subbundle Q c L(H, H), which locally can be spanned by I, J, and 10 J for two anti-commuting almost complex structures I and J on H. To define the twistor space for such a structure, Biquard used an analog of the construction for quaternionic structures we have described in 4.4.9. The subbundle Q carries a natural norm, which is characterized by the fact that an element of Q is a unit vector if and only if it defines a complex structure on the corresponding fiber of H. Then the twistor space M of the quaternionic contact structure (M, H) is defined as the unit sphere sub bundle of Q. By construction, M is the total space of an S2-bundle over M, so in particular, its dimension equals 4(p + q) + 5. Next, Biquard directly defines a corank one subbundle iI c TM and an almost complex structure jon iI. Basically, these can be described as follows. For a point x E M, elements of Qx act on Hx as scalar multiplications by purely imaginary quaternions. The induced action on TxM/Hx ~ im(lHI) is given by the commutator with the given imaginary quaternion. Hence, any nonzero element in Qx determines a two-dimensional subspace in TxM / Hx, namely the orthocomplement of the kernel of this commutator map. The elements of iI in that point are then those tangent vectors, which under the composition of projections T M ~ T M ~ T M / H land in this codimension one subspace. The complex structure on iI is defined directly using the footpoint acting as a complex structure on various spaces. Next, one shows that iI defines a contact structure on M, so j defines a natural almost CR-structures on the twistor space M of (M, H). It turns out that the signature of this CR-structure is (2p + 1, 2q + 1). Using a notion of preferred connetions for quaternionic contact structures, which we will discuss further in Chapter 5, Biquard then proves directly that for definite quaternionic contact structures of dimension at least eleven, this is an integrable and hence a CR-structure. The case of dimension seven is more involved and was sorted out later by Biquard's student D. Duchemin; see [Du06]. It turns out that the almost CR-structure on M in this case is integrable if and only if the quaternionic contact structure is torsion free; compare with 4.3.3. We will next describe a construction of a natural CR-structure on the same space via an analog of the Fefferman construction, which works for general signatures. While we expect that the result coincides with the structures constructed by
490
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Biquard and Duchemin in the positive definite case, to our knowledge this has not been proved until now. The basic construction is closely parallel to the classical Fefferman construction as discussed in 4.5.1 and 4.5.2. We first have to describe the groups G and P in more detail. The Lie algebra 9 of Gis sp(p + 1, q + 1) and G is the automorphism group Aut(g). Since the Dynkin diagram of sp(p + 1, q + 1) is of type C, it has no automorphisms, so Aut(g) = Int(g), the group of inner automorphisms. Hence, we can realize G as the quotient of Sp(p + 1, q + 1) by its center, which consists of ±id only. The parabolic subgroup PeG is the stabilizer of an isotropic quaternionic line in lHI p + q + 2 • Now we have to interpret quaternionic vector spaces as special complex vector spaces and quaternionic Hermitian forms as special complex Hermitian forms. So let us view lHI p+q+2 as C 2p +2q +4 with an additional almost complex structure j which anti-commutes with i and a quaternionic Hermitian form of signature (p + 1, q + 1) as an extension of a complex Hermitian form of signature (2p + 2, 2q + 2) for which j is orthogonal. This gives rise to an inclusion Sp(p + 1, q + 1) '--+ U(2p + 2, 2q + 2), and since Sp(p + 1, q + 1) is simple, the image must be contained in SU(2p + 2, 2q + 2). Projecting to the quotient G := PSU(2p + 2, 2q + 2), the result factorizes to a homomorphism i : G ~ G, which is easily seen to be injective. Choose a complex isotropic line C C C2p+2q+4 and let PeG be the stabilizer of c. Elementary linear algebra shows that Sp(p + 1, q + 1) acts transitively on the space of nonzero null vectors in C 2p+2q+4. Hence, G acts transitively on the projectivized null-cone which can be identified with GI P. The subgroup GnP of course is the stabilizer of C in G. Now a quaternionic linear map which stabilizes C also stabilizes the quaternionic line CIHI generated by C. Hence, using CIHI to define the parabolic subgroup PeG, we see that GnP c P, and we can apply the theory developed in 4.5.1 and 4.5.2. With the tools we have available at this stage, we can easily prove that in the torsion free case, we get a partially integrable almost CR-structure. To prove integrability, a small bit of input from BGG-sequences is needed. PROPOSITION 4.5.5. Let (M, H) be a quaternionic contact structure (with vanishing harmonic torsion ifdim(M) = 7) of signature (p,q) with twistor space M. Then M inherits a natural integrable CR-structure of signature (2p + 1, 2q + 1). PROOF. Let (p : 9 ~ M, w) be the parabolic geometry of type (G, P) determined by the quaternionic contact structure. We first claim that 9 I (G n p) can be identified with the twistor space M of M. We have noted in 4.3.3 that the group Go ~ PI P+ can be identified with the subgroup of GL(g-l) generated by elements of Sp(p, q) and quaternionic scalar multiplications. The subbundle Q c L(H, H) is the associated bundle to 9 corresponding to the natural representation of P (which factors through Go) on the space of purely imaginary quaternions acting on g-l. This is given by conjugating imaginary quaternions by the part of Go which acts by scalar multiplications on g-l. The stabilizer of the imaginary quaternion i under this action is is simply given by C \ {O} c 1HI \ {O} and hence coincides with the stabilizer of a complex line in the natural representation of 1HI \ {O} on 1HI. Thus, we can view GnP c P as the stabilizer of a unit quaternion in im(lHI) and since P acts transitively on the unit vectors in im(lHI), we conclude that PI (G n p) can be viewed as the space of this unit vectors. Passing to associated bundles,
4.5. ANALOGS OF THE FEFFERMAN CONSTRUCTION
491
we obtain an identification of Q/(G n p) ~ Q Xp P/(G n p) with the space of unit vectors in Q, which is the twistor space M. By Theorem 4.5.1, we obtain a canonical Cartan geometry (g - t M, w) of type (G, p) on M. To proceed, we need some facts about the compatibility of the inclusion 9 = sp(p+ 1, q+ 1) G induces a diffeomorphism GI P - t GI P, so the theory developed in 4.5.1 and 4.5.2 applies. The parabolic subgroup PeG corresponds to a Ill-grading, so parabolic geometries of type (G, p) are automatically regular and determine an underlying conformal structure of (split) signature (2,3); see 4.1.2. Hence, Theorem 4.5.1 directly implies existence of a canonical conformal structure of signature (2,3) induced by a generic rank two distribution on a five-dimensional manifold. Using ideas from the theory ofWeyl structures, one can derive explicit formulae for metrics in the conformal class based on a notion of generalized contact forms; see [CSa07j. Finally, similar arguments as in in the proof of Proposition 4.5.6 show that the extended Cartan connection is automatically normal; see [SagOSj. The case of generic rank three distributions in dimension six is closely parallel. Compared to the discussion in 4.3.2, we have to replace the group SOo(4, 3) used to describe the geometry by its two-fold covering Spin(4, 3) =: G. At least locally, such an extension is always possible and uniquely determined, so it causes no problems. Doing this, we have the spin representation at our disposal, which for this signature can be chosen to be real, of dimension eight and canonically endowed with an inner product of split signature (4, 4). Hence, the spin representation defines an inclusion i : G - t G := SO( 4,4). From 4.3.2 we know that the grading on g we are dealing with corresponds to the last simple root. From Theorem 3.2.12 and 3.2.15 we conclude that a parabolic subgroup PeG for this grading is given as the stabilizer of a highest weight line in the spin representation. This highest weight line is isotropic, so its stabilizer P in G is a parabolic subgroup, and GI P can be identified with the space of null lines in ]R(4,4). Using the fact that G IP and GIP have the same dimension and are both compact, one concludes similarly as before that the spin representation G "-> G induces a diffeomorphism G IP ~ GIP. Hence, we may apply the theory developed
4.5.
ANALOGS OF THE FEFFERMAN CONSTRUCTION
495
in 4.5.1 to conclude that a generic rank three distribution H on a smooth manifold M of dimension six induces a canonical conformal structure of split signature (3,3) on M. One can also prove in this case, that the extended Cart an connection wis automatically normal. More information on this case can be found in [Br06] and
[Arm07e]. REMARK 4.5.7. (1) The representations of split G2 in 80(3, 4) and of 8pin(4, 3) in 80(4,4) have analogs for compact real forms. These are given by a homomorphism from the compact real form of G2 to 80(7) and the spin representation 8pin(7) ---t 80(8). Remarkably, these two homomorphisms correspond to two special Riemannian holonomies. (2) Bryant's example of a conformal structure associated to a generic rank three distribution in dimension six generalizes to higher dimensions in an unexpected way. From 4.3.2 we know that replacing 800 (4,3) by 800 (n+ 1,n) and taking P to be the parabolic subgroup given by the last simple root, one obtains generic rank n distributions in dimension n(n2+1). The parabolic subgroup P can be realized as the stabilizer of an isotropic subspace of (maximal) dimension n in jR(n+1,n). Now we can simply include jR(n+l,n) into jR(n+1,n+1) thus defining an inclusion G := 80 0 (n + 1, n) '---+ 80 0 (n + 1, n + 1) =: G. Choosing an isotropic subspace of (maximal) dimension n + 1 in jR(n+l,n+1) and letting P be its stabilizer, one easily verifies that GnP = P is a parabolic subgroup determined for the grading on .50 (n + 1, n) we are dealing with. One further shows that the G-orbit of eP in GI P is open, so the theory from 4.5.1 and 4.5.2 applies. From 3.2.12 and 3.2.15 we can see that the parabolic subalgebra peg corresponds to either the last but one or the last (depending on whether our (n + 1)dimensional isotropic subspace is self-dual or anti-self--dual) simple root. From 4.1.12 we see that normal parabolic geometries of type (G, p) are equivalent to almost spinorialstructures, so this is the structure on a manifold of dimension n(n2+1) , which we canonically get from a generic distribution of rank n. The compatibility of the construction of the extended Cartan geometry with normality in this case is subtle. The problem is that for both geometries in question the harmonic curvatures are of torsion type. Hence non-flat geometries always have nontrivial torsion, which makes it difficult to apply BGG-sequences to the problem. This example is studied in [DS]. The reason why in the special case n = 3 a different structure is obtained is triality. In this case, the group G can be taken to be 8pin(4, 4). Triality tells us that the group of outer automorphisms of G which is the permutation group 6 3 can be realized as permuting the two spin representations and the standard representation. Correspondingly, the three parabolic sub algebras coming from these representations are all conjugate by an outer automorphism and almost spinorial structures in this dimension are equivalent to conformal structures. (3) The examples discussed in this subsection together with the one from 4.5.6 exhaust all analogs of the Fefferman construction for which the Fefferman space coincides with the original space. This is proved in [DS] based on deep results on generalized flag manifolds due to A. Onishchik; see §15 of [On94]. Onishchik proved that for a complex generalized flag manifold G I P, the group of biholomorphisms of the complex manifold GIP coincides with G, except in (the complex versions
496
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of) the three cases considered here. In these cases, the group of biholomorphisms coincides with G.
CHAPTER 5
Distinguished connections and curves In Chapter 3 we have discussed how to associate to a parabolic geometry (p :
g -; M, w) of some fixed type (G, P) an infinitesimal flag structure (Po: go -; M,O) of the same type. In Chapter 4 we have explicitly described this underlying infinitesimal flag structure in a variety of examples. In view of the equivalences of categories proved in 3.1.14, 3.1.16, and 3.1.18, this provides an equivalent description of regular normal parabolic geometries in almost all cases. In the two exceptional cases (see 4.1.5 and 4.2.6), one may specify a regular normal parabolic geometry by choosing additional data on the level of the infinitesimal flag structure. In any case, the bundle Po : go -; M is easy to construct from the underlying geometric structure. We have also noticed in Chapter 3 that for topological reasons the principal P+-bundle 7r : g -; go is always trivial. In this chapter, we will systematically use this fact to give (in the regular normal case) equivalent descriptions of the Cartan connection w in terms of objects defined on go. The motivating example for these general developments comes from conformal structures, where the bundle Po : go -; M is simply the conformal frame bundle. In 1.6.4 the canonical principal bundle p : g -; M is constructed in terms of Weyl connections on M, i.e. torsionfree linear connections on the tangent bundle T M, which are compatible with the given conformal class. The value of the canonical normal Cartan connection in a point is then determined by the soldering form, the connection form of the Weyl connection, and the Rho tensor as described in 1.6.7. These classical constructions for conformal structures admit close analogs for all parabolic geometries. The description which generalizes most easily is the interpretation of Weyl connections as equivariant sections of the bundle 7r : g -; go from Proposition 1.6.4. Such sections exist for all smooth parabolic geometries, and in view of the conformal case we call them Weyl structures. Similarly, as in the conformal case, one may view a general Weyl structure as consisting of a soldering form, a Weyl connection, and a Rho tensor. Technically, the main point about Weyl structures is that they form an affine space modelled on 01(M) and, although it is involved, the affine structure can be described explicitly. Using this affine structure we obtain a generalization of the interpretation of conformal Weyl structures in terms of scales (see 1.6.5) to general parabolic geometries. There is an obvious abstract version of the data (soldering form, Weyl connection, and Rho tensor) associated to a Weyl structure. In Section 5.2 we show that among these abstract objects, the ones obtained via Weyl structures from a regular normal parabolic geometry can be characterized by an analog of the normalization conditions for Cartan connections. This normalization condition can be expressed in terms of torsion and curvature quantities that are naturally associated to the abstract objects. 497
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This then allows us to directly construct the soldering form, Weyl connection, and Rho tensor associated to a Weyl structure from the underlying geometry. We obtain an explicit description of tractor bundles, which are equivalent to the Cartan bundle and Cartan connection as discussed in 3.1.19-3.1.22. Hence, this is an alternative way to describe objects that are equivalent to the Cartan geometry. We work out these ideas in detail for structures corresponding to 111-gradings and for parabolic contact structures. In the latter case, we also obtain a version of WebsterTanaka connections for arbitrary parabolic contact structures. In Section 5.2 we discuss several applications of the concepts developed so far, for example, to questions of affine holonomy theory and of Einstein rescalings of pseudo-Riemannian metrics. The last part of the chapter is devoted to the study of distinguished curves, as introduced in 1.5.17 for general Cart an geometries, in the special case of parabolic geometries. We discuss the various classes of distinguished curves. Any such curve is determined by some finite jet in one point, and we derive general results on which order of the jet is needed for each class of curves. Particular emphasis is put on cases in which distinguished curves are uniquely determined up to parametrization by their direction in one point. The basic example of this situation is provided by chains in hypersurface type CR--structures. We closed the chapter with some results on the geometry of chains. 5.1. Weyl structures and scales
The basic theme of this section is describing the Cartan connection associated to a parabolic geometry of some fixed type in terms of data on the underlying infinitesimal flag structure. The model case is the family of Weyl connections on a manifold endowed with a conformal structure, so we use the term Weyl structures for the general concept. Weyl structures form an affine space modelled on the space of one-forms on the underlying manifold. Understanding the behavior of the various objects associated to a Weyl structure under these affine changes is a key to many results. For conformal structures, the Levi-Civita connections of the metrics in the conformal class provide a nice subclass of Weyl connections. Replacing the bundle of metrics in the conformal class by appropriate line bundles called bundles of scales, this concept generalizes to all parabolic geometries. This leads to the concept of closed and of exact Weyl structures. A more subtle class of local Weyl structures is provided by the so-called normal Weyl structures, which are closely related to normal coordinates and provide the best approximations to the Cartan connection in a point. Fixing a Weyl structure, one may identify any natural bundle with a bundle associated to a completely reducible representation. Usually this can be nicely phrased as an identification with the associated graded bundle (with respect to a natural filtration). Specialized to tractor bundles, this leads to an explicit description of tractor calculus in terms of a Weyl structure as well as formulae for the effect of a change of Weyl structures. This topic will also be taken up in Section 5.2 below, where we will have more explicit descriptions of Weyl structures at hand. 5.1.1. Weyl structures. Let g = g-k EI1 ... EI1 gk be a Ikl-graded semisimple Lie algebra, G a Lie group with Lie algebra g, let PeG be a parabolic subgroup for the given grading and Go c P the Levi subgroup; see 3.1.3. Let (p : g -> M, w)
5.1.
WEYL STRUCTURES AND SCALES
499
be a parabolic geometry of type (G, P), and consider the underlying principal Gobundle Po : go ----+ M introduced in 3.1.5. By definition, go = g / P+, so there is a natural projection 1T : g ----+ go, which is a principal bundle with structure group P+.
g
DEFINITION 5.1.1. A (local) Weyl structure for the parabolic geometry (p : M, w) is a (local) smooth Go--equivariant section a : go ----+ g of the projection g ----+ go.
----+
IT:
As a first basic result we want to prove that global Weyl structures always exist and that they form an affine space. From 3.1.5 we know that the associated graded gr(TM) ofthe tangent bundle TM is the associated bundle go xG o (g-k61" ·61g-1). The action of Go on each gi is induced by the restriction of the adjoint action of G to the subgroup Go. By part (2) of Proposition 3.1.2, for each i ~ 0 the G omodule gi is dual to g-i, so we conclude that the associated graded gr(T* M) of the cotangent bundle is isomorphic to go xG o (gl 61 ... 61 gk). The dualities are induced by the Killing form of g. In particular, smooth sections of gr(T* M) can be identified with smooth functions f = (ft, ... , /k) : go ----+ gl 61···61 gk such that f(u· g) = Ad(g-l)(f(U)) for all u E go and g EGo. PROPOSITION 5.1.1. For any parabolic geometry (p : g ----+ M, w), there exists a global Weyl structure a : go ----+ g. Fixing one Weyl structure a, there is a bijective correspondence between the set of all Weyl structures and the space r(gr(T* M)) of smooth sections of the associated graded of the cotangent bundle. Explicitly, this correspondence is given by mapping Y E r(gr(T* M)) with corresponding functions Y i : go ----+ gi for i = 1, ... , k to the Weyl structure a(u):= a(u)exp(Y1(U))" ·exp(Yk(u)). PROOF. By topological dimension theory, any fiber bundle over the smooth manifold M admits an atlas with finitely many (usually disconnected) charts; see [GHV72]. Hence, there is a finite open covering {U1 , ... ,UN} of M such that both g and go are trivial over each Ui . Via a chosen trivialization, the inclusion Go ~ P induces a smooth Go--equivariant section ai : P01(Ui ) ----+ p-1(Ui ) for each i = 1, ... , N. Moreover, we can find open sets VI"'" VN such that Vi C Ui for all i, and such that {VI, .. " VN} still is a covering of M. By Theorem 3.1.3, the exponential map restricts to a diffeomorphism p+ ----+ P+, where p+ = gl 61··· 61 gk. Thus, there is a smooth map \lI : POl(Ul n U2) ----+ p+ such that a2(u) = al (u) exp(\lI(u)) for all u E POl (Ul n U2). Equivariance of a1 and a2 immediately implies that \lI(u·g) = Ad(g-l )(\lI(u)) for all g E Go. Now let f : M ----+ [0,1] be a smooth function with support contained in U2 , which is identically one on V2 and define a : POl (U1 U V2) ----+ p-1 (U1 U V2) by a(u) = a1 (u) exp(f(po(u))\lI(u)) for u E U1 and by a(u) = a2(u) for u E V2. Then obviously these two definitions agree on U1 n V2 , so a is smooth. Moreover, from equivariancy of the ai and of \lI one immediately concludes that a is equivariant. In the same way, one next extends the section to U1 U V2 U V3 and by induction one reaches a globally defined smooth equivariant section. Fixing a global equivariant section a, any other section a of 1T : g ----+ go can be written uniquely as a(u) = a(u)CP(u) for some a smooth function cP : go ----+ P+. The section a is Go--equivariant if and only if cp(u . g) = g-lcp(U)g for all u E go and all g E Go. By Theorem 3.1.3, the map (Zl,"" Zk) f-> exp(Zt}···exp(Zk) is a diffeomorphism p+ ----+ P+, so we may uniquely write cP
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as ~(U) = exp(T 1(u))·· ·exp(Tk(u)) for smooth functions Ti : go --+ gi. Since g-1 exp(T l(U)) ... exp(Tk(U))g = g-1 exp(T 1 (u))g . .. g-1 exp(Tk(u))g = exp(Ad(g-1)(T 1(u))) ... exp(Ad(g-1)(T k (u))),
uniqueness of the representation as a product implies that ~(u . g) = g-1~(u)g is equivalent to Ti(U' g) = Ad(g-1)(Ti(u)) for all i = 1, ... ,k, and the result follows. 0 Fixing one Weyl structure q, there is another bijection between the set of all Weyl structures and the space r(gr(T* M)). Namely, for T E r(gr(T* M)) corresponding to the functions T i : go --+ gi one may also put u(u)
= q(u) exp(T1(u) + ... + Tk(U)),
The proof that this indeed defines a bijection is completely parallel to the proof in the proposition. The advantage of the convention used there is that it makes it easier to separate homogeneous degrees.
5.1.2. Weyl connections, soldering form and Rho tensor. Given a Weyl structure q : go --+ g for a parabolic geometry (p : g --+ M, w), we can consider the pullback q*w E {}1 (go, g) of the Cartan connection. Equivariancy of q reads as r g 0 q = q 0 r g for all 9 E Go. Using this, we obtain (rg)*(q*w) = q*«rg)*w) = Ad(g-1)
0
(q*w),
so the one-form q*w is Go-equivariant. As a Go-module, the Lie algebra 9 decomposes as g-k EB ... EB gk, so decomposing q*w = q*W-k + ... + q*Wk accordingly, each of the components is Go-equivariant. PROPOSITION 5.1.2. Let q : go --+ g be a Weyl structure on a parabolic geometry (p : g --+ M, w). Then we have: (1) The component q*wo E {}1(go,go) defines a principal connection on the bundle Po : go --+ M. (2) The components q*W-k, . .. , q*W-1 can be interpreted as defining an element of {}1(M,gr(TM)). This form determines an isomorphism TM --+ gr(TM) = T-kM/T-k+lM EB··· EBT- 1M
which is a splitting of the filtration of T M. This means that for each i = -k, ... ,-l, the subbundle ~M is mapped to E9j~igrj(TM) and the component in gri(TM) is given by the canonical surjection TiM --+ ~M/Ti+lM.
(3) The components q*W1,.'" q*Wk can be interpreted as a one-form P E {}1(M, gr(T* M)). PROOF. For an element A E go, the corresponding fundamental vector field on go is by definition given by (~O(u) = ftlt=ou, exp(tA) for all u Ego. Equivariancy of q immediately implies that q(u·exp(tAH = q(u) ·exp(tA), which in turn implies that Tuq·(~O(u) = (~(q(u)), the fundamental vector field on g. By definition, the Cartan connection w reproduces the generators of fundamental vector fields, so we conclude that q*wo reproduces the generators of fundamental vector fields. Since we have already observed that q*wo is Go-equivariant, this proves (1). On the other hand, the components q*Wi for i =F 0 vanish upon insertion of a fundamental field, so they are horizontal. Since we have already observed that
501
5.1. WEYL STRUCTURES AND SCALES
a*wi is Go-equivariant, Corollary 1.2.7 shows that it determines an element of 01(M, go xGo gi). For i > 0 we have go xGo gi ~ gri(T* M) and the proof of (3) is complete. For i < 0 we get go x gi = gri(TM), so we can interpret a*w_ as an element of 01(M,gr(TM)). To verify the remaining claims in (2) consider w(a(u)) for a point u E go. This is a linear isomorphism Tu(u)g - g which reproduces the generators of fundamental vector fields, so it descends to a linear isomorphism Tu(u)g/ker(Tup) ~ g/p. The map Tua induces a linear isomorphism Tugo/ker(Tupo) - Tu(u)g/ker(Tup). Hence, we conclude from the construction that the element of 01(M,gr(TM)) defined by the negative components of a*w restricts to an injection on each tangent space. Since the bundles T M and gr(T M) have the same rank, we must obtain an isomorphism TM - gr(TM). From 3.1.5 we know that the identification of gri(TM) with gOXGogi is obtained in terms of the frame form () = ((}-k, ... , (}-d as follows. For u E go and E T~go, the pair (u, (}i(U)(e)) E go x gi represents the class of TuPo' in Ti M/Ti+1 M. But recall that the component (}i of the frame form was obtained by choosing any lift of to a tangent vector on g and take the gi-component of the value of w on that lift. This immediately shows that (}i(U)(e) = a*wi(u)(e) for all E T~go, which implies the last claim in (2). 0
e
e
e
e
DEFINITION 5.1.2. Let 0' : go - g be a Weyl structure for a parabolic geometry (p: g -M,w). (1) The principal connection a*wo on the bundle go - M is called the Weyl connection associated to the Weyl structure a. (2) The gr(T M)-valued one-form on M determined by the negative components of a*w is called the soldering form associated to the Weyl structure 0'. (3) The one-form P E 01(M, gr(T* M)) induced by the positive components of a*w is called the Rho tensor associated to the Weyl structure 0'.
5.1.3. Weyl connections and soldering forms on natural bundles. Let go - g be a Weyl structure for a parabolic geometry (p : g - M, w) of type (G, P). Then the corresponding Weyl connection a*wo is a principal connection on the bundle go. As discussed in 1.3.4, this principal connection gives rise to an induced connection on any fiber bundle associated to the principal bundle go. In the case of an associated vector bundle, the resulting connection is automatically linear. All these induced connections will be referred to as the Weyl connections corresponding to the Weyl structure 0'. Assume that P x S - S is a smooth left action of the group P on a smooth manifold S. Since Go by definition is a subgroup of P, we can restrict this action to a smooth left action g : Go x S - S. Via the action we can form the associated bundle g Xp S - M, while g gives rise to the fiber bundle go x Go S - M. From above we know that we have a Weyl connection on the associated bundle go x Go S - M. 0' :
e:
e,
PROPOSITION 5.1.3. Let (p : g - M,w) be a parabolic geometry of some fixed type (G, P), and let S be a smooth manifold endowed with a smooth left paction. Then choosing a Weyl structure 0' : go - g induces an isomorphism g x p S ~ go x Go S and thus gives rise to a connection on the natural bundle g x p S. In the case of a natural vector bundle this connection is automatically linear.
502
5.
DISTINGUISHED CONNECTIONS AND CURVES
PROOF. Consider the smooth map (J" x ids: go x S --+ g x S. Composing with q : g x S --+ g x p S we obtain a smooth map, which by equivariancy of (J" factors to a smooth map go xGo S --+ g Xp S. By construction, this is a fiber bundle map covering the identity on M and it restricts to diffeomorphisms on the fibers. Hence, it is an isomorphism of fiber bundles. We have already observed above that (J" gives rise to a connection on the bundle go xGo S, which is linear in the case of a natural vector bundle. 0 For a large class of natural vector bundles, we can interpret this result in a more conceptual way, which shows that the soldering form on the tangent bundle generalizes to this class of vector bundles. COROLLARY 5.1.3. Let V be a finite-dimensional representation of P, which is completely reducible as a representation of Go. (1) There is a P-invariant filtration V = VO :J VI :J ... :J V N :J {O} such that for each i, the action of p+ maps Vi to Vi+I. (2) For a manifold M endowed with a parabolic geometry, let V M --+ M be the natural vector bundle induced by V. Consider the filtration {Vi M} ofVM by smooth subbundles induced by the filtration of V from (1). Then we can naturally identify gr(V M) with go x Go V, so any Weyl structure induces an isomorphism between V M and its associated graded bundle, which defines a splitting of the filtration. PROOF. (1) We have proved in Proposition 3.2.12 that one obtains ap-invariant filtration on V by putting V N := {v E V : Z . v = 0 VZ E P+} and then inductively V i - I := {v E V : Z· v E Vi VZ E p+}. We have also seen there, that each of the sub quotients Vi jVi+1 is completely reducible as a representation of 90. Now clearly the filtration is also invariant under the action of the group P and the subquotients are completely reducible as Go-modules. Since p+ . Vi C Vi+1 by construction, this completes the proof of (1). (2) The associated graded gr(V) to the filtered vector space V carries a natural representation of Pi see 3.1.1. Since p+ . Vi C Vi+I, we see that p+ and thus P+ acts trivially on gr(V). On the other hand, since the Go-action on V is completely reducible, we can find for each i ~ 0 a Go-invariant subspace Vi c Vi such that Vi = Vi EB Vi+I. In particular, Vi ~ VijVi+1 = gri(V) and as a Go-module V = Vo EB··· EB VN , so V is naturally isomorphic to gr(V) as a Go-module. Now let 7r : g --+ go = g j P+ be the natural projection. Since P+ acts trivially on gr(V) and PjP+ ~ Go, the map 7r x id : g x gr(V) --+ go x gr(V) factors to a diffeomorphism g x p gr(V) --+ go x Go gr(V), and the latter bundle can be identified with go x Go V. Hence, we can naturally interpret go x Go V as the associated graded vector bundle gr{9 Xp V). From the construction it is clear, that this is a splitting of the filtration. 0 REMARK 5.1.3. (1) From 3.2.12 we know that on the level of the Lie algebra 90, complete reducibility of a complex representation is equivalent to the center 3(90) acting diagonalizably. Hence, the assumption that a representation of P is
completely reducible as a representation of Go is a very weak condition. (2) Denoting the largest filtration component by V O is just one possible convention. In some cases (for example for the adjoint tractor bundle) other conventions are more natural and we will use these. (3) The filtration from part (1) of the corollary can be described on the level of bundles. The infinitesimal action of p+ defines a map p+ x V --+ V, which
5.1. WEYL STRUCTURES AND SCALES
503
is a P-homomorphism. Since g Xp p+ = T* M, we get an induced bundle map T* M x VM -+ VM, which makes VM into a bundle of modules over the bundle T* M of Lie algebras. We will write this bundle map as (¢,8) I--t ¢.8. The filtration of the bundle VM is then given by V: M = {v E VxM : ¢. v = 0 V¢ E T;M} and V~-lM = {v E VxM: ¢.V E V~M V¢ E T;M}. (4) Choosing a Weyl structure (j induces an isomorphism VM -+ gr(VM), which can be used to transfer the Weyl connection on gr(VM) to VM. Of course, this generalizes the isomorphism T M to (j, see 5.1.2.
~
gr(TM) given by the soldering form associated
EXAMPLE 5.1.3. Consider the adjoint tractor bundle AM = g Xp 9 and its associated graded gr(AM) = gr_k(AM) E& ••• E& grk(AM). Each fiber of gr(AM) is a graded Lie algebra isomorphic to g, so in each fiber gr(AxM) there is a unique element E(x) such that {E(x),_} is multiplication by j on the grading component grj(AxM); see Proposition 3.1.2. These elements fit together to define a smooth section E E r(gro(AM)) called the grading section of gr(AM). Indeed, viewing gr(AM) as go xGo g, the section E corresponds to the constant function mapping all of go to the grading element of g. Choosing a Weyl structure (j : go -+ g we obtain an isomorphism AM -+ gr(AM) and thus a section EI7 E r(AM) corresponding to the grading section E. By construction, EI7 is a section of the filtration component A OM, which maps to E under the natural projection A OM -+ gro(AM). Otherwise put, the choice of the Weyl structure (j gives us a lift EI7 E r(AoM) of the grading section. We shall see later, that (j I--t EI7 defines a bijection between the set of Weyl structures and the set of all such lifts.
5.1.4. Bundles of scales. Our next aim is to prove that a small part of the data associated to a Weyl structure (j in 5.1.2 is already sufficient to uniquely pin down (j. More precisely, we want to show that there is a class of natural line bundles, such that the induced Weyl connection on one of this line bundles determines (j uniquely. A natural line bundle associated to go is given by a homomorphism A : Go -+ JR, which defines a one-dimensional representation of Go on R The bundle is oriented (and hence can be trivialized) if and only if the representation has values in JR+. The (oriented) line bundle L>' = go x>.1R can be equivalently described by its (oriented) frame bundle CA, which is a principal bundle with structure group IR (respectively JR+) on M. The frame bundle can be most easily described as the orbit space gol ker(A) of the normal subgroup ker(A) eGo. A homomorphism A : Go -+ IR gives rise to a Lie algebra homomorphism A' : go -+ R This homomorphism vanishes on the semisimple part of go, so it is just a linear functional on the center 3(go). To describe the homomorphisms which are appropriate for our purposes, recall from 3.1.2 that the Killing form B of 9 restricts to a nondegenerate bilinear form on go. The splitting of go into center and semisimple part is orthogonal with respect to B, so the restriction of B to 3(go) still is nondegenerate. Given a homomorphism A : Go -+ IR we therefore get a unique element E>. E 3(go) such that A'(A) = B(E>., A) for all A Ego. DEFINITION 5.1.4. (1) An element F E 3(go) is called a scaling element if and only if the restriction to p+ of the adjoint action adF : 9 -+ 9 is injective. (2) A bundle of scales for parabolic geometries of type (G, P) is a natural principalJR-bundle C>. associated to a homomorphism A : Go -+ JR, such that the
504
5.
DISTINGUISHED CONNECTIONS AND CURVES
corresponding element EA E 3(90) is a scaling element. If ). has values in lR+, then we obtain an oriented bundle of scales. (3) Having chosen a bundle £A of scales, a (local) scale for a parabolic geometry (p : 9 ---+ M, w) of type (G, P) is a (local) smooth section of the principallR+ -bundle £A ---+ M. PROPOSITION 5.1.4. (1) For any type of parabolic geometry there exist natural oriented bundles of scales. (2) A bundle of scales admits global smooth sections, if and only if it is oriented. PROOF. (1) The grading element E E 3(90) (see 3.1.2) acts on 9i by multiplication with i, so it is a scaling element. For g E Go the adjoint action Ad(g) : 9 ---+ 9 by definition respects the grading of 9, and we denote by Adj (g) the restriction of Ad(g) to 9j. Now define)' : Go ---+ lR by k
).(g) :=
IT Idet(Adj (g))1 2j .
j=l This clearly is a homomorphism with derivative>.' : 90 ---+ lR given by ),'(A) = 2:~=12j tr(adj(A)), where adj(A) denotes the restriction of ad(A) to 9j. By definition, B(E, A) = tr(ad(E) 0 ad(A)), and this composition acts by j adj(A) on each component 9j. Hence, B(E,A) = 2:~=_kjtr(adj(A)), and since from Proposition 3.1.2 we know that ad_j(A) is the dual map of adj(A), we conclude that >.'(A) = B(E, A). Thus the one-dimensional representation). gives rise to a natural bundle of scales. (2) This is just due to the fact that orient able real line bundles and thus principallR+ -bundles are automatically globally trivial and hence admit global smooth sections. 0 REMARK 5.1.4. For maximal parabolic subalgebras, i.e. those corresponding to Satake diagrams with just one crossed root, the center of 90 has dimension one and is generated by the grading element E. Hence, any nonzero element of 3(90) is a scaling element in this case, and the bundle of scales constructed in the proposition is the unique oriented bundle of scales up to forming roots or tensor powers (which poses no problem for trivial bundles). In all applications we know of it is sufficient to restrict to bundles of scales corresponding to multiples of the grading element (however, some nontrivial multiples maybe particularly convenient). We have chosen to present the theory in a more general version, since this causes no additional difficulties. 5.1.5. The effect of a change of Weyl structures on soldering forms. The key step towards the basic results on Weyl structures is understanding the effect of a change of Weyl structure on the various data associated to it. We start by analyzing the effect of the change on soldering forms, and we will only deal with the case of natural vector bundles. Considering two Weyl structures a and a we will always denote quantities corresponding to a by unhatted symbols and quantities corresponding to a by hatted symbols. For a representation V of P, which is completely reducible as a representation of Go, and the corresponding natural vector bundle V M, we will denote the isomorphism V M ---+ gr(V M) induced by the Weyl structure a by v r-t (v)" := (vo, ... , VN) (if the filtration indices start with 0) and the isomorphism corresponding to a by v r-t (v)". = (vo, ... , VN).
5.1. WEYL STRUCTURES AND SCALES
505
To formulate the results efficiently, we introduce some notation for multiindices. We will consider multi-indices consisting of k components (in the case of a Ikl-grading), and write 1 = (il, ... , ik) with il, ... , ik ~ O. For such a multiindex 1 we put 1! := il! ... ik!, IIiII := il + 2i2 + ... + kik, and (-l)i := (_l)il +"+ik. As we have noted in Remark 5.1.3, the restriction to p+ of the infinitesimal action of p on V induces a bundle map gr(T* M) x gr(V M) -+ gr(V M). This bundle map is homogeneous of degree zero, and we will denote it by (¢, II) I--t p(¢)(II), or by ¢. II if the action is clear from the context. Using this, we now formulate: PROPOSITION
5.1.5. Let a and a be two Weyl structures related by a(u) = a(u)exp(YI(U))·· . exp(Ydu)) ,
with corresponding section Y = (Y I, ... , Y k) of gr(T* M). For a representation V of P which is completely reducible as a representation of Go let p denote the corresponding action of gr(T* M) on gr(VM). Then the isomorphisms VM -+ gr(V M) corresponding to a and a are related by Vi =
,,(-l)i. L...J -.,_p(YkYk Ilil!+j=i
0···0
. p(Y I )'l(lIj).
!.
PROOF. We view VM as 9 Xp V and gr(VM) as 90 XCo V. By definition, the fact that (II)CT = (110, ... , liN) means that given u E 90 over the same point as II, and the unique elements Vi E Vi for i = 0, ... , N such that IIi = [u, Vi], we have II = [a(u), v], where v = Vo + ... + VN. Again by definition, this means that
II = [a(u),exp(-Yk(U))·· ·exp(-YI(U))· v], (11).7 = [u,exp(-Yk(u))···exp(-YI(u)) . v]. Hence, we get (11).7 = exp(-Yk(u))···exp(-YI(U)) ·V, from which the claimed formula follows by collecting pieces of fixed homogeneity. 0 The low grading components can be easily spelled out explicitly. If the filtration index starts with 0, then we get
Vo = Vo, VI = VI -
YI
V2 = V2 - Y I
•
vo,
• vI -
Y 2 • Vo
+ ~Y I •
YI
•
Vo,
and so on. Of course, the first line simply expresses the fact that gr0 (V M) is naturally a quotient of V M. We may apply the result of the proposition, in particular, to the isomorphism gr(T M) -+ T M induced by a choice of a Weyl structure. In this case, the action of gr(T* M) on gr(T M) comes from the adjoint action, so this is given by the algebraic bracket { , }. We can also use the proposition to obtain a first alternative interpretation of Weyl structures. Recall from Example 5.1.3 that for a given Weyl structure a we obtain a section ECT E f(AO M) such that grO(ECT) = E E f(gro(AM)), the natural grading section. COROLLARY 5.1.5. The map a I--t ECT defines a bijection between the set of all Weyl structures and the set of all sections s E f(AOM) such that gro(s) = E E f(gro(AM)).
506
5.
DISTINGUISHED CONNECTIONS AND CURVES
PROOF. The natural action of gr(T* M) on gr(AM) comes from the adjoint action, so we denote it by ad. By definition, E'T is the image of E under the isomorphism gr(AM) -+ AM induced bya. For a lift s E f(AO M) of E we must have (s).,. = (0, ... ,0, E, Sl!"" sn), so fixing a we obtain a bijection between the set of all such lifts and r(grl (AM)EB" ·EBgrk(AM)). For an arbitrary Weyl structure fT we obtain a section Y E f(gr(T* M)) such that a(u) = fT(u) exp(Y 1 (u)) ... exp(Y k(U)). By the proposition we obtain (E it ).,. = (0, ... ,So, ... , Sk) with Sj =
' " (-1)-~. ~ -.,- ad(Y k)'k 11111=j 1·
0 ... 0
. ad(Y 1)'1 (E).
In particular, So = E and SI = -{Yl,E} = Y 1 • Next, S2 = -{Y2,E} + HY 1 , {YbE}} = 2Y 2 - HY 1, Yd· Inductively, we conclude that each Sj is given by the sum of jY j plus some expression in the Y i for i < j. This shows that we can compute each Y j from (Eit).,. and conversely, any choice for Sj with So = E can be realized by choosing an appropriate section Y = (Y b ... , Yk). 0 REMARK 5.1.5. The modern treatment of tractor bundles was initially based on using Proposition 5.1.5 as a definition. In [BEG94J, the standard tractor bundles for conformal and projective structures were defined as being given by a direct sum for each choice of a metric in the conformal class, respectively, a connection in the projective class with the appropriate behavior under changes of these choices. This approach was extended to more general geometries and bundles in [CGo02] and [CGoOOj. The idea to look at grading sections and use them to parametrize Weyl structures is taken from [CDS05]. 5.1.6. The effect of a change of Weyl structure on Weyl connections. We derive the complete formula only in the case of a completely reducible representation V of P, so that V M = g x p V = go X Go V. The infinitesimal action of go on V induces a bundle map gro(AM) x VM -+ VM, which we denote by •. PROPOSITION 5.1.6. Let a and fT be two Weylstructures related by
fT(u) = a(u)exp(Yl(U))" ·exp(Yk(U)), with corresponding section Y = (Y 1, ... , Y k) of gr(T* M). For a smooth section v of a bundle V M associated to a completely reducible representation of P, the Weyl connections \7 and V are related by
V~v=\7~v+
L 111I1+j=o
(~~)i(ad(Yk)iko ... oad(Yl)i1(~j)).v, -
where (~).,. = (~-k"" ,~-d· PROOF. Let us also denote by v the Go-equivariant function go -+ V representing the section v. By definition of the Weyl connection, the value of the function representing \7~v in u E go is given by ~. v - wo(Tua.~) • v, where ~ is any lift of~. Denoting by l' the principal right action of P+ on g -+ go and putting (u) := exp(Y 1 (u))·· ·exp(Ydu)), we have fT = 1'0 (a, g is called closed if and only if the induced connection yo on LA is flat. (2) A Weyl structure 0" : go -> g is called exact if and only if the induced connection yo on LA comes from a global trivialization of LA. By definition, exact Weyl structures are automatically closed by definition. By part (2) of Proposition 5.1.4, the frame bundle £A of a bundle LA of scales admits global smooth sections, if and only LA is oriented. Since global smooth sections of £A are exactly global nonzero sections of LA, global exact Weyl structures exist precisely in the oriented case. For conformal structures, the obvious choice for £A is the bundle of metrics in the conformal class (see 1.6.5), so exact Weyl structures are in bijective correspondence with these metrics. The reason for the names "closed" and "exact" becomes apparent when looking at the affine structure on the space of Weyl structures. PROPOSITION 5.1. 7. Let LA be a fixed bundle of scales. Then the space of all Weyl structures for (p : g -> M, w) is an affine space modelled on the vector space Ol(M) of one-forms on M. If they exist, then the spaces of closed (respectively exact) Weyl structures are affine subspaces modelled on closed (respectively exact) one-forms. Denoting by )..' also the bundle map induced by the infinitesimal representation, the one form T; fr describing the change from 0" to 0-( u) = 0"( u) exp(Y 1 (u)) ... exp(Y k (u)) is give~ by
T;,fr(~) =
L Ilill+j=O
(~~)\/(ad(Yk)ik o ... oad(Ydl(~j)). -
PROOF. The affine structure on the space of Weyl structures is simply obtained by pulling back the affine structure on the space of linear connections on LA. The formula for T;,fr directly follows from Proposition 5.1.6. Changing a linear connection on a line bundle by a one-form T, the change of curvature is given by dT. Hence, starting from a closed Weyl structure 0", the Weyl structure 0- is closed if
5.1.
WEYL STRUCTURES AND SCALES
509
and only if dT; u = O. If the structure a is exact, then it is given by a global nonvanishing secti~n ¢ E r(LA). Any other such section can be written as ¢ = ef ¢ for a smooth function f : M --4 Itt A smooth section 'I/J of LA then can be written as 'I/J = h¢ = he- f ¢, and by definition we have '\7'I/J = dh Q9 ¢ and V'I/J = d(he- f) Q9 ¢, which immediately implies V'I/J = '\7'I/J - df Q9 'I/J. D REMARK 5.1.7. (1) Let us specialize to the case of III-gradings for illustration. There the formula for from the proposition simply reads as u = - A' ([Y, W. As we have noted in Remark 5.1.4, any scaling element is a multiple of the grading element E in the Ill-graded case. Denoting by B the Killing form, this means that there is a nonzero number a such that
T
T;
-A'([Y,W = -aB(E, [Y,W =
-aB(Y,~),
T;
and hence u = -aY. (2) Ther~ is another useful point of view for exact Weyl structures which generalizes the case of conformal structures discussed in 1.6.5. The frame bundle .c A of LA can be naturally identified with the quotient go/ ker(A). By Lemma 1.2.6, global smooth sections of go/ ker(A) --4 M are in bijective correspondence with reductions of the principal bundle go --4 M to the structure group ker(A) C Go. Hence, we can interpret exact Weyl structures as reductions of the structure group of go to ker(A). For conformal structures we exactly recover the reductions to the structure group O(n) C CO(n) given by the choices of metrics in the conformal class. 5.1.8. The change of the Rho tensor. Let us next clarify the dependence of the Rho tensor on the choice of Weyl structure.
PROPOSITION 5.1.S. Let u and 0- be two Weyl structures related by
o-(u) = u(u) exp(Y 1(u)) ... exp(Yk(u)), with corresponding section Y = (Y 1, ... , Y k) of gr(T* M). Then the Rho tensors P = (P 1, ... , Pk) and I' = (1'1, ... , Pk) associated to u and 0- are related by
k
+2:
m=l
+
(-I)l..
-C:j}--:'-(j:-m-'-+-I-:-:-) ad (Y k)1 k
.
0 ••• 0
ad (Y m)1'" ('\7 eY m)
IItll+m=i
2: lIillH=i
(-.~)tad(Yk)jko ... oad(Y1)11(P£(~)). 2:
where (~)u = (~-k' .. · '~-1). PROOF. Fix u E go and ~ E Tugo. By definition, Pi(~) corresponds to the gi-component of w(o-(u))(Tuo- . ~). Since this depends only on the projection of ~ to M we may assume that ~ is horizontal with respect to u. Putting <J>(u) = exp(Y 1(u)) ... exp(Yk(u)) we know from the proof of Proposition 5.1.6 that
Tuo-· ~ = Tu(u)r(u) . Tuu· ~ + T(u)ru(u) . Tu<J>·~.
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Denoting by A left translations in the group P+, it follows from the definition of the principal right action that r 0'( u) = r &( u) 0 A.p( u) -1. Differentiating this, we obtain T.p(u)rO'(u)
0
Tu
= Ter&(u) 0 T.p(u)A(u)-l 0 Tu.
Now Ter&(u) is the fundamental vector field map in a( u) while the composition of the other two factors by definition is 8 (u), where 8 E n1(90, p+) is the left logarithmic derivative of : go -+ P+; see 1.2.4. Hence, we conclude that
= Ad((U)-l)(w(TuO" ()) + 8(u)((). Using wo(TuO"() = 0, inserting Ad( (u)-l) = ead(-ld u )) o· .. oead(-ll(U)), and colw(Tua· ()
lecting the terms of the right degrees, we see that Ad((u)-l )(w(TuO"()) represents the first line and the last line in the claimed formula for Pi. We have noted in 1.2.4 that the left logarithmic derivative satisfies a Leibniz rule of the form 8(Jg)(x) = 8g(x) + Ad(g(X)-l )8f(x). Iteratively, this implies that k
8
= I::ead(-lk)
0'"
0
ead(-lj+l)
o8(expoTj).
j=l
Finally, 8(expoTj)(u)(() = 8(exp)(Tj(u))(Tu T j . (). Since ( was chosen to be horizontal with respect to 0', the function Tu T j .( represents the covariant derivative of T j in direction of the vector field on M underlying (. The formula for the right logarithmic derivative of the exponential mapping from [KMS, Lemma 4.27] can be easily adapted to the left logarithmic derivative, showing that 8(exp)(X) 2:::0 «;:l~! ad(X)". Inserting this, the result follows. 0 REMARK 5.1.8. Together with Propositions 5.1.5 and 5.1.6 we now have a complete description of how a change of Weyl structure affects the associated data. For many applications, one does not need the complete changes but the linearized changes are sufficient. More precisely, suppose that we have given a Weyl structure 0' and a smooth section T = (T 1 , ... , T k ) of gr(T* M). Then for t E ~ we can consider the Weyl structure O't defined by O't( u) = 0'( u)
exp(tY 1 (u)) ... exp(tY k( u)),
to obtain a smooth family {O't : t E IR} of Weyl structures. This leads to smooth families of soldering forms, Weyl connections, and Rho tensors, and the linearized changes are by definition the derivatives at t = 0 of these curves, which we indicate by the symbol 8. From Propositions 5.1.5, 5.1.6, and 5.1.8 one immediately reads off that (in the setting and notation of the respective proposition), they are given by k
8ve = - LTi eVe-i, i=l
k
8'f'h.s = -
L {Ti' (-i}
e
s,
i=l
k
8Pe(() = Y'~T£ -
L i=e+l
£-1
{Ti,(£-i} - L{Ti , P£-i(()}' i=l
5.1. WEYL STRUCTURES AND SCALES
511
5.1.9. The Rho-corrected derivative. In 5.1.3 we have defined Weyl connections on arbitrary natural bundles by combining the principal connection a*wo and the identification of a natural bundle with an associated bundle to 90 provided by a. For bundles associated to actions of P coming from actions of Go this is the best one can do, but for general bundles there is an alternative possibility, which was suggested in [8197] and formally introduced in [CD805]. Given a Weyl structure a : 90 -7 g, we can consider the image a(Qo) C 9 and use w- 1 ({I_) as a horizontal distribution along this subset. Then we can extend this horizontal distribution equivariantly to all of g, hence defining a principal connection on 9. Alternatively, this can be formulated as follows. 5.1.9. Let a : 90 -7 9 be a Weyl structure. (1) The principal connection on 9 associated to a is defined by the connection form 'Y~ E 0 1 (Q, p) defined by DEFINITION
'Y~(a(u)
. g)(e) := wp(a(u))(Tr 9
-1
.
e)
for u E 90 and 9 E P +, where wp is the p-component of w E 0 1 (Q, {I- EB p). (2) For a natural vector bundle the covariant derivative induced by the principal connection 'Y~ is called the Rho-corrected derivative V'P associated to the Weyl structure a. Using the facts that a is Go equivariant and that w reproduces the generators of fundamental vector fields, one immediately verifies that 'Y~ indeed defines a principal connection on g. It is easy to relate the Rho-corrected derivative to the Weyl connection. This relation also explains the choice of the name "Rho-corrected derivative" . PROPOSITION 5.1.9. Let a: go -7 9 be a Weyl structure with Weyl connection V' and Rho-corrected derivative V'P. For a natural vector bundle V M = 9 x p V let • : T* M Q9 V M - 7 V M be the action induced by the p+ -action on V. (1) For any section s E r(VM) and any vector field E .t(M) we have
e
k
(V'~S)i = V'eSi +
I: Pj(e). Si-j, j=l
where (s)~ = (so, ... , SN) and (V'~s)~ = ((V'~s)o, ... , (V'~S)N)' Otherwise put, V'~s = V'es + p(e). s. (2) For a(u) = a(u)exp(Yt{u))·· ·exp(Yk(U)) with associated Rho-corrected derivative v P and s E reV M) we have
v~s = V'~s +
I: Ilill+j~O
(~~)i (ad(yk)i
k 0'"
0
ad(Yd ' (ej)) • s.
-
PROOF. (1) Let us also denote by s : 9 - 7 V the equivariant function corresponding to s E reV M). By definition, the section Si E r(gr i (V M)) is represented by the \Ii-component (s 0 a)i of the function so a : go ..... V. Let us denote by h E .t(Qo) the horizontal lift of E .t(M) with respect to the Weyl connection of a. Then the section V'eSi is represented by eh.(soa)i = ((Ta·eh)·s);. On the other hand, choosing a lift ~ E .t(g) of the section V'~s E r(VM) is represented by the function ~. s - wp(~) • s. The component (V'~S)i is then obtained by composing with a and looking at the component in \Ii. Along the image a, we can use Ta· e h
e
e e,
512
5. DISTINGUISHED CONNECTIONS AND CURVES
e,
as { Since e h is the horizontal lift of we have wo(Ta· e h ) = 0 and by definition the p+-component of w(Ta . e h ) represents p(e). Equivariancy of s then implies V'~s = V'es + p(e). s along the image of a, and the formula follows by collecting terms in Vi. (2) Let ~ E X(9) be horizontal lift of E X(M) with respect to "t'. Then V'~ s and V~ s are represented by ~ . s and ~ . s + 'Y'" (~) • s, respectively. By definition w(a(u))(~) has values in g_. Putting 0 and W E IRn*. Consider the closed normal subgroup Q c P consisting of all elements with C E S L (n, IR) . Defining M# := 9 /Q, we see that the obvious projection 7r : M# - t M is a bundle with fiber P/Q ~ IR+, while the natural projection 9 - t M# is a principal bundle with structure group Q. In particular, the tangent bundle TM# can be realized as TM# = 9 xQ (g/q), where Q acts on g/q via the adjoint action. But now for C E SL(n,IR), a E IR and X E IRn, we get
G~) (~ ~) G_~S-l)
=
(a6~X
:).
On the one hand, this shows that the elements with X = 0 form a trivial Qsubmodule in g/q, which gives rise to a natural trivial line subbundle in TM#. Evidently, this is the vertical subbundle ker(T7r) of 7r : M# - t M. On the other
5.2. CHARACTERIZATION OF WEYL STRUCTURES
529
hand, g/q is isomorphic, as a representation of Q, to the restriction of the defining representation IR n+1 of C. In particular, this shows that TM# 9! 9 xQ IR n+1, so from the general theory developed in 1.5.7 we know that the Cartan connection w on 9 induces a linear connection \7# on T M#, called the cone connection. At first glance, it is not obvious, why constructing a natural affine connection associated to a projective structure would be a big simplification. One should keep in mind, however, that it is very easy to construct invariants of such a connection, using the curvature and its iterated covariant derivatives, and any such expression will automatically be an invariant of the projective structure. Also, it is rather easy to construct invariant differential operators using the cone connection. Next, we can easily describe M# explicitly. The subgroup Q c P by construction is the stabilizer of a nonzero point in the one-dimensional representation of P defined by the action on the P-invariant line in the standard representation IR n+1 of C. The P-orbit of this point is given by all positive multiples, so we can view P / Q as the space of positive elements in this representation. The line bundle over M corresponding to this representation is the density bundle £' ( -1). Hence, M # = 9 x p (P / Q) can be either viewed as the space £'+ (-1) of positive elements in £' ( -1) or as the oriented frame bundle of this line bundle. In particular, 11" : M# --+ M is a principal bundle with structure group 1R+, so the vertical subbundIe is canonically trivialized by the Euler vector field X E X(M#), the fundamental vector field corresponding to 1 E R Now clearly £'+( -1) --+ M can be used as a bundle of scales, so any (local) smooth section of this bundle defines a (local) Weyl structure for (Q --+ M, w). To formulate the properties of the ambient connection, we need one more observation. Suppose that s : M --+ M# is a local section of 11" : M# --+ M. Then for a vector field 'T/ E X(M), we can choose a vector field ij E X(M#) which is s-related to 'T/, i.e. such that ij(s(x)) = Txs, 'T/(x) for all x E M. Since s(M) C M# is a smooth submanifold, the value of \7#ij along s(M) in directions tangent to s(M) depends only on 'T/ and not on the choice of ij. Using this, one immediately concludes that there is a well-defined linear connection \7s on T M such that for all ~,'T/ E X( M) and any ij E X(M#) which is s-related to 'T/ we have (5.11)
\7~'T/(x) = Ts(x)1I"' (\7~:cs.eij(s(x))).
PROPOSITION 5.2.6. Let M be an oriented smooth manifold of dimension n 2: 2, and let 11" : M# --+ M be the oriented frame bundle of the density bundle £'( -1) --+ M. For t E 1R+ let pt : M# --+ M# be the principal right action of t. Let X E X(M#) be the Euler vector field, and let [\7] be a projective equivalence class of torsion-free linear connections on T M. Then the associated cone connection \7# on T M# --+ M# has the following properties: (i) For each t E 1R+, pt preserves \7#. (ii) \7t X = ~ for all ~ E X(M#). (iii) \7# is torsion free. (iv) The Ricci type contraction of the curvature R# of\7# vanishes identically. (v) For any smooth section s : M --+ M# of 11" : M# --+ M, the linear connection \7s defined by (5.11) lies in the projective class [\7]. PROOF. We have seen that T M# 9! 9 x Q IR n +1, so vector fields on M# correspond to smooth Q--equivariant functions 9 --+ IRn+1. If'T/ E X(M#) corresponds to f : 9 --+ IR n+1 and ~ E X(M#) is another vector field, then from 1.5.7 we know
5. DISTINGUISHED CONNECTIONS AND CURVES
530
that the function corresponding to vt'fJ is given by t· f + w(t) 0 f where t E X(9) is any lift of and in the second summand w(t) acts algebraically on the values of
e
f. The Euler vector field X by construction corresponds to the constant map ]Rn+1 with value the first unit vector e1 E ]Rn+1. Now again by construction, for E X(M#) and a lift E X(Q), the function 9 --t ]Rn+1 corresponding to is given by W(e)(e1), so (ii) holds. Next, we know from 1.5.7 that the curvature and torsion of V# are induced by the action the curvature of the Cartan connection won ]Rn+1. From 4.1.5 we know that, viewed as a Cartan connection on 9 --t M, w has vanishing torsion, and the homogeneity two part of its curvature is the totally tracefree part of the curvature of any connection in the projective class. In particular, the curvature of w has values in q, and hence is torsion free as a Cart an connection on 9 --t M#. Hence, the linear connection V# is torsion free, and its curvature R# satisfies ix R# = O. Expanding the latter equation we obtain
9
--t
t
e
e
O=V~vt'fJ-vtv~'fJ-V~~'fJ =V~vt'fJ- vtvt x - vt[X,'fJJ - V~,~l'fJ, where we have used torsion freeness. Using (ii) twice, to rewrite the second summand as - V# # X, we can use torsion freeness once more to get 'V'~'I/
But this exactly says that X is an infinitesimal automorphism of V#, so its flow preserves V#, and (i) follows. To compute the Ricci type contraction of R#, we can use a local frame {Xi} for T M# consisting of Xl = X and the pullback of a local frame of T M. Denoting the dual frame for T* M# by ¢i, we see that {¢2, ... , ¢n+1} descends to the dual of the above frame for T M. Then the Ricci type contraction is given by
Since ixR# = 0, the summand for i = 1 vanishes, and the rest descends to an expression on T M which vanishes since 8* '" = 0, so (iv) holds. To verify (v), fix a smooth section s : M --t M#. For a point x E M choose a local smooth section i of 9 --t M# defined in an open neighborhood U# of s(x). Then we can consider the pullback i*w which is a local g-valued one-form on M#. For any smooth function ¢ : U# --t Q also T(y) = i(y) . ¢(y) is a smooth section U# --t g, and T*W(y) - Ad(¢(y))-1 0 i*w(y) has values in q. In particular, if we assume that ¢(y) = exp(Z(y)) for a smooth function Z : U# --t gl, and for some E X(M#) the first column of i*w(e#) has the form (~~~)), then vector field
e#
the first column of T*W(e#) has the form (a(y)-:«~?X(Y)). For y E s(M) we have the codimension one subspace formed by the image of Ts, and by construction, the g_l-component of i*w(y) restricts to a linear isomorphism on this subspace. Otherwise put, elements of this subspace are characterized by the fact that a(y) = 1jJ(y)(X(y)) for some linear functional 1jJ(y) on g-l. This functional can be written as X(y) 1-+ Z(y)X(y) for an element Z(y) E gl. Since the whole construction is smooth, this defines a smooth function Z : U# --t g-l. Using this function to
5.2. CHARACTERIZATION OF WEYL STRUCTURES
531
modify f to r we see that for any x E M with s(x) E U# and ~ E TxM, the first column of r*w(Txs .~) has the form (~) for some X E g-1. Putting U := s-l (U#), r 0 s : U ~ 9 is a local smooth section of 9 ~ M. This gives rise to a Co-equivariant smooth section (j : P01(U) ~ p-1(U) of 1i' : 9 ~ 90 which is characterized by (j(1i'( ros(u))) = r(s(u)), and hence a local Weyl structure. We want to show that the local Weyl connection determined by (j coincides with V S , which completes the proof. Let ~ be a vector field on M. Consider T( r 0 s) . ~ along r( s( U)), and extend it to a vector field E X(9), which is projectable to a local vector field ~ E X(M#). Then by construction ~ is s-related to~. Applying the same construction to another vector field "7 on M, we also see that the local function f : 9 ~ 9 defined by f(u)w(i)(u)) induces both the P-equivariant function 9 ~ g/p corresponding to "7 and the Q-equivariant function 9 ~ g/q corresponding to fl. Now by construction, the Q-equivariant function 9 ~ g/q corresponding to V1fl is given by ~ . f(u) +
e
w(~)(f(u)) + q. Applying Tn simply corresponds to taking the image in g/p, so V E"7 corresponds to the function ~. f(u) + w(~)(f(u)) + p. By construction, both
w(~) and f(u) have the form (1 :). Using this, it is evident that w(~)(f(u)) +13 = wo(~)(f( u)+p), and since the horizontal lift of ~ with respect to the Weyl connection determined by (j is given by ~(u) - Co«((u))(u), this completes the proof. 0 REMARK 5.2.6. There is no direct analog of this construction for the other geometries corresponding to Ill-graded Lie algebras. While there is always a subgroup Q c P such that C/Q is the space of nonzero elements in a scale bundle, the representation g/q does not extend to 9 in general. We will see in 5.3.10 below that a direct analog of the ambient description of projective structures exists for contact projective structures. Consider for example the case of conformal structures. Viewing elements of P as block upper triangular matrices of the form
A -AwtITp.qC
(
o o
C 0
_~(wt, W t ))
w
A-I
(see Proposition 1.6.3), the subgroup Q corresponds to those matrices for which A = 1. This easily implies that g/q can be identified (as a Q-module) with the subspace in lRn +2 spanned by the first n + I vectors in the standard basis. Since this subspace is not invariant under g, we cannot obtain an ambient connection in this way. This idea still leads to an interesting construction in the case of conformal structures containing an Einstein metric. For such structures, Proposition 5.2.5 implies existence of a parallel standard tractor. Hence, one obtains a linear connection on the orthocomplement of this parallel standard tractor. In a similar way as discussed for projective structures above, this can be used to construct a linear connection on the tangent space of 9/ Q. This construction is also referred to as the cone construction for Einstein conformal classes; see [Arm07a]. There is an ambient description for general conformal structures, which, however, is much more complicated. This was introduced in [FG85], the details of the construction have been worked out in [FG07]. To obtain that description, one has to artificially enlarge the total space of a density bundle (usually one takes the ray subbundle of S2T* M which defines the conformal class) by one dimension. One
532
5.
DISTINGUISHED CONNECTIONS AND CURVES
then shows that on this enlarged space there exists a Ricci-flat pseudo-Riemannian metric (formally along the density bundle up to some order) which induces the given conformal class in a certain sense.
5.2.7. From the cone description to the Cartan description. We next want to prove that the ambient description of projective structures from 5.2.6 is actually equivalent to the Cartan description. This will also prove that the cone connection is uniquely determined by the properties (i)-( v) in Proposition 5.2.6. This equivalence can be nicely formulated via abstract tractor bundles. Let M be an oriented smooth n-dimensional manifold and let 'IT : M# ~ M be the space of nonzero elements in the bundle of n~1-densities of M. (From 4.1.5 we know that for any projective structure on M, the bundle £(-1) will be isomorphic to the bundle of n~1-densities.) Then we can construct a canonical volume form on M# as follows: Since M is oriented, for a point z E M# one may interpret the 1density zn+l as a nonzero element in AnT;(z)M. Hence, we can define a tautological n-form a E nn(M#) by a(z)(6, ... , ~n) = zn+l(Tzp·6, ... , TzP'~n)' and we claim that II := da E nn+1(M#) is a volume form. By construction, 'IT : M# ~ M is a principal fiber bundle with structure group 1R+, and we denote by pt the principal right action of t E 1R+. Then by construction (pt)*a = tn+1a. Denoting by X the Euler vector field on M#, the flow of X is given by Fl? = pet. For the Lie derivative of a along X, we thus get exa = (n + l)a, and since ixa = 0, this shows that ixll = (n + 1) a, which proves the claim. Note that (Pt)*11 = t n + 111 by construction. Next, for t E 1R+ and ~ E TzM# put ~. t := t-1Tzpt .~. This evidently defines a smooth right action on T M#, which by construction lifts the principal right action on M#. Consequently, the action is free and the orbit space T := TM#/IR+ is a smooth manifold and a vector bundle over M#/IR+ = M. (A vector bundle atlas for 7 can be easily constructed from a principal bundle atlas for 'IT : M# ~ M and the induced atlas for TM.) From the definition of 7 we conclude that smooth sections of T ~ M are in bijective correspondence with vector fields ~ E X(M#), which are homogeneous of degree -1 in the sense that ~(z . t) = t- 1 T zp t . ~(z). This can be equivalently characterized infinitesimally as [X, ~l = -~. PROPOSITION 5.2.7. (1) The vertical bundle of 'IT : M# ~ M descends to a line subbundle 7 1 C 7 such that 7 1 ~ £(-1) and T/T1 ~ TM ® £(-1). The canonical volume form II on M# descends to a canonical nonvanishing section of An+lT. (2) Let '\1# be a linear connection on TM# which preserves the volume form II and satisfies conditions (i)-(iii) from Proposition 5.2.6. Then '\1# descends to a tractor connection '\1 T on T, making it into an abstract standard tractor bundle of projective type. In particular, we obtain an induced (torsion free) projective structure on M. (3) The curvature R# of '\1# descends to the tractor curvature of '\1 T . The tractor connection '\1 T is normal if and only if the Ricci-type contraction of R# vanishes. PROOF. (1) Since 1R+ is commutative, we have pt 0 pS = pS 0 pt, and differentiating this, we get X(z . t) = Tzpt . X(z). Since X spans the vertical bundle of 'IT : M# ~ M, this shows that we get an induced subbundle T1 c T. Sections of
533
5.2. CHARACTERIZATION OF WEYL STRUCTURES
this bundle are isomorphic to vector fields of the form I X, where I : M # -) IR is a smooth function such that I(z· t) = C 1/(z). But then 8(7r(Z)) := I(z)z (scalar multiplication in £ ( -1)) is a well-defined section of £ (-1). Conversely, given a section 8 E r( £ (-1)) this equation defines a function I with the right equivariancy property. This shows that 7 1 ~ £ ( -1). In the same way, smooth functions I : M# -) IR such that I(z . t) = t W I(z) for some fixed wEIR can be identified with sections of £(w). Given a function I corresponding to a section of £(1) and a vector field which is homogeneous of degree -1, the product Ie is homogeneous of degree zero and hence projectable to M. The kernel of this projection is given by the vertical fields. This induces an isomorphism from 7/71 to the bundle of linear maps from £ (1) to T M, i.e. to TMQ9£(-1). For sections 81, ... , 8n +l E r(7), we can consider the corresponding vector fields 6, ... , en+1 E X(M#), which are homogeneous of degree -1. As we have observed above, the volume form v E nn+1(M#) satisfies (Pt)*v = tn+1v. Hence, the function v(6, ... , en+l) : M# -) IR is homogeneous of degree zero, so it descends to a smooth function on M. Assigning this function to 81, ... ,8 n +1 defines a section of An+17*, which is nowhere vanishing by construction.
e
,,#
(2) If'rf E X(M#) is homogeneous of degree -1, then torsion freeness of implies that "~'rf = X + [X, 'rfl = O. Consequently, for a vector field E X(M) and a lift ~ E X(M#) of the vector field depends only on and not on the
"#
"1'rf
e,
e
e
choice of the lift. Further, any lift ~ is homogeneous of degree zero, so [X, ~l = O. By assumption, for each t E 1R+ the principal right action pt by t preserves ,,#. In particular, this implies that is compatible with homogeneities, so for ~ and as above, the vector field is homogeneous of degree -1. Thus, it again corresponds to a section of 7, and if'rf corresponds to 8 E r(7), then we denote that section by "[8. It is straightforward to verify that this defines a linear connection on 7. Now consider G = SL(n + 1, 1R) and let PeG be the stabilizer of a line in IRn+1. Then via the line subbundle 7 1 c 7 and the nonvanishing section of An+17 constructed in (1), we get a natural frame bundle g for 7 with structure group P. As discussed in 3.1.21, this gives rise to an abstract adjoint tractor bundle via A = g Xp g. A linear connection on 7 then is a g-connection as defined in 3.1.21 if and only if it preserves the distinguished section of An+17. Since this is evidently the case for "T, we only have to verify the nondegeneracy condition from 3.1.21 to complete the proof of (2). As we have noted in (1), sections of the subbundle 7 1 are represented by vector fields of the form I X, where I : M# -) 1R is a smooth function which is homogeneous of degree -1. But then we get IX = (~. f)X + I~, so if I is nonzero in a point (and hence along the corresponding fiber), the class of this in 7/71 ~ TM IS> £(-1) is simply IS> I.
'rf
,,#
"1'rf
"1
e
(3) It follows directly from the construction that R# descends to the curvature The computations done in the proof of Proposition 5.2.6 read backwards show that ix R# = O. This easily implies that the Ricci-type contraction of R# descends to 8* R. 0
R of the tractor connection
"T.
REMARK 5.2.7. The passage from the cone description of projective structures to the Cartan description via abstract tractor bundles was motivated by a similar
534
5. DISTINGUISHED CONNECTIONS AND CURVES
construction in conformal geometry, which was worked out in [CGo03]. There the alternative description is provided by an ambient metric, which is a weakening of the Fefferman-Graham ambient metric as introduced in [FG85]; see also Remark 5.2.6. 5.2.8. Geometries corresponding to 111-gradings and affine holonomies. We have briefly discussed holonomy in part (2) of Remark 5.2.5. While the concept of holonomy is defined for connections on general bundles, the case of linear connections on the tangent bundle is of particular interest. It turns out that, under a few additional restrictions, the possible holonomy groups are not arbitrary but confined to a rather short list. Describing the possible holonomy groups was a long term project in differential geometry, starting from the basic work of Marcel Berger in the 1950s. We have seen in 5.2.5 that for a linear connection on a vector bundle E ---t M, the holonomy groups are subgroups of GL(Ex ), where Ex is the fiber over x E M. Likewise, the holonomy groups of a principal connection are subgroups of the structure group of the principal bundle. Now it turns out that in both cases the holonomy group is a closed subgroup and hence a Lie subgroup. Consequently, there is the Lie algebra of the holonomy group, which is called the holonomy Lie algebra of the connection in a point. If the base of the bundle is connected, then the holonomy group and the holonomy Lie algebra is essentially independent of the given point. Indeed, the parallel transport along any path between the two points induces an isomorphism between the two groups. Hence, the holonomy group can be viewed as a subgroup of the general linear group of the standard fiber, respectively, the structure group of the principal bundle defined up to conjugation. Likewise, the holonomy Lie algebra can be viewed as a Lie subalgebra of the corresponding Lie algebra defined up to the adjoint action of a group element. In particular, the holonomy Lie algebra can be viewed as a Lie algebra endowed with a fixed representation defined up to isomorphism. Now let us specialize to the case of the tangent bundle T M of a smooth manifold M. Then the holonomy group is a closed subgroup of GL(n,JR) and the holonomy Lie algebra is a Lie subalgebra of g[(n,JR), where n = dim(M). It turns out that arbitrary holonomy Lie algebras can occur if one allows connections with torsion. Likewise, locally symmetric connections (those, which have parallel curvature) form a special class, which can be classified by other means. Hence, in holonomy theory one is usually only interested in torsion free connections, which are not locally symmetric. The first step in determining possible holonomy groups or Lie algebras is to look at the case of Levi-Civita connections of Riemannian metrics. In that case, a theorem of de Rham ensures that a decomposition of a holonomy representation into a direct sum of two invariant subspaces is induced by a local direct product decomposition of the manifold into corresponding factors. Hence, it suffices to study the case that the holonomy representation is irreducible, i.e. that 9 C g[( n, JR) does not admit a nontrivial invariant subspace in JRn. In his classical work [Be55], M. Berger not only determined a list of possible holonomy Lie algebras of LeviCivita connections but also extended this to a list of possible irreducible holonomy representations of arbitrary affine connections (with the above restrictions). (Later on, a few small corrections to the list were found.) Obtaining the list is a purely algebraic task, the main step is to analyze the consequences of the Bianchi identities.
5.2. CHARACTERIZATION
OF
WEYL STRUCTURES
535
Berger's original claim was that his list is complete up to possibly finitely many exceptions. For some of the holonomy algebras in Berger's list, there is a simple geometric interpretation, which easily leads to examples. In general, however, proving existence of examples (and in particular of compact or complete examples) of such holonomies turned out to be a difficult task. A variety of different methods for constructing connections with certain holonomies was developed, some of them leading to interesting relations to other parts of mathematics. Step by step, this lead to extensions of Berger's list, which became known as exotic holonomies, and it turned out that there even is an infinite family of holonomy Lie algebras, which is not in Berger's list. The whole program was completed by S. Merkulov and L. Schwachhofer in [MS99], where the last remaining cases were sorted out, so the classification of affine holonomies was complete. The list of possible non-Riemannian holonomy Lie algebras has surprisingly close relations to the classification of certain types of parabolic subalgebras in simple Lie algebras. We will now use the theory of Weyl structures for geometries corresponding to III-gradings to prove existence of a class of holonomy representations. Suppose that 9 = 9-1 EB 90 EB 91 is a Ill-graded simple Lie algebra. Then from 3.2.3 and 3.2.10 we know that the adjoint action restricts to an irreducible representation of 90 on 9-1' Further, we know that 90 = ~E EB 9~s, where E is the grading element and 93s is the semisimple part of 90' We want to prove that both the natural representation of 90 on 9-1 and its restriction to 9~s are holonomy representations. Except for the full symplectic algebra (which is easy to realize), this exhausts all of Berger's original non-metric list, as well as some exotic holonomies related to the exceptional algebras of type E6 and E 7 • LEMMA 5.2.8. Let 9 = 9-1 EB 90 EB 91 be a Ill-graded simple Lie algebra of rank larger than one. Then the mapping (P, X A Y) f-t 8P(X, Y) induces surjections 8 2 91 ® A2 9 _ 1 ---. 9~s as well as 91 ® 91 ® A2 9 _ 1 ---. 90. PROOF. Note that both maps under consideration are evidently 90--equivariant, so it suffices to show that their image meets each irreducible component of the target space. By the assumption on the rank, we know that dim(9-1) > 1, so we can choose linearly independent elements X, Y E 9-1 and a linear map P : 9-1 ---. 91 such that P(Y) = 0 and B(P(X), Y) =f. 0, where B denotes the Killing form. But then 8P(X, Y) = -[Y, P(X)] and hence B(8P(X, Y), E)
= B(P(X), [Y, ED = B(P(X), Y) =f. O.
Since the decomposition 90 = ~E EB 9~s is orthogonal for B, this shows that the image of the second map is not contained in 9~s, so it suffices to prove surjectivity of the first map. Complexifying if necessary, we may assume that 9 is a complex Ill-graded semisimple Lie algebra (with each of the simple ideals being Ill-graded), and then we can use the root decomposition. In each of the simple ideals of 9, there is a unique simple root 0: such that the root space 90: is contained in 91. From the description of the Dynkin diagram of 90 in Proposition 3.2.2, we see that for any simple ideal of 9~s there is a simple root 13, which is not orthogonal to one of the roots 0:, such that the root space 913 is contained in the given simple ideal. Since 13 is not orthogonal to 0:, 0: + 13 is a root and by construction 90:+13 E 91. Now choose a basis {X')'} of 9-1 such that each X')' lies in the root space 9_')' and let {Z')'} be
536
5.
DISTINGUISHED CONNECTIONS AND CURVES
the dual basis of g1. Then consider P = ZOI V ZOI
+ ZOI+{3 V ZOI+{3
E 8 2 g1. Then
which is the sum of a nonzero element of g{3 and a nonzero element of g-{3. Hence, the image of our map meets the simple ideal containing g{3. 0 THEOREM 5.2.8. Let 9 = g-1 EB go EB g1 be a simple Ill-graded Lie algebra of rank larger than one. Then the representations go ~ g((g-1) and g08 ~ g((g-1) can be realized as holonomy representations of torsion free linear connections on the tangent bundle of a compact manifold, which are not locally symmetric. PROOF. Let G be a Lie group with Lie algebra g, PeG a parabolic subgroup corresponding to p = go EB 91, and consider the generalized flag manifold G I P. Putting P+ := exp(g1) c P as usual, we know that all Weyl connections for the canonical parabolic geometry of type (G, P) on GI P are induced from principal connections on the principal Go-bundle G I P+ -+ GI P, and they are all torsion free by Theorem 5.2.3. Since we are dealing with a Ill-graded simple Lie algebra here, the center of go is generated by the grading element E. Hence, for any functional >. defining a bundle of scales, the kernel of >. coincides with the semisimple part gos. From 5.1.7 we thus see that in the special case of an exact Weyl structure, we get a further reduction to a principal bundle whose structure group has Lie algebra gos. In particular, the holonomy Lie algebra of any exact Weyl connection is contained in goS, while the holonomy Lie algebras of arbitrary Weyl connections are contained in go (compare with 5.2.5). To start our construction, we need a specific exact Weyl structure on GI P. First of all, we know that there always exist global exact Weyl structures, and we choose one of those. On the other hand, from Example 5.1.12 we know that there is the very flat Weyl structure defined on the open subset G_ c GIP. This is also an exact Weyl structure, its Rho tensor vanishes identically, and the corresponding Weyl connection on any induced vector bundle is flat. Now on the open subset G_, the change from the restriction of the globally defined Weyl structure to the very flat Weyl structure is described by an exact one-form T = df· Multiplying f by a function which has support in a ball around zero and is identically one on a smaller ball, the product can be smoothly extended by zero to all of GIP. Using the corresponding exact one-form to modify our initial global Weyl structure, we obtain a globally defined exact Weyl structure CT, which coincides with the very flat Weyl structure locally around 0= eP. Now we modify the Weyl structure CT using a one-form T, call the result a and use notation as before. The following computations are done on the open neighborhood of a on which \7 is flat. There we have R = 0 and P = 0, and hence R = o(P), P(~) = \7~T + HT, {T,~}} (see 5.1.8) and V~s = \7~s - {T,e}. s (see 5.1.6) for any ~ E X(GI P) and any section s of an associated bundle. In particular, assume that for some k 2: 0, we have j~T = 0, where j~ denotes the k-jet in o. Then j~-1p = 0, j~P = j~(\7T) and, for any s we get j~(Vs) = j~(\7s). Iterating this, we conclude that Vkp(o) = \7k+1T(o). Finally, since 0 is a bundle map on a vector bundle associated to GI P+ -+ GI P, which comes from a Go-equivariant map between the inducing representations, it is parallel for any of the Weyl connections. Using this, we conclude that \7 k R(o) = (id 0 o)(\i'k+1T(o)).
5.2. CHARACTERIZATION OF WEYL STRUCTURES
537
Now it is well known that all values of \lk R(o) lie in the holonomy Lie algebra. Hence, we can complete the proof by showing that we can find a one-form Y such that j~Y = 0 and the values of (id (8) 8)(\lk+1y(o)) span all of 90, respectively, the same statement for 90s with an exact one form Y. Now of course we can find a one-form (respectively an exact one-form) Y with j~Y = 0, such that \lk+1y(o) is an arbitrarily prescribed element of Sk+1T;(GjP) (8)T;(GjP) (respectively of Sk+2T;(Gj P)). Passing to the inducing representations and using the lemma, we see that it suffices to construct elements of Sk+1 91 (8) 91. respectively, of Sk+2 91 , which are surjective when viewed as maps from Sk 9_ 1 to 91 (8) 91, respectively, to S2 91 . For the first case, we can use k = 4 and, using a basis {ed of 91 with dual basis {e i } of 9-1, consider the element Li,j e~e~ (8) ei E S5 91 (8) 91. Evaluating on (ei)4, this gives ei (8) ei, while evaluating on (e i )3ej , we get ej (8) ei. For the second case, we can also use k = 4 and consider the element Li<j ete~ E S6 91 . Putting n = dim(9-1), we can get e~ for i < n as the value on (e i )2(ei+1)2 and e; as the value on (e n - 1)4. For i < j, we can obtain eiej as the value on (e i )3ej . D 5.2.9. Weyl forms for general parabolic geometries. Let us return to the general theory of Weyl forms. To generalize the results of 5.2.3 to arbitrary parabolic geometries, we first define appropriate replacements for the torsion and curvature quantities used there. In 5.2.2, we have already observed that for a Weyl form 7 E 0 1(90,9), the components L + 70 E 0 1(90,9_ EB 90) define a Cartan connection on 90. Hence, it is a natural idea to look at the curvature of this Cartan connection. Since the splitting 9_ EB 90 is 90-equivariant, this curvature splits accordingly into a torsion part and the curvature of the principal connection 70. Since L identifies TM with gr(TM), the torsion part can be naturally interpreted as a form T E 02(M, gr(TM)). We call T the torsion of the Weyl form 7. The curvature of the Weyl connection 70 can be naturally viewed as R E 0 2 (M, Endo(gr(T M))). The appropriate definition for the analog of the tensor Y is a bit more difficult to guess, since there is the possibility to include various terms that automatically vanish in the Ill-graded case. Recall that we view the Rho tensor associated to 7 as P E 01(M, gr(T* M)). The Weyl connection \l on gr(T* M) extends to the covariant exterior derivative d'V on gr(T* M)-valued forms. Second, using the identification T M s::: gr(T M) provided by the soldering form L , we can carryover the algebraic bracket to { , } : T M x T M -+ T M. Finally, we have the algebraic bracket on T* M, which we also denote by { , }. Using these ingredients, we now define Y E 02(M,gr(T*M)) by Y(~, 1]) := d'VP(~, 1])
+ P( {~, 1]}) + {P(~), P(1])}.
Motivated by the conformal case (compare with 1.6.8), we call Y the Cotton-York tensor of the Weyl form 7. We can now easily describe the relation between the curvature kr of 7 and the triple (T, R, Y). Using the identification T M s::: gr(T M) provided by L , we can apply the map 8 to P, and view the result as 8P E 02(M, gr(TM) EB Endo(gr(TM)) EB gr(T* M)). Using this, we can now generalize a part of Theorem 5.2.3:
5. DISTINGUISHED CONNECTIONS AND CURVES
538
5.2.9. Let r be a Weyl form on a regular infinitesimal flag structure M,O). Then
THEOREM
(Po: 90
-+
k.,. = (T, R, Y)
+ 8(P) E 02(M, gr(TM) E9 Endo(gr(TM» E9 gr(T* M)).
In particular, the harmonic part of k.,. coincides with the component in ker(D) of (T, R, Y), so for normal Weyl forms, this component represents the harmonic curvature ,.. H •
Take vector fields" ry E X( M) and let ,h, ryh E X(90) be their horizontal lifts with respect to ro. Then the section k.,.(" ry) E r(90 xGo g) is by definition represented by PROOF.
dr('h, ryh)
+ [r(,h), r(ryh)].
Since ,h is horizontal we have r(,h) = L(,h) +r+(,h) and likewise for ry. Splitting the bracket term accordingly, we see that by definition
dr('h, ryh)
+ [L(,h), L(ryh)]
represents (T(" ry), R(" ry), d'7p(" ry». The term [r+ ('), r+(ry)] of course represents {PC'), P(ry)}. Finally, the expression
[r+(,h), L(ryh)]
+ [L(,h), r+(ryh)]
evidently represents
{P(O, ry} - {', P(ry)} = 8P(" ry) + P( {', ry}), and the result follows by definition of Y.
D
From this result, in particular, we see how to describe the Cartan curvature and the harmonic curvature of a regular normal parabolic geometry in terms of a Weyl structure. 5.2.10. Constructing normal Weyl forms. The part of Theorem 5.2.3 that we have not generalized so far is the characterization of normal Weyl forms. For general parabolic geometries, this is significantly more complicated than in the \1\graded case. The main reason is than in the Ill-graded case the tensors T, R, and Y coincide with the homogeneous components of (T, R, Y) and 8P is concentrated in one homogeneity. For general geometries, it is still possible to split the forms r, T, R, and Y according to their values. However, the conceptually more important splitting is according to homogeneities. The objects we consider here are one-forms and twoforms with values in gr(TM) E9 Endo(gr(TM» E9 gr(T* M). Hence, these values naturally split into components of degree -k, ... , k. For the one form rand r 2: 0, we mean by the "component of homogeneity ::; r" of r, the restrictions of the forms Ti to Ti-r M for i = -k, ... , r - 1. Note, in particular, that the component of homogeneity ::; 0 of any Weyl form r coincides with the frame form of the infinitesimal flag structure, so this is always known in advance. For the two-forms T, R, and Y, we use an analogous terminology. For example, the component of T of homogeneity ::; r consists of the restrictions of the forms Te E 0 2 ( M, gre(T M) ) to Ti M X Tj M where i + j + r = f.. and f.. = -k, ... ,-1. Note that the components of R of homogeneity ::; 1 and of Y of homogeneity ::; 2 vanish by definition. The main technical result now is to describe the dependence of the homogeneous components of T, R, and Y on the homogeneous components of r. This exhibits an important difference between Y and the other two components.
5.2. CHARACTERIZATION OF WEYL STRUCTURES
539
LEMMA 5.2.10. Let (p : go --+ M,8) be a regular infinitesimal flag structure, and let T be a Weyl form. (1) The torsion T of T is given by
a,b 0, the component of homogeneity :S r ofT depends only on the component of homogeneity :S r of T. (2) For each r ~ 2, the component of homogeneity :S r of R depends only on the component of homogeneity :S r of T. (3) For each r ~ 3, the component of homogeneity :S r of Y depends only on the component of homogeneity :S r - 1 of T.
°
PROOF. (1) As in the proof of Theorem 5.2.9, consider vector fields~, 1] E x(M) and let ~h, 1]h E x(Qo) be their horizontal lifts with respect to TO. In that proof we have seen that T(~,1]) is represented by dL(~h,1]h)
+ [L(~h),L(1]h)].
Inserting the definition of the exterior derivative and taking into account that differentiating by a horizontal lift represents a covariant derivative, the formulae for the components of T follow by splitting the values into components. Now suppose that i,j < 0 and e < i + j. By definition of a Weyl form, Te vanishes on THl M and thus, in particular, on TiM, Tj M and Ti+j M. Hence we see that the first three terms in the formula form Te vanish on Ti M x Ti M. Moreover, if a+b = e, then either i > a or j > b, so the last term cannot contribute either. Hence, the component of homogeneity :S -1 of T vanishes automatically. If e = i + j, then i > and j > and for ~ E Ti M and 1] E Ti M the formula for Tf(~, 1]) reduces to -Te([~, 1]]) + {Ti(~)' Tj(1])}. Since [~, 1]] is a section of TiM, we may replace in this expression each T by the frame form 8 and the vanishing follows from Proposition 3.1.7. Thus, the component of homogeneity :S 0 of T vanishes automatically. Finally, suppose that e = i + j + r for some r > O. Then i = r + j < r, e- j = r + i < r, and e- (i + j) = r, so the first three summands in the formula for Te depend only on the components of homogeneity :S r of T. For the last term, we must have a+b = but only summands with a :S i and b :S j can produce a nonzero contribution on Ti M x Ti M. From a + b = i + j + r we obtain a - i = j - b + r :S r and likewise b - j :S r. (2) Since the curvature R(~, 1]) is represented by dTo(~h, 1]h), we see that the restriction of R to Ti M x Ti M depends only on the restriction of TO to Ti+i M and the result follows. (3) From the definition of Yin 5.2.9 we know that it only depends on the Rho tensor, which is represented by T +. Expanding the definition, we see that
e
e
e-
e,
a,b>O;a+b=e
Assume that for some r ~ 2, i, j < 0 we have e = i + j + r > O. Then for sections 1] of Tj M, the expression {~, 1]} E Ti+ j M depends only on the restriction of T_ to Ti+j M, and this concerns only the components Ti+j, ... , L l '
~ of Ti M and
540
5. DISTINGUISHED CONNECTIONS AND CURVES
Hence, {~,17} only depends on the component of homogeneity -(i + j) - 1 < r of T. Via the covariant derivatives, we have a dependence on the restrictions of TO to Ti M, respectively, Tj M, but this again only gives homogeneities < r. In the first two terms and the last sum in the formula for Y above, we only use restrictions of Pa to Ti M and Tj M for a ~ e, so we meet at most homogeneities e- i < r, respectively, e- j < r. Finally, by regularity [~, 17]- {~, 17} is a section of Ti+j-l M so in the last remaining term we only need the restriction of Pe to Ti+j-l M, which depends only on the component of homogeneity r - 1 of T. 0 Now we describe in principle how to use these results in order to determine normal Weyl forms. The case of parabolic contact structures will be discussed in more detail below. First make any choice for 7_ E nl(M,gr(TM)), which only amounts to choosing a splitting TM ~ gr(TM) of the filtration of TM. Further, choose a principal connection 70 E nl(Qo, 90). This gives rise to linear connections on all gri(TM). For a start, we put 7+ = O. The strategy is now to work homogeneity by homogeneity, on the one hand correcting the choices and at the same time computing parts of the Rho tensor P E n l (M, gr(T* M)), which are then used as 7+. Recall from Theorem 5.2.9 that k7' = (T, R, Y) + 8P, so the basic equation describing normality is 0 = 8*(T, R, Y)+8*8P. Recall further that 8* is compatible with homogeneities; see 3.1.12. Hence, this equation can be analyzed homogeneity by homogeneity. Since 8 is also compatible with homogeneities, and the components of homogeneity ~ 2 of P evidently vanish, we conclude that the same is true for 8* 8P. For homogeneity ~ 1 we are therefore left with 8*T(1) = O. Here and below we use upper indices in braces to indicate homogeneous components. From the lemma we see that this depends only on the component of homogeneity ~ 1 of 7, i.e. the restriction of each 7i to T i - l M and the restriction of 70 to T- l M. The general theory tells us that we can rearrange these data in such a way that 8*T(1) = O. Let us denote the Weyl form obtained in this way again by 7. Then 8*(/'\,7') is by construction homogeneous of degree ~ 2, and similarly to the proof of Theorem 5.2.2, one shows that we can normalize 7 by changing the part of homogeneity ~ 2. Hence, the homogeneous components of degree one of 7 already coincide with the components of some normal Weyl form. Next, we look at the component of homogeneity 2. The normalization equation has the form 8* (T(2) + R(2)) + 8* 8P(2) = O. In this step, we first have to correct the components of homogeneity 2 of 7 that we have chosen so far, and then compute p(2). For the given choice of 7 we can compute 8* (T(2) + R(2)) and this depends only on the components of homogeneity ~ 2 of 7 by the lemma. Now we can rewrite the normalization equation as (5.12)
8* (T(2)
+ R(2)) + DP(2) -
88* p(2) = 0,
where 0 is induced by the Kostant Laplacian; see 3.1.11. Since 8*T(1) = 0, the map 8*(T(2) + R(2)) induces bundle maps gri(TM) --4 gri+2(TM) for i ~ -3, gr_2(TM) --4 Endo(TM), and gel (TM) --4 grl (T* M). We claim that (5.12) admits a solution if and only if there is a section a2 E r(gr2(T* M)) such that for the map induced by 8*(T(2) + R(2)) + 8a2 only the component gr_l(TM) --4 grl(T*M) is nonzero. By Proposition 3.3.4, the Kostant Laplacian acts by a scalar on each isotypical component, so it is a linear combination of projections to isotypical components and thus commutes with any Go-equivariant
5.2. CHARACTERIZATION OF WEYL STRUCTURES
541
map. Moreover, from Proposition 3.3.7 we know that 0 acts invertibly on bundle maps gr(TM) --t gr(TM) Ef) Endo(gr(TM)) Ef) gr(T* M), which are homogeneous of degree ~ 2. In particular, if is such a bundle map, then has values in gr(T* M) if and only if O has values in gr(T* M). If (5.12) has a solution, then p(2) has values in gr(T* M), so the same is true for OP(2), and then Q2 := -()* p(2) has the required property. Conversely, given Q2 such that ()* (T(2) + R(2») + ()Q2 has values in gr(T* M) we define p(2) := _0- 1 (()* (T(2) + R(2») + ()Q2), and this has values in gr(T* M), too. This also implies that ()*{)p(2)
+ (){)*p(2)
= Op(2) = _()*(T(2)
+ R(2») -
()Q2.
Since im({)*) and im({)) are complementary, we get ()Q2 = _(){)*p(2), and hence (5.12) is satisfied. In fact, we can do even better, since there is only one possibility for Q2, which can be computed in advance. Let ¢: gr_2(TM) --t Endo(gr(TM)) be the restriction of ()* (T(2) + R(2»), and let B denote the bilinear form induced by the Killing form of g. Then gr2(T* M) is dual to gC2(T* M) via B, so Q2(~) = B(~, (2) for ~ E f(gC2(TM)). By assumption, ¢ must coincide with -()Q2 on gr_2(TM), so we must have ¢(~) = -{~, Q2}' Now if E E r(Endo(gr(TM))) is the canonical grading section (see 5.1.5), then we obtain B(E,¢(~)) = -B(E,{~,Q2}) =
-B({E,{},Q2) = 2B(~,Q2)'
Thus, Q2(~) := ~B(E, ¢(~)) is the only possible solution. Now we can collect what has to be done for homogeneity 2: We can compute the candidate for Q2 by the above formula, and then we have to modify the component of homogeneity 2 of 7 H, so we must have \jH J = O. Likewise, for Lagrangean contact structures as discussed in 4.2.3, \jH must respect the decomposition H = E EEl F into a sum of Legendrean subbundles, so it actually is induced from connections \jE and \jF on the two factors. THEOREM 5.2.11. Consider a parabolic contact structure on a smooth manifold M with contact subbundle H c T M and quotient bundle Q := T M / H, and let q : T M -> Q be the natural bundle map. Let be a contact form for H with Reeb field r. Then we have: (1) The isomorphism T M ~ HEEl (T M / H) determined by the exact Weyl structure corresponding to e is given by ( f-+ (rr( (), q( ()), where rr( () = ( - e( ()r. (2) The Weyl connection determined by the exact Weyl structure corresponding to e has the property that for the induced connection \j H on H, the bundle map A2H ---.. H induced by
e
(~, 1])
f-+
\jr 1] - \j~ ~ - rr([~, 1]])
has values in ker(D). This uniquely determines the restriction of the Weyl connection to directions in H. PROOF. For the normal Weyl form corresponding to the exact Weyl structure determined bye, \jQ has to be the flat connection induced bye. Hence, for ~,1] E X(M), the section \j~q(1]) must correspond to ~ . e(1]). For 1], ( E r(H), in particular, we have .e(1], () = q([1], (]) and e([1], (]) = -de(1], (). Hence, the defining equation for \jQ above can be rephrased in the form that viewing de as a section of A2 H*, we have \j de = 0 for the induced connection. Next, the homogeneity one component T(l) of the torsion splits into two parts, one mapping A2 H to H and the other mapping H ® Q to Q. The explicit formula for these components follows immediately from part (1) of Lemma 5.2.10. For the first component we have to consider ~,1] E r(H). Then L1(~) = ~ and similarly for 1], but L1([~,1]]) corresponds to rr([~,1]]) E r(H). For the second component take ~ E r(H) and 1] E X(M). Then L2(~) = 0, while L2(1]) represents q(1]), and L1(1]) represents rr(1]). Similarly, L2([~' 1]]) represents q([~, 1]]). Hence, for the two components we obtain
(5.14) (5.15)
T- 1 (e,1]) = \jr1] - \j~e -rr([e,1]])
T- 2 (e, q(1])) = \j~q(1]) - q([e, 1]]) + .e(e, rr(1]))
e,1] E r(H),
e E r(H), 1] E X(M).
Now we know that for a normal Weyl form, T(1) coincides with the homogeneous component of degree one of the Cartan curvature. In particular, it must have values in ker(D) by Theorem 3.1.12, which proves the first part of (2). In Lemma 4.2.2 we have seen that the component of ker(D) in homogeneity one is contained in A2g~1 ® g-l, so (5.15) has to vanish identically. Rewriting this in terms of the contact form e we obtain
0= e . e(1]) - e([~, 1]]) - de(e, rr(1])) = de(~, 1] - rr(1])),
544
5. DISTINGUISHED CONNECTIONS AND CURVES
for all ~ such that O(~) = O. Since H has codimension one, it follows that this equality must be satisfied for all ~ E X(M). But this exactly means that TJ - 7r(TJ) must be a multiple of the Reeb field, which immediately implies the formula for 7r in (1). Having described the projection 7r : TM -> HM, the remaining information about the component of the Weyl form in homogeneity one is contained in the restriction of \/H to r(H) x r(H). We know by the general theory that this restriction is uniquely determined by the requirement that T-1 : A2 H -> H lies in ker(D). 0
5.2.12. Webster-Tanaka connections. Rather than proceeding directly to the determination of a Weyl connection, we will take a slight detour here and generalize one of the standard tools of CR-geometry to all parabolic contact structures. In 5.2.11, we have associated to a contact form 0 on a parabolic contact structure (and the corresponding isomorphism T M = H EB Q) a unique partial connection on H, or equivalently, a partial principal connection on the infinitesimal flag structure Po : go -> M which describes the geometry. This was characterized by the fact that the homogeneous component of degree one of the associated torsion is harmonic. The Weyl connection associated to the contact form 0 is a canonical extension of this partial connection to a connection. There is, however, a second natural extension to such a connection, which is determined by a normalization condition that involves only the torsion. This was introduced (in the setting of integrable CR-structures) in [Tan75] and [We 78] , and is known as the Webster-Tanaka connection. This idea extends to general parabolic contact structures and we have kept this name in the more general setting. To formulate the normalization condition, we need a bit of algebraic background. Consider a contact grading 9 = 9-2 EB··· EB 92. Then 9_ is a Heisenberg algebra, so by Lemma 4.2.1 any automorphism of the graded Lie algebra 9_ is determined by its restriction to 9-1 and the group of all these automorphisms is isomorphic to CSp(2n, 1R), where 2n = dim(9-1). On the Lie algebra level, this means that each derivation of 9_ which is homogeneous of degree zero is determined by its restriction to 9-1, and the algebra of all such derivations is isomorphic to csp(2n, 1R). Now we have the Lie algebra differential 8 : L(9-,9) -> L(A2 9 _, 9), and we may restrict it to maps which are homogeneous of degree zero. Then by definition (see also the proof of Proposition 4.3.1) ker( 8) C L(9-, 9-)0 is the space of derivations of 9_, which are homogeneous of degree zero. On the other hand, 8 : 90 -> L(9-, 9-)0 is injective, so we may view 90 as a subspace of either L(9-,9-)0 or L(9-1.9-1)' From Theorem 3.3.1 we further know that ker(D) is a natural complement to im(8) ~ 90 in ker(8) C L(9-,9-)0. Notice that ker(D) is easily computable explicitly using Kostant's version of the Bott-Borel-Weyl theorem (Theorem 3.3.5) and the algorithms for determining the Hasse diagram from 3.2.18. Now all this has a geometric counterpart. Let M be a manifold endowed with a parabolic contact structure of type (G, P) where G has Lie algebra 9 and P is a parabolic subgroup for the given contact grading. Then we have the corresponding regular infinitesimal flag structure p : go -> M, and passing to associated bundles we have ker(8) = im(8) EB ker(D) C L(gr(TM),gr(TM)), and each map in this space is determined by its restriction in L(H, H).
545
5.2. CHARACTERIZATION OF WEYL STRUCTURES
Now if we extend the unique partial connection compatible with () to a principal connection on go, then we can form the component of homogeneity two in the torsion. This is the restriction of T -1 to a tensor T -1 : Q x H ---+ H, so we can view it as a section of Q* ® L(H, H). Note that the Reeb field r associated to () by definition satisfies rr(r) = 0 and [r,~l E r(H) for all ~ E r(H). Hence, this torsion component is determined by T-1(q(r),~) = V'r~ - [r,~l. PROPOSITION 5.2.12. Let Po : go ---+ M be a regular infinitesimal flag structure of type (G, P), where P corresponds to a contact grading on g. Let He TM be the corresponding underlying contact structure and let () be a contact form for H. Then there exists a unique principal connection on go which is compatible with () and whose torsion T has the following properties: • Its homogeneous component of degree one is harmonic. • Its homogeneous component of degree two is a section of Q* ® ker(D) C Q* ®L(H,H). PROOF. Let V' : r(H) x r(H) ---+ r(H) be the partial connection compatible with () as determined in 5.2.11. Choose any extension of the corresponding partial principal connection on go to a true principal connection, and let V' : X(M) x r(H) ---+ r(H) be the induced linear connection on H. This is a contact connection, and hence induces a linear connection on Q. Now any other extension of this form is given by
(5.16) for some section A of Q* ® Endo(H) , where Endo(H) = go On the other hand, consider
~c:~:= ()(()[r,~l
xGo
go
c L(H, H).
+ V'71"(C:)~'
for ( E X(M) and ~ E r(H). Here r is the Reeb field associated to () and rr(() = (- ()(()r. By definition of the Reeb field, we have [r,~l E r(H), so ~c:~ E r(H). Further, the expression is evidently linear over smooth functions in ( and since ( = ()(()r + rr((), it satisfies a Leibniz rule in (. Hence, ~ defines an extension of V' to a linear connection on H. Expanding the equation 0 = dd()(r,~, 17) for~, 17 E r(H) and using the definition of r, we get
0= r . d()(~, 17) - d()([r, ~l, 17) - d()(~, [r, 17]) = (~rd())(~, 17)· This means that ~ is a contact connection on H, which is compatible with (). Consequently, we get ~c:~ - V'c:~ = B(q(())(~) for some section B of Q* ® csp(H) c Q* ® L(H, H). From Lemma 4.2.1 we know that CSp(g-l) is the automorphism group of the Heisenberg algebra g_, viewed as a graded Lie algebra. Hence, csp(g-d is the Lie algebra of derivations of g_, which are homogeneous of degree zero. This is exactly the kernel of 8 : L(g_, g-)o ---+ L(A2 g_,g_)0; compare with the proof of Proposition 4.3.1. The bundle Endo(H) corresponds to the image of 8 : go ---+ L(g_, g_ )0, which is injective. By Proposition 3.1.11, the subspace ker(D) C L(g_, g-)o is complementary to im(8) within ker(8). Translating this back to geometric terms, we see that ker(D) C csp(H) is complementary to the subspace Endo(H). Hence, we see we can uniquely choose
546
5. DISTINGUISHED CONNECTIONS AND CURVES
A E r(Q* ® Endo(H)) in such a way that the connection the property that
~(~
V defined by
(5.16) has
= V(~ - C(q(())(~)
for a section C of Q* ® ker(O). But this exactly means that Vr~ - [r, ~l -C(q(r))(~), so V satisfies the claimed torsion conditions. Conversely, if a principal connection on go satisfies the two torsion conditions, then the underlying partial connection has harmonic torsion in homogeneity one, so it coincides with the one constructed in 5.2.11. Hence, the induced linear connection on H can be written in the form of (5.16) above. Then the remaining torsion condition is evidently equivalent to the fact that the difference to ~ is a section of Q* ® ker(O), which completes the proof. 0
5.2.13. From Webster-Tanaka connections to Weyl connections. To determine the Weyl connection, we next need a bit of information on the homogeneity two components of the curvature and torsion of a Webster-Tanaka connection. For the torsion, we simply have T-l which is a section of Q*®ker(O) C Q* ®H*®H as determined above. The homogeneity two part of the curvature is the section R of A2H* ® Endo(H) determined by the usual formula. Now depending on the structure in question, this splits into several components, but some components are available for all parabolic contact structures. Namely, the bracket A21h ~ g2 induces { , }: A2 H* ~ Q*, and applying this to R, we obtain a tensor Ricw E r(Q* ® Endo(H)) called the Webster-Ricci curvature. Next, we have to analyze the normality condition in homogeneity 2, and we do this for a general extension of the partial connection determined in 5.2.11. According to 5.2.10, what we need to compute is the restriction of 8*(T(2) + R(2)) to gr -2 (T M), so in our situation we have to evaluate it on the Reeb field r only. Here T(2) represents the part Q ® H ~ H of the torsion, and R(2) represents the part A2 H ~ Endo(H) of the curvature, so we can always see from the entries which quantity is needed. Now we can derive a formula for the relevant component of 8*(T(2) + R(2)) and use the result to compute Weyl connections from WebsterTanaka connections. THEOREM 5.2.13. Consider a parabolic contact structure on a smooth manifold M with contact sub bundle H c T M. (1) For a given Weyl form, let T(2) E r(Q* ® H* ® H) and R(2) E r(A2 H* ® Endo(H)) be the homogeneity two components of torsion and curvature. Then in terms of the maps 8* : H* ® H ~ Endo(H) and { , }: A2 H* ~ Q* the restriction to Q of 8* (T(2) + R(2)) is given by
-(id ® 8*)(T(2)) - ({ , } ® id)(R(2)) : Q ~ Endo(H). (2) Let V' be the Webster-Tanaka connection associated to a contact form () on M, and let Ricw E r(Q* ® Endo(H)) be its Webster-Ricci curvature. Further, let s E IR be the number characterized by ({ , } ® id)(C) = s· idQ, where we view C as a section of A2 H* ® Q. Then the Weyl connection associated to () is given by V' (~+ A(q(())(~), where A E r(Q* ® H* ® H*) is characterized by (id ® (0 + s . id))(A) = Ricw , with o : Endo(H) ~ End o(H) induced by the K ostant Laplacian on go.
547
5.2. CHARACTERIZATION OF WEYL STRUCTURES PROOF. (1) Recall from 3.1.12 that for a decomposable element of AM the map 8* acts by
8* (¢1 1\ ¢2 @ 8) = -¢2 @ {¢I. 8} + ¢1
@
{¢2, 8} - {¢1, ¢2}
@
A2T* M @
8.
Now T(2) can be written as a sum of terms of this type with ¢1 E f(Q*), ¢2 E f(H*) and 8 E r(H), while R(2) can be written as a sum of terms with ¢I. ¢2 E f(H*) and 8 E r(End o(H)). Since we only want to compute the restriction of the result to Q, we only have to look at those terms for which the first factor in the tensor product lies in Q*. For R(2) these are exactly the terms of the form {¢1. ¢2} @ 8. For T(2) these bracket terms always vanish identically, and we only have to consider the terms ¢1 @ {¢2, 8}. However, by definition 8* (¢2 @ 8) = {¢2, 8}, so the result follows. (2) By construction, the Webster-Tanaka connection \7 associated to () extends the partial connection associated to a normal Weyl form. Since the Weyl connection ~ associated to () has the same property, there must be a section A of the bundle Q* @ Endo(H) such that
~(~ = \7(~+A(q((»(~) holds for all ( E X(M) and ~ E f(H). Since \7 and ~ both leave () parallel, A(q(()) must induce the zero map on Q. Let us denote by T(2) and R(2) the relevant torsion and curvature quantities of \7, and by f(2) and k(2) the ones for ~. Then from the definitions we immediately get f(2)(q((),~) = T(2)(q((),~)
+ A(q(()(~),
k(2)(~, TJ)
= R(2)(~, TJ) - A(C(~, TJ). Since A(q((» is a section of Endo(H) c H*@H and induces the zero map on Q, we may actually write the first equation as i q (Of(2) = i q (OT(2) + 8(A(q(())), where 8 is induced by 8 : go - t g"'..l @ g-l. By definition of the Webster-Tanaka connection, T(2) is a section of Q* @ ker(O) so, in particular, (id @ 8*)(T(2)) = O. Hence, from part (1) we conclude that the restriction of 8*(f(2) + k(2) to Q is given by -(id @ O)(A) - 8A + ({
, } @ id)(R(2)) =
-(id @ (0 + 8' id))(A)
+ Ricw .
By construction, both Ricw and A have values which are orthogonal to the grading element E (since they act trivial on Q). In the notation of 5.2.10 this implies that 02 = 0 and hence we know from there that 8* (f(2) + k(2) has to restrict to zero on Q. 0 Note that in each example the condition characterizing the difference between the Webster-Tanaka and the Weyl connection in part (2) of the theorem becomes very simple. The Kostant Laplacian 0 acts by a multiple of the identity on each irreducible component of go, and there are very few such components. 5.2.14. Example: Distinguished connections for Lagrangean contact structures. We next show how the constructions of 5.2.11-5.2.13 above can be made explicit for specific structures. We do this in detail for Lagrangean contact structures, other examples are briefly discussed below. Suppose that H = E EB F c T M is a Lagrangean contact structure on a smooth manifold M of dimension 2n+ 1. As we have noted in 5.2.11 above, the (partial) contact connections on H which
5. DISTINGUISHED CONNECTIONS AND CURVES
548
come from the infinitesimal flag structure are exactly those which respect the two Lagrangean subbundles, or equivalently are of the form \lE EB \IF. Note that by definition, the Reeb field r associated to a contact form () has the property that [r, ~J E r(H) for all ~ E r(H). In particular, the splitting H = E EB F defining the Lagrangean contact structure allows us to further split [r, ~J = [r, ~JE + [r, ~JF' We also need a bit of information about the Webster-Ricci curvature. By definition, this is a section of Q* Q9 Endo(TM). Since the Webster-Tanaka connection leaves () parallel, its curvature has to act trivially on Q = T M / H. Now recall from 4.2.3 that the Lie algebra 9 = sl( n + 2, 1R) has the form
9
~ {(~
¢f)' a,b,~"
E JR;X, WE JR"; Y,Z E JR";a+ b+ h(A)
~ o}
with the evident grading coming from the distance to the main diagonal. The subalgebra of 90 which acts trivially on 9-2 is formed by block diagonal matrices with entries (a, A - ~ll, a) for a E IR and A E s[(n, 1R). For X, Y E 9-11 the action of this element is given by X I-t AX - ~aX and Y I-t -Y A + ~aY. Hence, we see that acting on 9~1 induces an isomorphism between that subalgebra and 9[(9~1)' for which A can be recovered as the tracefree part of an endomorphism of 9~1' while a can be recovered as n-+12 times the trace of the endomorphism. Consequently, we can split the Webster-Ricci curvature into a tracefree part Ric': and the Webster scalar curvature R W E r(Q*), which is defined as the trace of the action of Ricw on the bundle E. Using these observations, we can now describe all the distinguished connections for Lagrangean contact structures. PROPOSITION 5.2.14. Let (M,H = EEBF) be a Lagrangean contact structure, and let () E n1(M) be a contact form. (1) The partial connections \lE and \IF which induce the distinguished partial connection corresponding to the exact Weyl structure determined by () are given by
d()(\l~~, 1]2) = d()([1]1 , ~], 1]2),
d()(\l~ 6,1]) = 6· d()(6, 1])
(5.17)
d()(\l~ 1],6)
d()(\l~ 1]2'~)
+ d()(6, [6,1]]),
= d()([6, 1]], 6),
= 1]1' d()(1]2'~) + d()(1]2, [1]1,W,
for~,6,6 E
r(E) and 1],1]1,1]2 E r(F). (2) The Webster-Tanaka connection associated to () is characterized by \lWT r = o and \lr'T~ = ()()[r,~JE + \l~«)~, (5.18)
\lr'T 1]
= ()()[r, 1]JF + \l~((J1],
for ~ E r(E), 1] E r(F), and ( E X(M), tions from part (1). (3) Let Ricw be the Webster-Ricci tion from (2) and let R W be its Webster associated to () is characterized by \lr = (5.19)
\l (~
where \lE and \IF are the partial conneccurvature of the Webster-Tanaka connecscalar curvature. The the Weyl connection 0 and
= \lr'T~ - 2(n~1)Ricw (q())(~) + n(3~+2) R W (q())~,
\l (1] = \lr'T 1] - 2(n~1) Ricw (q())(1]) - n(3~+2) R W (q())1],
5.2. CHARACTERIZATION OF WEYL STRUCTURES
549
for ~ E r(E), "l E r(F), and ( E X(M), where V'WT is the Webster-Tanaka connection from {2}. PROOF. (1) Observe first that the right-hand sides of all claimed equations contain only known data. FUrther, on the left-hand side we always have inserted the result of a differentiation with values in one of the two subbundles with a general element in the other subbundle. By nondegeneracy of dO, we conclude that the equations always completely determine the values of V'E, respectively, V'F in Edirections, respectively, F -directions. Hence, the four equations together determine the partial connections V' E and V' F, and hence the partial connection V' H . One may also verify directly (although this is formally not necessary), that the right-hand side of each equation has the right behaviour under multiplications of the individual fields by smooth functions in order to guarantee that one really obtains partial connections. For example, consider the right-hand side of the first equation in (5.17). Since "l1, "l2 E r(F), we get dO("l1, "l2) = 0, which implies that the expression is linear over COO(M, 1R) in "l1. On the other hand, for f E COO(M, 1R) we get dO(["l1, f~]' "l2) = fdO(["l1,~], "l2) + ("l1 . f)dO(~, "l2). Hence, we get V'~f~ = fV'~ ~ + ("l1 . fK For the other equations, these properties are verified similarly. The lowest possibly nonzero homogeneous component of the Cartan curvature must be harmonic by Theorem 3.1.12. We have described the harmonic curvature for Lagrangean contact structures in 4.2.3. If dim(M) = 3, then the geometry is automatically torsion free, so there is no harmonic curvature in homogeneity one. If dim( M) > 3, then the harmonic curvature in homogeneity one has two irreducible components, one of which is given by a section of A2 E* ® F while the other is given by a section of A2 F* ® E. (These two torsions are exactly the obstructions against integrability of the Legendrean subbundles E and F.) In particular, this means that T-1(~'''l) = 0 for ~ E r(E) and "l E r(F). Expanding the formula (5.14) for T-1 from 5.2.11 and splitting into components in r(E) and r(F), we see that this is equivalent to
(5.20)
V'[ "l - ll'F([~' "l])
= 0,
-V'~~ -ll'E([~' "l]) = O.
Here we have split 1l'([~, "l]) E r(H) into its components in r(E) and r(F). Now we just have to observe that [~, "ll = dO(~, "l)r
+ ll'E([~' "l]) + ll'F([~' "l])·
By definition, multiples of r insert trivially into dO, and dO also vanishes if both its entries are either from E or F. Hence, we conclude that for "l2 E r(F) we get dO([~,"l],"l2) = dO(ll'E([~,"l]),"l2)' while for 6 E r(E), we get dO([~,"l],6) = dO(ll'F([~,"l]),6). Hence, the first and third line of (5.17) follow directly from (5.20). Next, we know that V'dO = O. Expanding this, we obtain
dO(V'~ 6, "l) =
6 . dO(~2' "l) - dO(6, V'fr "l)
for 6, 6 E r (E) and "l E r (F) . Using this, the second line in (5.17) follows immediately from the third line. Likewise, the last line in (5.17) is deduced from the first line.
550
5. DISTINGUISHED CONNECTIONS AND CURVES
(2) As in the proof of Proposition 5.2.12, one immediately verifies that the two lines in (5.18) define linear connections on E and F, so together they define a linear connection on H, which is induced by a principal connection on 90, It also follows directly from the construction, that the torsion component in homogeneity two maps Q ® E to F and Q ® F to E. Finally, (5.18) is the unique extension of the partial connection determined by V E and VF with that property. Now it is easy to see that ker(D) C L(H, H) is contained in those maps which map E to F and F to E, which completes the proof. (3) We compute the Weyl connection using Theorem 5.2.13. Therefore, we have to compute the number s E lR that occurs in this theorem, as well as the action of the Kostant Laplacian on 90, For the duality between 9_ and p+, we use the trace form of 9. Take the standard bases {e i } of 9~1' {ei} of 9~1' Ai of 9f and Ai of 9f, which are dual with respect to the trace form. Also, the elements ¢ E 9-2 and 'If; E 92 which have their unique nonzero matrix entry equal to one are dual with respect to the trace form. The Lie bracket [ , J : 9-1 x 9-1 ~ 9-2 maps (Xl, Yd X (X2' Y2 ) to (Yl X 2 - Y2X l )¢. Viewed as an element of A29l ® 9-2, this bracket is therefore given by ~i(Ai ® Ai - Ai ® Ai) ® ¢. Applying [ , J ® id to this element, we get -2n'lf; ® ¢, so the number s from Theorem 5.2.13 equals -2n. Since the Kostant Laplacian acts by a scalar on each irreducible component, it preserves the subalgebra of 90 which acts trivially on 9-2, As we have seen above, there are only two irreducible components in this subalgebra, one isomorphic to lR and one to sl(n, lR). We can compute the eigenvalue of D on each component by inserting some fixed element. The matrix corresponding to a = 1 and A = 0, acts on 9~1 as - n~2id and on 9~1 as ~id. Applying 8 simply means viewing this element as an endomorphism of 9-, i.e. as sitting in p+ ® 9-. This is clearly given by - n~2 ~i Ai ® ei + n~2 ~i Ai ® ei. To compute 8*, we have to apply the bracket to these elements, which immediately shows that the eigenvalue of D on this component is -(n + 2). For the other component, one can simply take the matrix for which a = 0 and A is the highest weight vector of sl(n, lR), i.e. the matrix with a 1 in the top right corner and zeros everywhere else. This corresponds to An ® e l - Al ® en, and applying the bracket we see that D acts by multiplication by -2 on that component. According to Theorem 5.2.13, we have to consider the map D + s· id, which acts by - 2( n + 1) on the tracefree part and by -3n - 2 on the trace part. Decomposing the deformation tensor A from Theorem 5.2.13 as Ao + aid when acting on E, we must have Ao = 2(;:~1) Rict and a = n(3~~2) RW. From this, the formula for the Weyl connection on E follows immediately. The formula on F then can be deduced from compatibility of V, V WT and Ricw with de. D
5.2.15. Tractor calculus for Lagrangean contact structures. Having at hand the Weyl connection determined by a contact form e, one can compute its torsion T, curvature R, and Cotton-York tensor Y. As described in 5.2.10, knowing these quantities, one can compute, homogeneity by homogeneity, the Rho tensor associated to the exact Weyl structure determined bye. Compared to the [1[graded cases we have studied before, the Rho tensor now has several independent components. In terms of the Rho tensor, we can then give a complete description of the tractor bundles associated to a Lagrangean contact structure. To formulate the results, we use an abstract index notation, which is particularly useful to compare our results to the tractor calculus for CR-structures
5.2. CHARACTERIZATION OF WEYL STRUCTURES
551
introduced in [GoGr05]. We denote the bundle E by £01. and the bundle F by £0. Further, using G = SL(n + 2,lR) as our basic group, we have the standard representation which gives rise to the standard tractor bundle T. The P-invariant one-dimensional subspace in the standard representation gives rise to a line subbundle T1 C T, which we denote by £ ( -1, 0). The P-invariant (n + 1)-dimensional subspace gives us a subbundle TO c T, and the quotient T /To is a line bundle, which we denote by £(0,1). For k, € E IE we then define £(k, €) via tensor products and duals of these bundles. From the matrix representation it is evident that TO /T1 ~ £01. ( -1,0), and that Q = TM/ H ~ £( -1,0)* ® £(0,1) = £(1, 1). The Levi bracket can then be viewed as an invertible section .cOl.j3 of £01.13(1, 1), and we denote the inverse as .cOl.j3 E r(£OI.j3( -1, -1)). We will use these two sections to raise and lower indices (at the expense of a weight). In these terms, we can now express the components of the Rho tensor. We choose the notation to simplify comparison to [GoGr05]. The homogeneity two part of the Rho tensor is a section of H* ® H*, so it splits into four components, which we denote by AOI.!3, A oj3, P01.13, and P o!3' As usual, we use the convention that the form index comes first, so for example, P represents the component of the Rho tensor which maps F to E*. In homogeneity three, we have a part mapping H to Q* and a part mapping Q to H*. The first part is represented by sections To. E r(£0I.(-1, -1)) and To E f(£o(-I, -1)) while the second part is represented by sections Sol. and So of the same bundles. Finally, the homogeneity four component of the Rho tensor is represented by a single section S E r( £ (- 2, - 2) ) . Having the ingredients at hand, we can now describe the standard tractor bundle and its dual.
013
PROPOSITION 5.2.15. Let (M, H = E Ef) F) be a Lagrangean contact structure, T - M the standard tractor bundle and T* - M its dual. (1) Any choice of Weyl structure (and in particular any choice of contact form) gives rise to isomorphisms
T
~
£(0,1)
T* ~ £(1,0)
£01. ( -1,0)
Ef) Ef)
£0(0, -1)
Ef) Ef)
£( -1,0), £(0, -1).
In terms of this splitting, the dual pairing is given by
((0', J.l0l., p), (T, v O , w))
I-t
O'W + pT + J.lOl.VOl.'
Changing the Weyl structure by T E f(gr(T*M)) with components To. E f(£OI.)' To E r(£o), and T E f(£(-l,-l)) these identifications change as
(O'--:;;;:p) = (0', J.l0l. - aT 01. , p - T 0I.J.l0l. + 0'( ~ YO. yo. - Y)), (T:0;W) = (T, VO + TYo,W + Y oVo + T(Y + ~ Yo. YOI.)). (2) Denoting all Weyl connections by V' and using the splittings from (1), the normal tractor connection on T in directions of H is given by
V'~ (:13) (V'0I.J.l!3+~8F+O'POI.{3), p V' OI.P + AOI.{3J.l!3 + O'TOI. =
V'r
(:13) p
= (
:J.l~0'
V' ++:Ao{3 ). V' oP + P0{3J.l{3 + O'To
552
5.
DISTINGUISHED CONNECTIONS AND CURVES
For a section s E r(Q)
~
'V;
£(1, 1), the covariant derivative in direction s is given by
(:f3)
= (
p
'V
:rtSSf3IJP ++::Sf3 ). + saS
'VsP +
PROOF. We have already observed the form of the composition series for T above. From this the composition series for T* follows immediately taking into account that £0(1, 0) ~ £0:(0, -1) via £0(3. Then part (1) follows from Proposition 5.1.5 and straightforward computations, similar to the ones in 5.1.11. Likewise, part (2) is deduced via straightforward computations from Proposition 5.1.10. 0 REMARK 5.2.16. (1) From the formulae for the tractor connection on T, the ones for T* can be easily deduced by duality. (2) The formulae in the proposition compare nicely to the ones in [GoGr05]. One can see that some of the curvature terms showing up in the formulae there simply express the difference between Webster-Tanaka and Weyl'connections, so the formulae in terms of Weyl connections are simpler. While the tractor calculus in [GoGr05] only works for integrable CR-structures, our results hold for arbitrary Lagrangean contact structures. In particular, the analogous results in the CR case work assuming only partial integrability of the CR-structure. Of course, in the non-integrable case, the expressions for the components of the Rho tensor (which we have not derived explicitly) will become more complicated than the formulae in [GoGr05].
5.2.17. Preferred connections for other parabolic contact structures. For the other parabolic contact structures, the Weyl connections associated to a choice of contact form can be computed similarly as for Lagrangean contact structures. Let us briefly outline the cases of partially integrable almost CR-structures, for which the results are known best, and the one of projective contact structures, which is slightly special but turns out to be easier in the end. In the CR case, there are two basic possibilities. Either one starts by complexifying the tangent bundle. Then the situation becomes very closely parallel to the Lagrangean contact case and one may directly follow the developments there. This leads to formulae for the complex linear and the conjugate linear parts of derivatives which is also the usual approach to Webster-Tanaka connections in CR-geometry. Alternatively, one may proceed as follows. Consider a contact form 0 on a partially integrable almost CR-structure (M, H, J). Then (C 1]) f-7 dO(t" J1]) defines a nondegenerate symmetric bilinear form on H. The partial connection 'V associated to 0 by definition satisfies 'V dO = 0 and 'V J = 0, so it preserves this bilinear form. Hence, one can imitate the procedure for obtaining an explicit formula for the Levi-Civita connection on a Riemannian manifold. First expand the expression 0= 'VdO( ,J) for three sections t" 1], ( E r(H). Then permute the entries cyclically and subtract one of the resulting terms from the other two to obtain (5.21)
o=t.. dO(1], J() -
(. dO(t" J1])
+dO('Ve1] + 'V I]t" J ()
+ 1]' dO((, Jt,)
+ dO('Ve( - 'V Ct"
J'f/)
+ dO('V1]( - 'V c1], Jt,).
Now from the definition of torsion, we know that 'V e( - 'V Ct, can be computed from
11'-1([t,,(]) = [t,,(j- dO(t,,()r
5.2. CHARACTERIZATION OF WEYL STRUCTURES
553
and the Nijenhuis tensor N of the CR-structure. Using this, we can directly rewrite the last two summands in (5.21) in terms of known quantities. Rewriting V'{1)+ V' 1/~ as 2V'~1) - (V'{1) - V'1/~)' we then end up with an explicit formula for 2dB(V'{1), J(), which completely determines the partial connection on H associated to B. Notice that this shows that the partial connection V' on H is determined by V'dB = 0, V' J = and the torsion conditions. The other two usual defining conditions for a Webster-Tanaka connection, V'B = and V'r = 0, simply say that the connection on the tangent bundle preserves H and how this restriction to H is extended to all ofTM. To determine the full Webster-Tanaka connection V'WT associated to B, we have to understand ker(D) C L(H, H). For Lagrangean contact structures, these were maps exchanging the two subbundles, so in the CR case we see directly from the complexified picture that ker(D) C L(H, H) consists of conjugate linear maps. Using V'WT J = 0, the equation that ~ ~ T(r,~) for ~ E X(M) has vanishing complex linear part immediately gives
°
°
V';VT~
=
~([r,~]- J[r, JW,
which together with the partial connection completely determines V'WT. Note that via the conjugate linear isomorphism 9-1 ~ 9~1 ® 9-2, and one can identify ker(D) with S~9~1 ® 9-2. The fact that the torsion is described by an element of S~H* ® Q is one of the usual ways to formulate the torsion condition on the Webster-Tanaka connection, compare with [GoGr05]. Other common versions of stating the condition involve the complex linear part of the connection, boiling down to the fact that no torsion in homogeneity two is visible in this complex linear part.
The Webster-Ricci curvature in our situation has values in the space of skew Hermitian endomorphisms of H. Hence, it splits into a part Rieli' with vanishing complex trace and a purely imaginary multiple of the identity, which defines the Webster scalar curvature RW. Using these curvature quantities, one computes the Weyl connection associated to V' similarly as in the proof of Proposition 5.2.14. Having this at hand, one can explicitly describe the standard tractor bundle for partially integrable almost CR-structures, and so on. As already indicated above, the situation for contact projective structures is slightly special. From the description in 4.2.6 it is clear that in this case the distinguished partial connections are exactly the ones in the projective equivalence class which defines the geometry. Let us verify that in a projective equivalence class, there is exactly one partial connection that is compatible with a given contact form. Recall from 4.2.6 that two partial connections are projectively equivalent if and only if there is a smooth section Y E r(H*) such that (5.22)
V{TJ =
V' {TJ + Y(~)TJ + Y(TJ)~ + y# (£(~, 1))),
for all ~,TJ E r(H). Here y# : TM/H ---- H is characterized by £(Y#({3),() = Y((){3. Now suppose that B is a contact form. For each partial connection V' in the projective class, we can consider V'dB E r(H* ® A2 H*). Since any (partial) contact connection leaves the subbundle A~H* invariant, we see that (V'{dB) (TJ, () = a(~)dB( TJ, £) for some a E r(H*). Replacing V' by a projectively equivalent connection V with the change corresponding to Y E r(H*), we conclude from (5.22) and the definition
554
5. DISTINGUISHED CONNECTIONS AND CURVES
of y# that ('~~de) = (V'~de) -2Y(~)de(7],(). This shows that the projective class contains a unique connection which is compatible with e as claimed. Obtaining the Webster-Tanaka connection and Weyl connection is particularly simple for projective contact structures. The unique irreducible component in HI (9-,9) is contained in homogeneity 1, so there is no harmonic piece in ker( 8) C Lo (9-,9-). Since we also know that projective contact structures have no harmonic torsion in homogeneity one, the Webster-Tanaka connections for projective contact structures must have vanishing torsion in homogeneity one and two. Hence, the Webster-Tanaka connection corresponding to is given by
e
V'rT~
=
e(()[r,~]
+ V'71'()~'
where in the right-hand side we use the unique partial connection in the projective class which is compatible with e. Finally, the Webster-Ricci curvature has values in an irn·ducible bundle, so there is no further splitting into components. Hence, by Propc'::lition 5.2.13 the Weyl connection associated to e is obtained from the Webster-Tanaka connection by adding an appropriate multiple of the WebsterRicci curyature. 5.2.18. Special symplectic connections. To conclude this section, we give a brief O1ltline of another application of Weyl connections for parabolic contact structureti. This comes from the beautiful article [CahS04], in which the authors describe a construction of connections compatible with a symplectic structure. They show that, among other interesting examples, this construction locally produces all connections with special symplectic holonomy, which leads to several striking cowiequences. While the construction in [CahS04] is based on generalized flag manifolds, it does not use Weyl structures. The relation to the latter was first observed ill [PZ08]. A full discussion of the results of [CahS04] would take us too far from tht, lllain line of development, so we only sketch the basic ideas and the relation to \Yeyl structures. The fir-.;r step is to define the class of so-called special symplectic subalgebras. Consider a "'al simple Lie algebra 9 endowed with a contact grading 9 = 9-2EB"'EB 92. In the c;;tssification of complex contact gradings in 3.2.4, we have seen that, for complex 9, one can choose a Cartan subalgebra and positive roots in such a way that 92 is til,' root space for the highest root, 9-2 is the root space for the lowest root, and [fl ~2, 92] C 90 consists of all complex multiples of the grading element E. The clas;-;ification in the real case proceeds via complexification, so we see that 5 := 9-2 EB lR . E EB 92 is a subalgebra of 9, which is isomorphic to 5[(2, ~). On the other hand, the restriction of the Killing form B to 90 is always nondegenerate, and B(E, E) > 0, so we conclude that I) := E.L C 90 forms a complementary subspace to ~ . E. Invariance of the Killing form and the fact that E lies in the center of 90 immediately imply that I) is a subalgebra of 90, which is automatically reductive. It also follows easily that [1),5] = {O}, so 9-2 EB 90 EB 92 ~ 5 EB I) as a Lie algebra. By the grading property, the subspace 9-1 EB 91 C 9, which evidently is complementary to 5 EB I), is invariant under the adjoint action of 5 EB I). We can determine 9-1 EB91 as a representation of 5EBI) from the fact that the only eigenvalues of E E 5 are ±l. It must be of the form ~2 I:8J V, i.e. an exterior tensor product of the standard representation of 5 ~ 5[(2,~) and some representation V of I). Fixing
5.2. CHARACTERIZATION OF WEYL STRUCTURES
555
an appropriate nonzero vector in JR2 leads to an identification of V with 91. Since I) acts trivially on 92, the bracket [ , ] : 91 x 91 ~ 92 gives rise to an I)-invariant symplectic form on V, so I) C sp(V). The subalgebras of the symplectic Lie algebras obtained in this way are called special symplectic Lie subalgebras in [CahS04], and the corresponding groups are called special symplectic Lie subgroups. The complete list of these, which is given in Table 1 of [CahS04], can be easily obtained from the classification (If contact gradings in 3.2.4 and 3.2.10. There also is an "abstract" definition of sIwcial symplectic Lie subalgebras. They can be characterized as sub algebras I) C sp(V) which are endowed with an I)-invariant nondegenerate bilinear form and a bilillf'ar map o : S2(V) ~ I), which satisfy certain compatibility conditions. If I) i~ , btained from a contact grading as above, these are obtained as the restriction of t]w Killing form, respectively, from the I)-component of the bracket [ , ] : 9-1 16' gl ~ 90. Conversely, given the abstract data, one uses them to construct a Lie bracket on JR2 ® V E9 (sl(2, JR) E9 I)) and proves that the result is simple and inherits a contact grading. Given a special symplectic Lie algebra I) C sp(V), the authors define (using the operation 0 from above) an I)-equivariant injection I) '---> A2 V* ® I). They show that the image R~ of this map is contained in the space of formal curvature:.;. i.e. its elements satisfy the first Bianchi identity. Moreover, the contraction defining Ricci curvature restricts to an injection on R~. In most cases, the subspace RI] exhausts all possible formal curvatures corresponding to I). Now let (M, r) be a symplectic manifold; see 4.2.1. A symplectic connection on M is a linear connection V' on the tangent bundle TM such that V'r = O. This implies that the holonomy of V' (see 5.2.8) is contained in the symplectic- group Sp(2n, JR) where 2n = dim(M). A symplectic connection V' is called special if it is torsion free and its curvature is contained in R~ for some special symplf'.) c Go, where>. : Go - 7 IR+ is the representation inducing the bundle of scales. Since our bundle of scales corresponds to (a multiple of) the grading element, we can identify ker(>.) with the special symplectic subgroup H. PROPOSITION 5.2.19. The image of the reduction to the structure group H of the principal bundle p-1 (Ca) -7 Ca coming from the exact Weyl structure described above is given by
ra = {g E G: Ad(g-l)(a) - 4J E p'} c G, := ~ EB gl EB g2 C I' with ~ c go denoting the special symplectic subalgebra
where 1" coming from the contact grading. Moreover, this Weyl structure is invariant under the one-parameter group of automorphisms induced by a. PROOF. First note that for 9 E ra by construction we get 'I/J(Ad(g-l)(a)) = 1/J(4J) = 1, which shows that ra c p-1(Ca). Second, for an element h E H the adjoint action Ad(h) preserves 1" and is trivial on g-2, which immediately implies that r a is invariant under right translations by elements of H. We claim that pJr a : r a -7 Ca is surjective with the fibers being the H -orbits. For 9 E p-1(Ca) and b E P, we have w(Ra(gb)) = Ad(b- 1 ) (w(Ra(g))). First, we take b = exp(tE), where E is the grading element. This acts by multiplication by e- 2t on g-2, so we can uniquely choose t in such a way that 'I/J(Ad(b- 1)(w(Ra(g)))) = 1.
5.2. CHARACTERIZATION OF WEYL STRUCTURES
557
Assuming that w(Ra(g)) already has this property, we can take b = exp(Z) for Z E 91. Then acting by Ad( b- 1 ) on w( Ra (g)) leaves the 9 _rcomponent unchanged, and since that component is nonzero, we can uniquely choose Z in such a way that Ad(b- 1)(w(Ra(g))) has vanishing 9-1--component. Assuming again that w(Ra(g)) already has both properties, we can finally consider b = exp(t~). Since [¢,~] is a nonzero multiple of the grading element E, we can uniquely choose t in such a way that Ad(b-1)(w(Ra(g))) Era' This also shows that 9 Era, bE P and gb E ra implies b E H, thus completing the proof of the claim. Finally, we can start with a local smooth section of p-1(Ca) ~ Ca and apply the above construction depending smoothly on the base point, to obtain smooth local sections of pir a : r a ~ Ca. Hence, r a ~ Ca is a principal fiber bundle with structure group H and the inclusion r a '--+ p- 1 (C a ) is a reduction of structure group to H c P. Now let us denote by 7r : G ~ GIP+ and by Po : GIP+ ~ GIP the natural 1(Ca ) to the projections. Then we see that 7r restricts to a reduction j : r a ~ structure group H c Go. Now it is easy to see that any element 9 E Go can be written as hg' with h E Hand g' in the center of Go. This immediately shows that the unique map a : G I P+ ~ G characterized by a(j (g)) = 9 for all 9 Era is Go-equivariant and hence defines a Weyl structure for the parabolic geometry (p-1(C a) ~ Ca,w). By construction of r a, the equivariant function p-1(Ca) ~ 9_ corresponding to the vector field a* E X(Ca ) restricts to the constant function ¢ on r a' But this exactly means that the vector field a* is parallel for the Weyl connection corresponding to a, so the induced section of gr -2 (T M) must be parallel, too. But the latter condition pins down the Weyl structure uniquely. It remains to prove that the Weyl structure a is invariant under the oneparameter subgroup of automorphisms induced by a. For 9 E G and t E JR, we get
Po
w(Ra(exp(ta)g)) = Ad(g-l) 0 Ad(exp(-ta)) (a) = Ad(g-l)(a). This shows that both p-1 (Ca) and r a are invariant under left translations by elements of the one-parameter subgroup of G generated by a. As we observed above, any element of P01(Ca ) can be written as j(g)g' with 9 E Ca and g' in the center of Go. But then exp(ta) . j(g)g' = j(exp(ta)g)g', and hence a(exp(ta) . j(g)g') = exp(ta)gg' = exp(ta)a(j(g)g'). 0 To prove their first main result, the authors of [CahS04] directly define the set G exhibited above and consider the restriction of the components in 9_ and 90 of the Maurer-Cartan form of G to this subset, so these are exactly the soldering form and the Weyl connection of the Weyl structure determined by a. Then they consider open subsets of Ca , which are small enough to form local leaf spaces for the foliation determined by the nowhere-vanishing vector field a*. Of course, such a leaf space is just the quotient by the one-parameter group of automorphisms induced by a*, so from the last part of the proposition it follows that all data associated to our Weyl structure descend to the quotient. In particular, the contact structure descends to a symplectic structure on the leaf space, and r a descends to a reductions of the symplectic frame bundle to the structure group H. The authors then show that the Weyl connection descends to a special symplectic connection on this leaf space. The results in [CahS04] actually go much further than that. Namely, it is proved that locally the connections constructed from parabolic contact structures
rae
558
5. DISTINGUISHED CONNECTIONS AND CURVES
as above exhaust all possible special symplectic connections. This has strong consequences like automatic analyticity, finite dimensionality of the moduli space of symplectic connections, and the fact that special symplectic connections always admit nontrivial infinitesimal automorphisms. Finally, the authors use these results to obtain a complete classification of special symplectic connections on simply connected compact manifolds.
5.3. Canonical curves In this section, we specialize the concept of canonical curves for Cartan geometries as discussed in 1.5.18 to the case of parabolic geometries. We will restrict our attention to the simplest classes of curves, namely those coming from exponential curves in the homogeneous model, which still leads to a very rich and diverse theory. We will mainly address the question how many distinguished curves emanate from a given point in a given direction. Otherwise put, we study how many derivatives in a point are needed to pin down a distinguished curve uniquely or uniquely up to reparametrization. As we shall see, this heavily depends on the direction in question, so understanding the possible types of geometrically different directions will be an important task, too. Via the concept of development from 1.5.17 these local questions need to be studied for the homogeneous model only, and we will reduce them to purely Lie algebraic considerations. The basic source for this section is
[CSZ03]. 5.3.1. The basic setup. Let us fix some type (G, P) of parabolic geometry. Given a geometry (p : [i ~ M, w) of this type, Cartan's space SM is defined as [i x p (G I P), the associated bundle with fiber the homogeneous model of the geometry. In 1.5.17 we have shown that the Cartan connection w induces a connection on the fiber bundle SM ~ M. Using this connection, we defined the development of curves. To a point x EM, a curve c : I ~ M defined on some open interval with o E I and c(O) = x, and a frame U E [ix, this associates a curve deve : I' ~ GIP defined on a subinterval l' c I with deve(O) = 0 = eP E GIP. By Theorem 1.5.17, the map c ~ dev e induces a bijection between the space of germs in the point x of smooth curves in M and the space of germs in 0 of smooth curves in G I P. For each r E N, this relation is compatible with the notion of rth order contact, so two curves have the same r-jet in x if and only if their developments have the same r-jet in o. The definition of canonical curves in 1.5.18 was based on the notion of development. The starting point was a family C of curves in GIP, which is admissible in the sense that for "( E C, to in the domain of ,,(, and g E G such that "((to) = g-l. 0, also t ~ g. "((t + to) lies in C. Consider the bundle SM = [i Xp (GI P) and let us denote the natural projection [i x (GIP) ~ SM by (u,y) ~ [u,y]. Given M and x as above, canonical curves of type C through x are then defined to be the curves whose development in x can be expressed in the form t ~ [u, "((t)ll for some u E [ix and some "( E C. Changing the element u E [ix amounts to replacing "((t) by g,,,((t) for some g E P, and the latter curve lies in C by admissibility. The definition of canonical curves and the properties of the development map hence imply that understanding local properties of the family of canonical curves of type C through x in M is equivalent to understanding local properties of the family C of curves through 0 in G I P.
5.3. CANONICAL CURVES
559
We will mainly look at the case that the initial family C is a family of exponential curves. Let g = g_ EEl P = g_ EEl go EEl p+ be the decomposition of the Lie algebra g of G from the grading corresponding to the parabolic subgroup PeG. Take a G oinvariant subset Ao c g_, and define A := {Ad(exp(Z))(X) : X E Ao, Z E p+} c g. By Theorem 3.1.3, any element of P can be written as exp(Z)gO for some go EGo and Z E 1'+, so the subset A egis P-invariant. Denoting by p : G ---'> G I P the canonical projection, it follows immediately that CA := {t t---> p(exp(tX)) : X E A} is an admissible family of curves through 0 in G I P; see 1.5.18. The tangent vectors of curves from CA in 0 are described by the P-invariant subset A + pc gil'. Note that under the identification of gil' with g_, this subset may be much larger than the initial subset Ao. For a parabolic geometry (p: g ---'> M,w) and a point x E M, the set of possible tangent vectors of canonical curves of type CA is the subset of TxM determined by A + p via T M 9! g x p (g/p). The families of canonical curves induced by families of exponential curves also have the advantage that there is a general description of canonical curves of type CA which does not need the development and depends only on Ao. PROPOSITION
5.3.1. Let Ao C g_ be a Go-invariant subset and put A:= {Ad(exp(Z))(X): X E Ao,Z E p+} C g.
Consider a parabolic geometry (p: g ---'> M,w) of type (G, P). Then a curve c: I ---'> M is canonical of type CA if and only if it locally coincides up to a constant shift of parameter with the projection of a flow line in g of a constant vector field of the form w- 1 (X) with X E Ao. PROOF. In Corollary 1.5.18 we have seen in the realm of general Cartan geometries that c is a canonical curve of type CA if and only if it locally coincides up to a constant shift of parameter with the projection of a flow line in g of a vector field of the form w- 1 (y) for some YEA. By assumption, Y = Ad(exp(Z)) (X) for some Z E p+ and some X E Ao. But then for each u E g we get
w-l(y)(u)
= Trexp(-Z)
. w-l(X)(u· exp(Z)),
so w- 1 (y) and w- 1 (X) are (rexP(Z))-related. Hence, their flows are (rexp(Z))_ related, so for each flow line of w- 1 (y), we can find a flow line of w- 1 (X) which 0 has the same projection to M and vice versa. From this result, we immediately obtain a relation to normal Weyl structures as introduced in 5.1.12. COROLLARY 5.3.1. In the setup of the previous proposition, a curve c with c(O) = x E M is canonical of type CA if and only if there is a Weyl structure a, which is normal in x, and an element X E Ao with c'(O) = [a(x), X~, such that c is a geodesic for the Weyl connection corresponding to a. EXAMPLE 5.3.1. Let us illustrate the above description of canonical curves on the best known examples of homogeneous models. (1) Let us first consider conformal structures in Riemannian signature. Here the homogeneous model G I P is the Mobious sphere; see 1.6.2 and 1.6.3. Since Go = CO(n) and gil' 9! IR n as a Go-module, there is just one nontrivial G oinvariant subset A o, namely IR n \ {O}, and we get only one type of canonical curves. We can immediately compute their form explicitly using the formulae from 1.6.3. There we have used the flat coordinates coming from the exponential mapping
560
5. DISTINGUISHED CONNECTIONS AND CURVES
(which equal the coordinates from the very flat Weyl structure discussed in Example 5.1.12). We have computed there the action of Ad(exp(Z)) in these coordinates. For X E g-l and Z E gl, the element Ad(exp(Z))(X)P E GIP is given by (5.23)
X
+ ~(X,X)zt
To get the expression for the canonical curve, we simply have to replace X by tX in this formula. Of course, for Z = 0 we simply get the line through X with its natural parametrization, and if zt is a multiple of X, then we get a reparametrization of this line. In general, the curve always lies in the plane spanned by X and zt. We saw in 1.6.3 that the above mapping is actually obtained from an inversion on the unit circle, followed by translations by ~ zt and another inversion on the unit circle. This implies that our curve is a circle whose midpoint is given by a certain multiple of the projection of zt onto the line perpendicular to X. In particular, we see that we can obtain any circle through zero and tangent to the line spanned by X in this way. It was this description that led to the names conformal circles and generalized geodesics given to our canonical curves in the older literature on conformal Riemannian manifolds. Notice that we need the two-jet in 0 to fix the canonical curve, even if we are not interested in its parametrization. (2) For conformal structures in indefinite signature, the situation changes drastically. Namely, we obtain three different nontrivial Go-invariant subsets of vectors, corresponding to (X, X) being positive, negative or zero. The formula from (1) remains correct in general signatures (if we interpret the transpose as being taken with respect to the indefinite inner product), and for (X,X) =1= 0 the behavior is similar as above. However, for (X, X) = 0 the formula shows that for any choice of Z, we only get a reparametrization of the line spanned by X. Thus, we see that the canonical curves in null directions are determined by their first jets as unparametrized curves. This corresponds to the well-known fact that the null-geodesics of pseudo-Riemannian metrics are conformally invariant as unparametrized curves. (3) Let us finally look at projective structures, for which the homogeneous model G I P is the projective space JR.pn. As in the first example, there are no distinguished directions here and the available transformations in the very flat coordinates are just the projective transformations fixing the origin. Thus, we end up in a situation similar to the null directions in (2), and the canonical curves for projective geometries are just the geodesics of the connections in the projective class, with their distinguished projective parametrizations.
5.3.2. Reduction to algebra and a fundamental estimate. For Y E g let us denote by cY : JR. --t G I P the curve t ~ p(exp( tY)), so the canonical curves of type CA through 0 E GI P are the curves cY for YEA. Recall from 1.2.4 that for G-valued functions one has the left logarithmic derivative 8 f. Here we will be interested in curves c : JR. -; G. Via the usual trivialization of the tangent bundle of JR., we can consider the left logarithmic derivative of c simply as a smooth function 8c : JR. -; g. Explicitly, this is given by &(t) = Tc(t)Ac(t)-' . c'(t). Then we can form the iterated derivatives (8c)(i) E COO(JR., g). LEMMA 5.3.2. Let G be a Lie group with Lie algebra g, PeG a closed subgroup with Lie algebra p, and let p : G -; GI P be the canonical projection. Let CI, C2 :
561
5.3. CANONICAL CURVES
-> G be smooth curves with Cl (0) = C2 (0) = e, and let u : lR -> G be the curve defined by u(t) = C2(t)-lcl(t). (1) For each r E N with r ~ 1, the following conditions are equivalent: (i) Cl and C2 have the same r-jet in O. (ii) The curve u has the same r-jet in 0 as the constant curve e. (iii) The left logarithmic derivatives c5Cl and c5C2 have the same (r - I)-jet in
lR
O. (2) For each r E N with r ~ 1, the curves P 0 Cl and p 0 C2 have the same r-jet in 0 if and only if (c5u) (i) (0) E P for all i = 0, ... , r - 1. PROOF. (1) We can use exp : 9 -> G as a local parametrization for G. Hence, we write Ci(t) = exp(¢i(t)) for i = 1,2 and smooth g-valued functions ¢l, ¢2 defined locally around zero such that ¢l (0) = ¢2 (0) = O. (i) ==} (iii) By definition, (i) is equivalent to ¢l and ¢2 having the same r-jet in O. We have noted in the proof of Proposition 5.1.8 already that the left logarithmic ad(X)j. derivative of the exponential mapping is given by c5(exp) (X) = I:~o Clearly, this implies that
tj;i;,
00
(5.24)
c5(expo¢)(t) = c5(exp)(¢(t))· ¢'(t) =
L
(~;l;, ad(¢(t))j(¢'(t)).
j=O
This shows that c5(Ci(t)) is given by a universal formula in terms of ¢i(t) and ¢~(t), which shows that (i) implies (iii). (iii) ==} (ii) Observe first that equation (5.24) shows that c5(expo¢)(t) is the sum of ¢'(t) and a sum of iterated Lie brackets involving several copies of ¢(t) and one copy of ¢'(t). Inductively, this shows that c5(expo¢)(s) is given by ¢(s+1)(t) plus a sum of iterated Lie brackets with entries equal to ¢(£)(t) for 0 ~ e ~ s + 1 and the total number of derivatives in each term is s + 1. But this shows that if c5 (exp o¢) has vanishing (s - I)-jet in 0, then ¢ has vanishing s-jet in 0 and the s-jet of exp o¢ coincides with the s-jet of the constant curve e. Now the compatibility of the left logarithmic derivative with pointwise products and inverses observed in the proof of Theorem 1.2.4 implies that c5u(t) = c5Cl (t) Ad(u(t)-l )c5C2(t). By assumption, u(O) = e so we obtain c5u(O) = c5Cl(O) - &2(0) = 0, so the one-jet of u in 0 coincides with the one-jet of the constant map e. Hence, the one-jet in 0 of c5u(t) coincides with the one-jet of c5Cl (t) - c5C2(t). Inductively, we conclude that (iii) implies that c5u has vanishing (r -I)-jet in 0, and from above we see that this implies (ii). (ii) ==} (i) is obvious since Cl(t) = C2(t)U(t) and the right-hand side has the same r-jet in zero as c2(t)e = C2(t). (2) Let a be a smooth section of p: G -> G/ P defined on an open neighborhood U of 0 = eP E G / P. Then (x, b) t--+ a( x)b defines a local diffeomorphism U x P -> p-l(U), so locally around zero, we may write our curves uniquely as Ci(t) = a(p(ci(t)))· bi(t) for smooth P-valued curves bl , and b2 defined locally around zero. This means that, locally around zero, we get
Now if p 0 Cl and po C2 have the same r-jet in 0, then the same is true for a 0 po Cl and aopoC2. By part (1), this implies that a(p(c2(t)))-la(p(cl(t))) has the same
562
5.
DISTINGUISHED CONNECTIONS AND CURVES
r-jet in 0 as the constant curve e. Thus, the r-jet of u in 0 coincides with the r-jet of b2(t)-lb 1(t), which implies that (ou)(i)(O) lies in p for i = 0, ... ,r-1. Conversely, assume that (ou)(i)(O) lies in p for i = 0, ... , r - 1. Solving an ODE, we can find a smooth curve b in P (defined locally around zero) such that (ob)(i)(O) = (ou)(i)(O) for i = 0, ... , r - 1. This easily implies that also for the curves t I-t U(t)-l and t I-t b(t)-l, the left logarithmic derivatives have the same (r - l)-jet in O. By part (1), the curve t I-t u(t)b(t)-l has the same r-jet in 0 as the constant curve e. But by construction, C2(t) = Cl(t)U(t), so
P(C2(t)) = P(Cl(t)U(t)) = p(cl(t)u(t)b(t)-l),
o
and the latter curve has the same r-jet in 0 as P(Cl(t)).
Using this, we can now formulate an efficient algebraic criterion to check how many derivatives in a point are needed to pin down a canonical curve uniquely. DEFINITION 5.3.2. Let G be a semisimple Lie group, PeG a parabolic subgroup, 9 = 9- EB 90 EB p+ the corresponding grading of the Lie algebra of G, Go c P the Levi subgroup, and put P+ := exp(p+) c P. For r E N we say that a Go-invariant subset AD c 9_ is r-determined if and only if for all Xl. X 2 E AD and all g E P+ the condition that ad(Xd(Xl - Ad(g)(X2)) E P for all 0 ~ e ~ r implies that ad(Xd(Xl - Ad(g)(X2)) E P for all eE N. THEOREM 5.3.2. Let (G, P) be a type of parabolic geometry corresponding to 9 = 9-EB90EBP+, let AD C 9_ be a Go-invariant subset, and put A:= Ad(P)(Ao) C 9 as before. If for some r E N, the subset AD is r-determined, then any canonical curve of type CA in a parabolic geometry of type (G, P), which is defined on a connected open interval, is uniquely determined by its (r + 1) -jet in a single point. PROOF. Consider two elements Y1 , Y2 E A and the associated curves Ci(t) = exp(tYi) in G for i = 1,2, and put u(t) = C2(t)-lCl(t). From formula (5.24) in the proof of the lemma, we see that 00
OCi(t) =
L
(j-':l~! ad(tYi)P(Yi)
= Yi,
j=O
and as observed in that proof, this implies that ou(t) = Y1 E lR.. Now we claim that
-
Ad(u(t)-1)Y2 for all
t
(5.25) To prove this, we have to compute the derivative of t I-t Ad(u(t)-l), which clearly is given by TU(t)-l Ad .Tu(t)v, u'(t), where v denotes the inversion mapping. Now Ad' = Te Ad = ad and from AdoAg = Ad(g) 0 Ad we get Tg Ad = Ad(g) 0 adoTgAg-l. Likewise, Tev = -id and VOA g = pg-l ov imply that Tgv = _Tepg-l 0 TgAg-l. Since u'(t) = TeAu(t) . ou(t) and Tu(t)-lAu(t) 0 Tepu(t)-l = Ad(u(t)), we finally end up with
1t Ad(u(t)-l) =
(Ad(u(t)-l)
0
= -(Ad(u(t)-l)
adoTu(t)-lAu(t) 0
0
(_TePU(t)-l)) (ou(t))
ado Ad(u(t)))(ou(t))
and hence (5.26)
(ou)'(t) = Ad(u(t)-l)[Ad(u(t))(ou(t)), Y2] = - [Ad(u(t)-1)(Y2)' ou(t)].
5.3. CANONICAL CURVES
563
Since Ad(u(t)-1)(y2) = Yl - 8u(t), this proves (5.25) for r = 1. For general r, (5.25) then immediately follows by induction. Now suppose that the curves po Cl and po C2 have the same (r + I)-jet in O. By the lemma and our claim, this implies
(8u)(£)(0)
= (-1)£ ad(yd(Yl -
Y2 ) E P for .e
= 0, ... ,r.
Since Yi E A and Ao is Go-invariant, there are elements Xi E Ao and gi E P + such that Yi = Ad(gi)(Xi ) for i = 1,2. Putting 9 = g'1 l g2 E P+, we get ad(Yd(Yl - Y2 ) = Ad(gl)(ad(Xll(X l - Ad(g)(X2))) for all .e E N. Since gl E P+, we see that ad(Xd(Xl - Ad(g)(X2)) E P for all 0 S .e r. If Ao is r-determined, then this holds for all .e E N and hence ad(Yl)£(Yl - Y2 ) E P for all .e E N. Again by the lemma, po Cl and po C2 have the same infinite jet in 0, and since both curves are analytic by construction, they coincide locally. Together with the observations on developments from 5.3.1, this shows that two canonical curves of type CAin any manifold with a parabolic geometry of type (G, P), which are defined on an open interval I c lR. have the same germ in a point tEl if and only ifthey have the same (r+ I)-jet in t. Now the subset of I, in which two smooth curves have the same germ, is evidently open. But also the subset, on which two curves have different (r + I)-jet is evidently open. If I is connected, this implies that the two canonical curves coincide on I if they have the same (r + 1)-jet and hence the same germ in one point to E I. 0
s
Using this, we get a simple first fundamental estimate for the jet needed to pin down a canonical curve. COROLLARY 5.3.2. Let (G, P) be a type of parabolic geometry corresponding to a Ikl-grading of the Lie algebra 9 ofG. Suppose that Ao C (9-kE£)·· 'E£)9-j) for some j = 1, ... , k is Go-invariant, and put A = Ad(P)(A o ). If r is such that rj > k, then any canonical curve of type CA in a parabolic geometry of type (G, P), which is defined on a connected open interval, is uniquely determined by its (r + I)-jet in a single point. PROOF. It suffices to show that Ao is r-determined, so suppose that X l ,X2 E Ao and 9 E P+. By assumption C := Xl - Ad(g)(X2) E P and Xl E 9-k EB ... EB 9-j· Hence, by the grading property, ad(Xd(C) E 9-k EB ... EB 9k-£j' In particular, ad(Xd'(C) E 9_, so if this also lies in p, then ad(XIY(C) = 0 and hence ad(Xd(C) = 0 for all .e > r. 0
Example 5.3.1 shows that this bound is not sharp in generaL There we always had j = k = 1, so the corollary shows that any canonical curve is determined by its three-jet in a point. In Example 5.3.1, two-jets were always sufficient, and we will give a general proof of this fact below. 5.3.3. Improvements of the fundamental estimate. The bound on the number of derivatives needed to pin down a canonical curve proved in Corollary 5.3.2 only used the grading of 9 and no finer information on the algebraic structure. Going deeper into this algebraic structure, we can improve this result for subsets contained in one grading component. Example 5.3.1 shows that the resulting estimates are sharp in some situations.
564
5.
DISTINGUISHED CONNECTIONS AND CURVES
THEOREM 5.3.3. Let (G, P) be a type of parabolic geometry corresponding to a Ikl-grading on g. For some j = 1, ... , k let Ao c g_j be a Go-invariant subset and put A = Ad(P)(Ao). Then any canonical curve of type CAin a parabolic geometry of type (G, P) defined on a connected interval is uniquely determined by its r-jet in a single point provided that rj > k. We shall need the following lemma in the quite technical proof of our theorem. LEMMA 5.3.3. Let X and Z be elements of an arbitrary Lie algebra such that for some n > 0 we have ad(X)n+1(Z) = O. Then for each > n, there is a linear map ¢ such that ad(X)l 0 ad(Z) = ¢ 0 ad(X)l-n.
e
PROOF. The Jacobi identity says that ad(X)oad(Z) = ad(ad(X)(Z))+ad(Z)o ad(X). Inductively, this implies that for each > 0, the map ad(X)l 0 ad(Z) can be written as a linear combination of the maps ad(ad(X)i(Z)) 0 ad(X)l-i for i = 0, ... , e. In particular, if ad(X)n+1(Z) = 0 and e > n, then each of these summands can be written as the composition of some map with ad(X)l-n. 0
e
PROOF OF THE THEOREM. In view of Theorem 5.3.2, we only have to show that if rj > k, then Ao is (r - I)-determined. According to Theorem 3.1.3, any element g E P+ can be written as 9 = exp(Zl) ... exp(Zk) for some Zi E gi. Hence, for X E Q, we get (5.27)
Ad(g) (X) - X
=
L
il!.~.ik! ad(Zd i1
0'"
0
ad(Zk)ik(X),
where at least one of the it is nonzero. By the grading property, if X E g_ j, then a nonzero contribution can only come from summands for which i1 + 2i2 + ... + kik ::::; k+j.
Now consider Xl>X 2 E Ao C Q_j, and compute Ad(g)(X2) according to (5.27). Of course, the g_j component of this is X 2, so Xl - Ad(g)(X2) E P implies X 2 = Xl =: X. Further, the g_j+1-component of Ad(g)(X) equals [Zl>X], so if j > 1, then X -Ad(g)(X) E P implies Zl = O. In the same way, we get Z2 = ... = Zj-1 = 0, so the sum in (5.27) is actually only over ij,"" ik' Now for = 1, ... , k we write Wi for the sum of all terms in (5.27), for which il+1 = ... = ik = O. Hence, we know that if j > 1, then W{ = ... = Wj_1 = 0, while W£ = Ad(g)(X) - X. Now if rj > k, then the following claim in the case n = r - 1 implies that Ao is (r -I)-determined. (Notice that for n = r -1, the claim covers the case m = k by assumption. ) Claim: If ad(X)i(X - Ad(g) (X)) E P for i = 1, ... , n then for each m < (n + I)j, we have ad(X)n+1(Zm) = 0 and ad(X)l(W~) E p for each e> n. We prove this by induction on n. First, the component of [X, X -Ad(g)(X)] in Q_ evidently is [X, [Zj, X]]+·· +[X, [Z2j-1, X]]. Since the summands lie in different grading components, ad(X)(X - Ad(g)(X)) E P implies that ad(X)2(Zm) = 0 for m < 2j. We already know that Zm = 0 for m < j, and hence W~ = 0 for m < j, so Wj = I:i ad(Zj)i(X). Now if e > 1, then
e
it
ad(X)l(Wj) =
Lit ad(X)l ad(Zj)i(X). i
Summands with i > e automatically lie in p. From the lemma we know that ad(X)l 0 ad(Zj) can be written as the composition of some map with ad(X)t-1.
5.3. CANONICAL CURVES
565
Inductively, this shows that for i :::; f, we get
for some linear map '1/;, and this vanishes since f - i + 2 2:: 2. Still for n = 1, assume inductively, that for some j < m < 2j we have proved that ad(X)i(W:n_l) E p for all f 2:: 2. To prove that ad(X)i(W:n) E p it suffices to prove the same statement for W:n - W:n-l' which by definition can be written as
I:
ij! ..\m! ad(Zj)ij
0···0
ad(Zm)im(X),
ij, ... ,i m
with im > O. Applying ad(X)£, we only have to consider terms for which jij + ... + mim < (f + l)j, whence ij + ... + im :::; f. Now as above, we can write ad(X)i
0
ad(Zj)ij
0··· 0
ad(Zm)im = 'I/; 0 ad(X)£-ij_ ... -i",+1
0
ad(Zm)
for some linear map '1/;. Applying this to X, we obtain ad(X)i-ij_ ... -i",+2(Zm) = O. This completes the proof of the claim for n = l. Now assume that n > 1 and the claim has been proved for all s < n. This means that for s = 1, ... , n - 1 and m < (s + l)j we have ad(X)S+l(Zm) = 0 and ad(X)£(W:n) = 0 for all f > s. In particular, we know that ad(X)n(W~j_l) E p. Hence, ad(x)n(x -Ad(g)(X)) E P implies that ad(x)n(x -Ad(g)(X) - W~j_l) E p. Now modulo gn j (which is automatically mapped to p by ad(X)n), the element X - Ad(g)(X) - W~_l is congruent to [Znj,X]
+ ... + [Z(n+l)j-l,X],
Since the individual summands lie in different grading components, we conclude that ad(X)n+l(Zm) = 0 for m < (n + l)j. To prove that ad(X)i(W~j) E p for all f> n, it suffices to prove the same fact for W~j - W~j_l' But this can be written as "" - ._1_ ad(Z.)i j 0'" 0 ad(Z .)inj(X) ~ tj!··.tnj! J nJ' ij " .. ,inj
with the sum over inj > 0 and jij + (j + 1)ij+1 + ... + njinj :::; k + j. Summands for whichjij +(j+1)ij+l + .. '+njinj > fj are automatically mapped to p by ad(X)i, so we may ignore them. Now let a(m) EN be the unique number such that a(m)j :::; m < (a(m) + l)j. Then we conclude that a(j)ij +a(j + l)ij+1 + .. ·+a(nj)inj :::; f. Applying the lemma inductively, we conclude that ad(X/o ad(Zj )i j = 'I/;
0
0 ••• 0
ad (Znj )inj
ad(X)£-a(j)ij- ... -a(nj-l)inj-n(inj-l)
0
ad(Znj)'
for some linear map '1/;. Applying this to X, we get zero as before, since f - a(j)ij - ... - a(nj - l)i nj - n(i nj - 1)
+ 1 > n.
Having proved this, one concludes inductively in the same way that ad(X)i(W:n) = > nand m < (n + l)j. 0
o for f
566
5. DISTINGUISHED CONNECTIONS AND CURVES
5.3.4. Parametrizations of homogeneous curves. As we saw already in Example 5.3.1, there may be canonical curves which only differ by their parametrizations. Correspondingly, any canonical curve comes with a distinguished class of parametrizations. These can be studied by a direct computation in the spirit of 5.3.2, as worked out in [CSZ03]. This leads to the conclusion, that the possibilities for reparametrizations are very limited. Since the reason for that is much more general (and thus simpler), we take a small detour at this point, following [EaS104] and [Do05]. Let us for a while consider an arbitrary homogeneous space, i.e. a manifold M with smooth transitive left action>. : G x M -+ M of a Lie group G. Then there is the infinitesimal action of the Lie algebra g of G, given by the derivative>.' : g -+ X(M), >.'(X)(x) = !t(exp(-tX)x)lt=o. Given a connected unparametrized curve, i.e. an immersed I--dimensional submanifold C eM, we define its symmetry algebra as 5 = {X E g; >.'(X)(x) is tangent to C for all x E C}. By definition, 5 is a Lie subalgebra of g. Moreover, since C is one-dimensional, it is itself a homogeneous space (under the restriction of the action on M) if and only if for any x E C there is an element in >.'(5) which is nonzero in x. A canonical curve in a parabolic homogeneous spaces p : G -+ G / P is given as c(t) = p(u· exp(tX)) = exp(Ad(u) (tX» . 0, so its image C is a homogeneous curve with X in its symmetry algebra. Notice that in the cases of interest, X lies in g_ and hence is nilpotent in the sense that ad(X) : g -+ g is a nilpotent map. Since ad(Ad(u)(X» = Ad(u) 0 ad(X) 0 Ad(u)-I, also Ad(u)(X) is a nilpotent element of g, so the symmetry algebras of canonical curves always contain nilpotent elements. The image >.'(5) C X(C) is a nontrivial finite-dimensional Lie subalgebra. Fortunately, the finite-dimensional Lie algebras of vector fields on one-dimensional manifolds were already classified by Sophus Lie. For the convenience of the reader, we present an elementary proof of the local version of this result, which we will need in the sequel. PROPOSITION 5.3.4. Suppose 5 is a finite-dimensional Lie subalgebra of the Lie algebra of vector fields on a neighbourhood of 0 E R Suppose 5 contains a vector field that does not vanish at O. Then there is a neighbourhood U of 0 and a change of coordinates such that one of the following three possibilities holds on U
(5.28)
5 = span
Lfx }, _
5 = span { tx' x tx } ,
,{a
a
2a}
5 - span ax ' x ax ' x ax . In particular, the dimension of 5 is at most three. PROOF. If dim(5) = 1, then 5 is spanned by a vector field which does not vanish at 0, so after a change of coordinates 5 = span{ tx} and we are done. Next, if dim(5) = 2, then 5 = span{tx,g(x)tx} for some nonconstant smooth function g(x). Now [tx' g(x) tx] = g'(x) tx' so closure under the Lie bracket implies g'(x) = J.l + >.g(x). Solving this differential equation we get
g(x)
=
{ce + + AX
f.1X
C
D if>. =I- 0, if >. = O.
This describes all two-dimensional subalgebras up to coordinate changes. However, for the local change of coordinates y = l-e;"X around zero, we get eAX tx = t y and
5.3. CANONICAL CURVES
tx
567
= (1- AY) t y' so span { tx' e AX tx}
~ span { t y , Y t y }
•
In particular, we have proved the claimed classification of the algebras of dimension at most two. So suppose dim(s) = k + 1 ~ 3 and choose a basis tx ,gl(X) tx"'" gk(X) tx of s. From closure under Lie bracket by tx' we immediately deduce a system of ordinary differential equations with constant coefficients k
g~(x)
= /-ti + L Aijgj(X),
for i
= 1, ... , k.
j=l
We may conclude that the functions gi (x) and, therefore, all vector fields in s are real-analytic. Since dim(s) ~ 3, there is a vector field g(x) tx E s which vanishes to second order in zero, so
g(x) = x N + aX N+ 1 +...
for some N ~ 2.
Because s is finite-dimensional, we may choose g(x) in such a way that N is maximal. But then the vector field
[[tx,g(x)tx] ,g(x)tx]
=
[g'(x)tx,g(x)tx] = ((g'(X))2 - g(x)gl/(x)) tx = (Nx 2(N-l)
+ ... ) tx
lies in s. This contradicts maximality of N unless N = 2. Therefore, dim(s) = 3 and (5.29)
s
= span {tx,g(x)tx,g'(x)tx} ,
where (5.30)
g(x)=x 2 +ax 3 + ....
But then s contains the vector field
[g'(x) tx,g(x)tx] - 2g(x) tx = ((g'(x))2 - g(x)gl/(x) - 2g(x)) tx =
(2ax 3
+ ... ) tx'
which again contradicts maximality of N unless a = O. Now, in order for (5.29) to be closed under Lie bracket we must have
gll(X)tx = [tx,g'(x)tx]
E span {tx,g(x)tx,g'(x)tx}
and to be, in addition, consistent with a = 0 in (5.30), we conclude that gl/(x) = 2 + I/g(x), for some constant 1/. This differential equation, with initial conditions imposed by (5.30), has the solutions
(2/A 2)(COS(AX) -1) if 1/ < 0, g(x) = { x 2 if 1/ = 0, (2/ A2)( cosh(Ax) - 1) if 1/ > O. It is easy to check that (5.29) is, indeed, closed under the Lie bracket in these cases.
5. DISTINGUISHED CONNECTIONS AND CURVES
568
Now clearly there are no more than three different Lie algebras (5.29) with the above choices for g. Although these algebras are different globally, locally we may choose the coordinate change y = tan((Ax)/2), which gives
tx = ~ (1 + y2) ty' sin(Ax) tx = AY ty' COS(AX) tx = ~(1 - y2) ty whence span
{tx' sin(Ax) tx' COS(AX) tx} ~ span { ty'y ty'y2 ty}.
Similarly, y = tanh((Ax)/2) gives
. (AX)axa= AY ay'a cosh() a= 2A (1 + y2) ay'a axa= 2A (1 - y2) ay'a smh AX ax whence span
{tx' sinh(Ax) tx' cosh(Ax) tx} ~ span { ty'y ty'y2 ty} o
and the proposition is proved.
Let us apply this result to understand the possible reparametrizations of homogeneous curves. Guided by the example of canonical curves in parabolic homogeneous spaces, we are interested in parametrizations t 1--* exp(tX) . 0 of a curve C for nilpotent elements X, which lie in the symmetry algebra 5 of C. These will be called preferred parametrizations of C. Notice that we have not included the (evidently possible) constant shifts of the parameter by fixing the value at t = 0 to be o. THEOREM 5.3.4. Let C be an unparametrized homogeneous curve with a preferred parameter t. The freedom among the preferred parametrizations is one of the following types: (1) affine, t
1--*
at for a
(2) projective, t
1--*
bta:.l
:I 0, for a
:I 0
and b arbitrary.
Moreover, the freedom is affine if and only if the dimension of the image of the symmetry algebra 5 in X( C) has dimension one or two.
PROOF. Since the generator of a fixed preferred parametrization X is a nilpotent element of g, it is nilpotent in 5, and hence also A'(X) E A'(5) C X(C) is nilpotent. By inspection, we may find the nilpotent elements in each of the local forms of the symmetry algebras in the proposition above.
atx Espan{tx}' atx E dim { tx ' tx }, (p - qX)2 tx E dim { tx ' X tx ,x2 tx }. In the first two cases, a tx -Nt if and only if x at, which gives affine freedom, X
=
=
whilst in the third case 2
a a ax =at-
(p-qx) -
which gives projective freedom.
{:=?
x
p 2t 1 + pqt
= ---,
o
5.3. CANONICAL CURVES
569
5.3.5. Reparametrization of canonical curves. Having understood the symmetry algebras of homogeneous curves, we may easily derive the results for canonical curves. THEOREM 5.3.5. Let c be a canonical curve of any type on a manifold M with a fixed parabolic geometry. Then for the reparametrizations of c which are again canonical curves of the same type we have the following two possibilities: (1) affine, t ~ at for a =f. 0, (2) projective, t ~ bt~l for a =f. 0 and b arbitrary. PROOF. Clearly, it is enough to prove the result for canonical curves on the homogeneous models. Then the parametrizations of a canonical curve have the form t ~ exp(tY) . 0 where Y is P-conjugate to an element of g_. Certainly, there is at least an affine freedom in such parametrizations, because Y can be replaced by aY. But the possible Yare, in particular, nilpotent elements of g. Therefore, the parametrizations of the image C of c as a distinguished curve are preferred parametrizations of C as a homogeneous curve. Theorem 5.3.4 then implies that, if there is any additional freedom, it must be projective. But just one projective transformation, together with the affine transformations, generates the full projective freedom and the proof is complete. 0
Before we discuss various types of canonical curves in the individual parabolic geometries, we shall draw some more conclusions from the general setup of homogeneous curves in homogeneous spaces. Let us consider a fixed element X E g_ and look for Y in the symmetry algebra 5 C g of the curve C C GjP parametrized by c(t) = exp(tX)P. By definition (see 5.3.4), this amounts to the requirement that A'(Y)(exp(sX)) = 1tloexp(-tY) ·exp(sX) is tangent to C for all s in a neighborhood of the origin. Moving the tangent vector to the origin, we get
1tlo (exp( -sX) exp( -tY) exp(sX))p = - Ad(exp( -sX))(Y) + p E gjp = To(Gj P).
TAexp(_sX)(A'(Y)(exp(sX))) =
Thus, the condition for Y E 5 is simply (5.31)
Ad(exp( -sX))(Y) = Y - s[X, Y]
1 + 2's2[X, [X, Y]]- ... E (X) EB p,
where (X) is the line spanned by X. This provides an efficient iterative procedure to determine 5. LEMMA
5.3.5. Let ai be the non-increasing sequence of subspaces of p defined
by
ao = p, aiH = {Y E Oi : [X, Y] c (X) EB Oi}. Then there is the smallest number r for which the sequence stabilizes, i. e. ar = OrH, and 5 = (X) EB Or is the symmetry algebra of the canonical curve generated by X. PROOF. The sequence has to stabilize because p is finite-dimensional. Since ao is a Lie subalgebra of g it follows by induction that each Oi is a subalgebra of g, whence (X) EB ar is a subalgebra of g by construction. To prove that this equals 5, assume first that Y E 5. If (5.31) is satisfied for s close to zero, then ad(X)k(y) E (X) EB P for all k 2: O. For k = 0, this means
570
5. DISTINGUISHED CONNECTIONS AND CURVES
that Y = QoX + Yo for some Yo E ClO = p. But then ad(X)k(y) = ad(X)k(yo) for all k > 1, so Yo also satisfies (5.31), and it suffices to show that Yo E Clr. Now (5.31) for Yo and k = 1 says that Yo E ClI, so [X, Yo] = QIX + YI with YI E ClO' Then [X, [X, Yoll = [X, YI ] E (X) EB ClO shows that YI E ClI. Thus, Yo E Cl2, and inductively we arrive at Yo E Cl,: for all i. Conversely, for Y E Clr we evidently have ad(X)k(y) E (X) EB P for all k, so Y satisfies (5.31). Thus Clr C s, which completes the proof. 0 REMARK 5.3.5. As a homogeneous space, the curve C is locally described by the pair (s, Clr ), or its effective quotient (s/m, Clrlm). Here m is the maximal ideal of s contained in Clr. By Theorems 5.3.4 and 5.3.5, the reparametrization freedom on C is projective if and only if dim(s/m) = 3, while the dimension is one or two for the affine case. Since we have noted above that ad(X) will be nilpotent in sand hence in s/m we see that if ad(X)2 #- 0 on s/m, then s must be three-dimensional. Notice also that the algebra Clr describes the vector fields in the symmetry algebra, which vanish at the origin. By the arguments from the proof, we may conclude that the smallest number r from the lemma provides the smallest order of the contact element by which the unparametrized homogeneous curve C is determined.
The lemma is also in the background of the following result proved in [Do05). PROPOSITION 5.3.5. (1) Let X E 9 be an element not contained in p, and suppose that there is Z E P such that H := [Z, X] E P and [X, [X, Zll = 2X. Then there is an element YEp completing X and H to an s[(2, IR)-triple. Moreover, the curve exp( tX) . 0 C G I P admits a projective family of preferred parametrizations. (2) Let X E g-i be a homogeneous element of g_. Then there exists an s[(2, IR)triple (X, H, Y) satisfying the conditions in (1) with H E go and Y E gi. In particular, the canonical curves of type CA coming from a subset Ao C g_ which is contained in one grading component, always admit projective reparametrizations. PROOF. (1) By assumption [H,X) = 2X, so the span b = (H,X) is a solvable Lie subalgebra of g, and X spans the derived algebra [b, b). Thus, Lie's theorem (see 2.1.1) applied to the restriction of the adjoint action of b on 9 implies that X is a nilpotent element of g. We want to modify Z to an element YEp without changing the bracket [Z, X], so we consider the kernel n of ad(X) and, in particular, the subspace no = n n p. Clearly, [ad(X), ad(H)) = -2 ad(X), so both n and no are ad(H) invariant. Claim. The restrictions ofthe linear mapping ad(H)+2id to n and no are invertible. To verify this, let us consider Vn = ad(X)n(g) for n ;::: O. By construction [ad(Z), ad(X)] = ad(H) and by a straightforward induction using [ad(H), ad(X)) = 2 ad(X), this implies that
[ad(Z), ad(x)n]
=
n(ad(H) - (n - l)id) ad(x)n-l.
Consequently, for each W E Vn - l the element n[H, W]- n(n - I)W is congruent to [Z, [X, Wll modulo Vn . Moreover, n is invariant with respect to ad(H), and so (ad(H) - (n - l)id)(n n Vn -
l )
en n Vn .
Now X is nilpotent, and therefore Vn = 0 for sufficiently large n. For such a number n, we conclude that (ad(H) - (n - l)id) o· .. 0 (ad(H) - id) 0 ad(H) is nilpotent on n, so all eigenvalues of ad(H)ln are nonnegative integers. In particular, this implies the claim.
5.3. CANONICAL CURVES
571
Next, observe that [H, Z) + 2Z E no by assumption. By the claim, there is an element Y' E no such that [H, Z) + 2Z = [H, Y'] + 2Y'. But now Y = Z - Y' is in 1', [Y,X) = [Z,X) = H, and [H, Y) = [H,Z - Y') = 2(Y'- Z) = -2Y. This proves the first part of (1). To discuss the preferred parametrizations of the homogeneous curve exp tX . 0, observe that both Hand Y belong to the symmetry algebra .5; see the condition (5.31). Since the ideal m is contained in a..., we see that it cannot contain X. Hence, its intersection with the subalgebra spanned by X, Hand Y must be zero since otherwise it would be a proper ideal in .5[(2, JR). This shows that dim(.5/m) 2:: 3 and the rest of (1) follows from Theorem 5.3.4. (2) Since X E 9-i, it is a nilpotent element in 9. There is the well-known Jacobson-Morozov theorem (see subsection X.2 in [Kn96)), which says that every nilpotent element X in a semisimple Lie algebra over a field OC of characteristic zero can be completed to an .5[(2, OC) triple (X, H', Z'). Given the corresponding embedding of .5[(2, JR), we will adjust it to a triple (X, H, Z) for which H E 90 and Z E 9i. Let us write H' = L:;=-k Hj and Z' = L:;=-k Zj for the decompositions into homogeneous components. By homogeneity of X, we obtain [Ho, X) = 2X and [X, Zn = Ho. Thus, putting H = Ho and Z = Z: we arrive at the assumptions for (1). Thus, the projective reparametrization freedom has been proved. A direct inspection of the construction in the proof of (1) shows that the computed 0 adjustment Y of Z: also lies in 9i. In order to distinguish the projective parameters we need the second order jet of the curve. Thus, taking together the above proposition and Theorem 5.3.3, we obtain COROLLARY 5.3.5. Suppose M is equipped with a parabolic geometry of type (G, P) with Ikl-graded Lie algebra 9 and suppose Ao C 9-k is a Go-invariant subset. Then the order determining the canonical curves of type A = Ad(P)(Ao ) is two.
5.3.6. Examples corresponding to 111-gradings. By Theorem 5.3.3, the parametrized canonical curves of all types in geometries corresponding to 111gradings are determined by their two-jet in one point. Moreover, they all admit projective reparametrizations by Proposition 5.3.5, and so this estimate is sharp; see Corollary 5.3.5. Still, there is an interesting diversity in the behavior of the various types of canonical curves in the individual geometries. Conformal geometry. As a warm up, let us reconsider the canonical curves for n-dimensional conformal structures, n 2:: 3, which we have discussed already in 5.3.1. At this point, we also establish the procedure to be followed in the next examples. Step (A): First we determine the geometrically different types of directions. In the positive definite case, the Go-action on 9-1 = JRn is just the standard representation of CO(n,JR). Hence, every nonzero vector X E 9-1 generates the entire set Ao of nonzero elements in JRn = 9/1' and all directions are equivalent. In indefinite signature, we have three different Go-orbits in 9/1' corresponding to positive, negative, and isotropic vectors, respectively. Step (B): Next, we discuss how many parametrized canonical curves emanate from a point with given initial tangent vector, and deduce conditions on jets which make sure that two curves of the form c(t) = exp(Z) . exp(tY)P E
572
5. DISTINGUISHED CONNECTIONS AND CURVES
G I P with Z E p, Y E Ao coincide. Without loss of generality, we may fix any X E A o, fix the curve C1(t) = exp(tX)P E GIP and look for conditions on C2(t) = exp(tAd(exp(Z))(Y))P with Z E p+ and Y E Ao. This was done by means of the logarithmic derivative c5u from Lemma 5.3.2 in the proof of Theorem 5.3.2. In particular, formula (5.25) is the most useful tool. Indeed, in the Ill-graded cases 1 (5.32) Ad(exp(Z))(Y) = Y + [Z, Y] + 2[Z, [Z, Y]] and so the condition that
C1
and
C2
share the same tangent vector at the origin is
c5u(O) = X - Ad(exp(Z))(Y) E p, i.e. Y = X. Next, c5u'(O) = -[X, c5u(O)] E p means [X, [Z, X]] E P and so this has to vanish. At the same time we know that, if [X, [Z,X]] is a nonzero multiple of X, then there is a Z' E 91 such that the corresponding curve corresponds to a reparametrization. But then the curve corresponding to Z must be a reparametrization of C1(t), too. This very nicely corresponds to the intuitive expectation that [X, [Z, Xll describes the change of the acceleration of the curve at the origin. From the description of the brackets in formula (1.29) in 1.6.3, we see that for X E 9-1 ~ IR n and Z E 91 ~ IRM we obtain [X, [Z, Xll = 2X ZX - (X, X)zt, where the transpose is taken with respect to the indefinite inner product. This clearly demonstrates the difference between isotropic and non-isotropic directions. Namely, if (X, X) = 0, then the curve C2(t) coincides with cdt) if and only if ZX = 0, while the remaining values of Z lead only to reparametrizations of the same curve. In all other directions, [X, [Z, Xll = 0 is only possible if zt is a multiple of X. Step (C): Finally, we determine the size of the family of curves emanating in the given direction. It suffices again to analyze the linear mapping Z f--+ 2X Z X (X, X)zt. We have noted already that this is injective if (X, X) ¥- O. Hence, for a nonzero element Z, we always have C2 ¥- C1 as parametrized curves, while a one-dimensional subspace of 91 leads to reparametrizations of C1' Summarizing, there is a (n - 1)-parameter family of unparametrized canonical curves in every non-isotropic direction. Each of them carries a distinguished projective family of parametrizations. There is just one unparametrized canonical curve in every isotropic direction. By definition, these curves c(t) in null directions coincide with the isotropic geodesics of arbitrary Weyl connections as unparametrized curves; see formula (5.1) in 5.2.4. Of course, the null geodesics of the normal Weyl connections also provide the distinguished parameters. Almost Grassmannian structures. The discussion is very similar to the previous example. Let us fix two positive integers p, q and consider the corresponding group SL(p + q, IR) and its Lie algebra as discussed in 4.1.3. Step (A): The adjoint action of Go = S(GL(p,lR) x GL(q,IR)) on 9-1 = L(IRP, IRq) is given by the standard action on linear maps, so its orbits are classified by the ranks of matrices X E 9-1, Thus, we have got as many geometrically different directions in the tangent bundle as ranks of matrices of size p times q. Step (B): The computation is the same as in the conformal case, except this time the bracket [X, [Z, X]] is given by
[(~ ~), [(~ ~), (~ ~)]] = (2X~X ~).
5.3. CANONICAL CURVES
573
Hence, we have to understand the mapping 'l/Jx : g1 ----7 g-l, Z ~ ~(adx)2Z = X Z X. Clearly, this corresponds to the composition of linear mappings IRP ----7 IRq ----7 IRP ----7 IRq and therefore ad(X)2(gt) ~ L(IRP / ker(X), im(X)). If the rank of X is r, then the dimension of both domain and target of the above space of mappings is r. Thus, we obtain an r2-parameter family of different parametrized canonical curves with the same tangent vector. Step (C): The curves described by the target of the mapping 'l/Jx will produce the same unparametrized curve if and only if the image is in the one-dimensional subspace spanned by X. Thus, there is a (r2 - I)-parameter family of unparametrized curves emanating in directions corresponding to a matrix X of rank r. In particular, there is always the subclass of curves in the direction of decomposable tensors, i.e. matrices X of rank 1. In this case, the unparametrized canonical curve is determined by the direction itself. This is consistent with the results for four-dimensional split signature conformal structures (compare with 4.1.4). Null directions correpond to elements of rank one and give rise to a unique wlparametrized curve. Non-isotropic directions correspond to metrices of rank 2 and generate a 3-parameter family of unparametrized canonical curves.
Almost quaternionic geometries. These geometries were discussed in detail in 4.1.8. They correspond to the Lie algebra s[(n + 1, JH[) which has the same complexification as the algebra for Grassmannian structures with p = 2, so the discussion is parallel to the Grassmannian case. We are concerned with the action of Go = S(JH[* x GL(n,JH[)) on g-1 ~ JH[n, so there is only one nontrivial Go--orbit and all directions are geometrically equivalent. As before [X, [Z,X]] = XZX, but now ZX has to be interpreted as the result of the JH[-valued pairing between JH[n and its dual. In particular, ad(X)2(gt} is the quaternionic line spanned by X, and hence has real dimension 4, so we obtain a 3-parameter family of unparametrized canonical curves in any given direction. Again this is consistent with the results for four-dimensional conformal structures in definite signature, which are a special case by 4.1.9. The formula for the change of covariant derivative of Weyl connections in 5.1.6, specialized to the case that one differentiates a vector field ~ in direction ~, reads as
~~~ = ~~~ + {{Y,O,O· We have just observed that [[Z, X], X] always lies in the quaternionic line spanned by X, which of course implies that {{Y,~},O(x) always lies in the quaternionic line spanned by ~(x) (which is well-defined in a prequaternionic vector space). In particular, if a curve c has the property that ~c'(t)C'(t) lies in the quaternionic line generated by c'(t) for one Weyl connection, then the same holds for all Weyl connections. These curves were studied in the literature under the name quaternionic curves. In this context, it is interesting that the quaternionic curves on an almost quaternionic manifold M are exactly the geodesics of all Weyl connections on M (as parametrized curves). This is almost obvious from the above formula, since we just have to check, that there are enough one-forms to make any quaternionic curve a geodesic by deforming a given Weyl connection. Some consequences of this
574
5.
DISTINGUISHED CONNECTIONS AND CURVES
observation were discussed in [HS06]. In particular, in dimensions ;::: 8, a diffeomorphism is a morphism of almost quaternionic manifolds if and only if it preserves the class of unparametrized quaternionic curves, i.e. the class of all unparametrized geodesics of all Weyl connections.
Lagrangean and spinorial geometries. These two geometries discussed in 4.1.11 and 4.1.12 are analogous to the Grassmannian case up to some minor adjustments. In the Lagrangian case, the tangent space is identified with symmetric matrices and so the above reasoning applies with p = q. If the rank of X is r = 1, ... ,p, then the dimension of the image of ad(X)2(g_1) equals the dimension of the space of all symmetric matrices of size r. Indeed, we have to consider only the restriction of the self-adjoint maps lR.P ~ lR.P to the image of X. Thus, there are ~r(r + 1) different parametrized curves sharing the same tangent vector X. Again, in direction of a decomposable tensor X, there is a unique unparametrized canonical curve. Dealing with spinorial geometries, we have to replace symmetric by skew symmetric matrices. Since there are no skew symmetric matrices of odd rank, the available ranks r for the directions X are all even numbers r ::; p. For the same reason as above, the image of ad(X)2(gl) corresponds to all skew symmetric mappings on a space of dimension r, thus giving ~ (r - I)r parameters. For the lowest possible rank 2, we again obtain a uniquely determined canonical unparametrized curve in the given direction. 5.3.7. Chains in parabolic contact geometries. The diversity of the behavior of canonical curves increases quickly with the length of the grading. Before we illustrate this explicitly by the simple example of Lagrangean contact structures, we introduce a particularly nice class of canonical curves, which are available for all parabolic contact structures. They behave similarly to the null-geodesics in conformal pseudo-Riemannian geometry, and have been studied intensively in the case of CR-structures under the name "chains". We use the same name and concept in general. DEFINITION 5.3.7. For all parabolic contact geometries, the canonical curves corresponding to the Go-invariant subset AD = g-2 C g_ are called chains. THEOREM 5.3.7. Let M be an arbitrary manifold equipped with a parabolic contact geometry. Then there is exactly one unparametrized chain in each direction transverse to the contact distribution. Each chain comes equipped with a distinguished projective family of parametrizations. PROOF. We shall see that for the choice Ao = g-2, the procedure from 5.3.6 becomes very similar to the Ill-graded case. In particular, Proposition 5.3.5 applies and the preferred parameters are always projective. Now consider a contact grading on 9 and corresponding groups Go C PeG, and let use determine the subset A c gil' induced by Ao. By Theorem 3.1.3, any element of P+ can be uniquely written in the form exp(Zl) exp(Z2) with Zi E gi for i = 1,2. Since we are dealing with a 12 I-grading, exp(Z2) acts trivially on gil'. On the other hand, the action of exp(Zt} maps X E g-2 to X + [Z, X]. Similarly, as in the proof of Lemma 4.2.2, nondegeneracy of the bracket gl x gl ~ g2 implies that for a nonzero element X E g-2, the map ad(X) : gl ~ g-l is a linear isomorphism. Hence, A is the whole complement of the P-invariant subspace g-l/l', so chains are available in all directions transverse to the contact distribution.
5.3. CANONICAL CURVES
575
Next, by Corollary 5.3.5 we know that the order of the jet determining each chain as a parametrized curve is two. Let us fix X E 9-2, Z = Z1 + Z2 E 91 EB 92, and Y E 9-2. We are going to compare the curves C1(t) = exp(tX)P and C2(t) = exp(t Ad(exp(Z))(Y))P in G/ P exactly as in 5.3.6. Due to our special choice, formula (5.32) gets only slightly more complicated than the one for Ill-graded geometries: 1 Ad(exp(Z))(Y) = Y + [Z1' Y] + [Z2' Y] + '2[Z1 + Z2, [Z1 + Z2, YJ] mod p+. The condition ou(O) = X -Ad(exp(Z))(Y) E p evidently implies Y = X. Moreover, the 9_1-component of X - Ad(exp(Z))(Y) equals -[Z1' Y]. From above we know that ad(Y) : 91 --? 9-1 is injective, so we also get Z1 = 0 and hence Z E 92. Then the above expression for Ad(exp(Z))(X) gets the same form as (5.32), and we may continue exactly as in 5.3.6. The final condition is ou'(O) = -[X, ou(O)] E p, which implies [X, [Z, X]] E p. Since we already know that Z E 92, this bracket lies in 9-2 and hence is always a multiple of X. Thus, the curves C1 and C2 coincide only for Z = 0, while for Z =I- 0, we get two parametrizations of the same curve. 0 Recall that the Weyl connections of the normal Weyl structures introduced in 5.1.12 form a distinguished class of affine connections on our manifold M. In the normal coordinates ¢ : 9-2 EB 9-1 --? M on M induced by any of these connections, the line defined by 9-2 X {O} is a chain. 5.3.8. Canonical curves for Lagrangean contact structures. As an illustration of general features, we shall now treat in detail the example of Lagrangean contact geometries. We shall proceed step by step as in 5.3.6. Step (A): Let us consider a manifold M of dimension 2n + 1 2 3 with contact distribution H = E EB F, where both E and Fare Lagrangean subspaces in H of dimension n. The splitting of H is conveniently expressed by the involutive automorphism ,lJ on H, which acts as the identity on E and as minus the identity on F. Then £ 0 (idH x,lJ) clearly is a nondegenerate symmetric bilinear bundle map H x H --? TM/H. We shall call vectors in H isotropic, if they are isotropic with respect to this bilinear form, and non-isotropic otherwise. Correspondingly, we will talk about isotropic and non-isotropic directions in H. Of course, all vectors in E and F are isotropic. We shall use the same notation for the elements in the graded Lie algebra a ( X
{3
Z
'Y)
A W E 9 = s[(n + 2, IR) Y b
as in 5.2.14. In particular, the elements of 9_ are the lower triangular matrices and we shall write X, Y, {3, and so on, for the elements of 9 completed by zeros at all other positions in the matrix. At the same time, the one-dimensional values {3 and 'Y will also be used as scalar multiples, but this will always be clear from the context. The concatenation of symbols means either matrix multiplications or tensor product if multiplication is not possible. A straightforward check shows that the following is the complete list of orbits of the adjoint action of Go on 9-. For each orbit, we also indicate for which directions in the tangent spaces representatives of the corresponding type are available.
576
5. DISTINGUISHED CONNECTIONS AND CURVES
(1) X E 9-1 corresponding to the directions in the distinguished subspace E in the contact distribution H c T M. (2) Y E 9-1 corresponding to the directions in the subspace F in the contact distribution H c TM. (3) X + Y E 9-1 with Y X = 0 and X oj:. 0, Y oj:. 0, corresponding to isotropic directions in H, which are neither in E nor in F. (4) X + Y E 9-1 with Y X oj:. 0 corresponding to non-isotropic directions in HcTM. (5) !3 E 9-2, available for all directions complementary to H. This corresponds to the chains from 5.3.7. (6) !3 + X E 9_ with X oj:. 0, available for all directions complementary to H. (7) !3 + Y E 9- with Y oj:. 0, available for all directions complementary to H. (8) !3 + X + Y E 9_ with Y oj:. 0, X oj:. 0 but Y X = 0, available for all directions complementary to H. (9) !3 + X + Y E 9_ with Y X oj:. 0, available for all directions complementary
toH. The first four orbits determine geometrically distinct directions, while for all other orbits representatives are available for any direction outside the contact distribution. The first five orbits give rise to different types of canonical curves, but they all correspond to Go-invariant subsets in one grading component, so the strong estimates on the orders from Theorem 5.3.3 apply. Next, it is easy to see that the Ad(P)--orbit of 9-2 contains the Go--orbits listed in (6), (7), and (8). Indeed, these three orbits are generated by elements !3 + X + Y such that Y X = O. Now assuming that Y X = 0, we compute
Ad
T~)) G~ n G~ n,
(exp G
since the matrix in the exponential has zero square. Hence, taking Ao to be one of the orbits from (6), (7), and (8), we always obtain the same set A, and hence the chains as for orbit (5). The last orbit (9) in our list is the first example of a subset Ao which is not contained in one of the grading components. As we shall see, the resulting canonical curves show completely different behavior than chains.
Steps (B) and (C): Next we check the jets of the curves of the individual types emanating in a fixed direction. Let us first formulate the result. PROPOSITION 5.3.8. Let M be a manifold of dimension 2n + 1 endowed with a Lagrangean contact structure E EEl F = H c TM. Then apart from the chains, there are the following types of (parametrized) canonical curves with a fixed tangent vector ~ E TM. (a) If ~ is in E or F, then the canonical curve is uniquely determined by its two-jet in a point. There is just one unparametrized canonical curve in direction (~), which carries a natural projective family of parametrizations and remains being tangent to the subbundle in question. (b) If ~ is an isotropic vector not contained in E or F, then the canonical curve is determined uniquely by its two-jet in a point. There is a one-parameter family of unparametrized canonical curves in direction (~). Each of these canonical curves
5.3. CANONICAL CURVES
577
carries a projective family of parametrizations, and its tangent direction remains isotropic along the curve. (c) IU E H is not isotropic, then the canonical curve is determined by its threejet in a point. There is a 2n-parameter family of unparametrized canonical curves in direction (e). All these curves carry a natural projective family of parametrizations. (d) If ~ H, then there is an additional (2n + I)-parameter family of unparametrized canonical curves in direction (e), all of which are distinct from the chain in direction (e). They carry only an affine family of parametrizations. These canonical curves are determined by their three-jet in a point.
e
PROOF. It follows from the discussion of the orbits above, that we have exactly these four classes of distinguished curves and the chains. Proposition 5.3.5 shows that for the types (a)-(c), we will always have a projective family of preferred parametrizations. By Theorem 5.3.3, curves of these three types are determined by their three-jet in one point. For type (d) we only have the rough estimate from Corollary 5.3.2, which shows that they are determined by their 4-jet in a point. Let us also notice that the description of the canonical curves as projections of flow lines of constant vector fields on the Cartan bundle 9 (see Proposition 5.3.1) implies directly our claims that the type of the tangent vector is preserved along the curve. Let us fix X, Y, and {3, and put =: = X + Y + (3 E g_. Let IJI be another element with components X, Y and /3, put IJ> = Z + W +')' E p+, and consider the curves C2(t) = exp(t Ad( exp( IJ») (w))P Cl(t) = exp(t=:)P, in the homogeneous model G / P. The condition for first order contact of the curves is easy to express. The only component of 8u(0) = =: - Ad(exp(IJ»(IJI)) in g_ is given by (5.33)
(3 -
/3 + X - X - /3w' + Y - Y - /3z',
where we add a prime to the name of an element of g±l to indicate its bracket with the matrix in g'f2 whose unique nonzero entry equals 1. For initial directions in H, i.e. for the orbits (1)-(4) of the above list, we have (3 = 0, so 8u(0) E p implies /3 = 0 and then X = X and Y = Y. In the remaining two cases, (5) and (9) in our list, 8u(0) E p is equivalent to /3 = {3 and the restrictions {3Z' = Y - Y, {3W' = X - X on IJ>. Curves of type (a). These correspond to the orbits (1) and (2) in the above list. Assume (3 = 0 and Y = 0, and recall the simple check for higher order jets deduced in (5.25) in the proof of Theorem 5.3.2. The condition that two curves Cl and C2 have the same two-jet in zero is [X, Ad(exp(lJ»w)] E p. Since we already know that X = X, /3 = 0, and Y = 0, this reduces to [X, [Z,X]] = O. Using this, one verifies that, modulo elements in p+, [X, Ad(exp(IJ»IJI)]
1
1
= 2[X, [Z + W + ')', [Z, X] + ')'X']] = 2[X, [Z, [Z, Xlll
= O.
Therefore, ad(X)2(Ad(exp(IJ»IJI)) E p is automatically satisfied. Hence, our curves have the same three-jet in 0, so they coincide by Theorem 5.3.3. Next, recalling that 8u'(0) = [X, [Z,X]] = 2(ZX)X modp, we see again that there is the projective reparametrization freedom; see Proposition 5.3.5. But since all available values of 8u'(0) coincide with one of reparametrizations of Cl(t), and
5. DISTINGUISHED CONNECTIONS AND CURVES
578
the parametrized curves are determined by their two-jet, there cannot be any other unparametrized curve C of type (1) in the same direction as Cl (t). Of course the curves corresponding to the orbit (2) can be treated in the same way, which completes the proof of part (a). Curves of type (b). These correspond to the orbit (3) in our list, so we suppose that both X and Yare nonzero but Y X = 0 and !3 = O. Proceeding as above, we get 8u'(0) = [X, [Z, X]] + [Y, [Z, X]] + [X, [W, Y]] + [Y, [W, Y]] modp. Computing the brackets explicitly, we arrive at (5.34)
2(ZX)X - (YX)W' = 0,
2(YW)Y - (Y X)Z' = O.
Since Y X = 0 by assumption, this means that ZX = 0 and YW = 0, and hence all four terms in the above expression for 8u'(0) vanish individually. Now we compute modulo p+ as follows:
8u'(0) =[X + Y,')'X' +')'Y' + ~[Z + W, [Z,X] + [W, Ylll =')'([X, Y'] + [Y, X']) - ~[[X + Y, [Z, X] + [W, Y]], Z + W] - ~[[Z,X] + [W, Y], [X + Y, Z + W]] (notice the last term in the first line vanishes by assumption) =')'([X, Y'] + [Y,X']) - ~[[Z,X] + [W, Y], [X,Z] + [Y, W]] =')'([X, Y']
+ [Y,X']).
Finally, still assuming that the second jets coincide,
8u" (0) = [X + Y, ')'[X, Y'] + ')'[Y, X']] = -,),(Y X) (X + Y) mod p+. As above, we conclude that curves of type (b) are determined by their two-jet in one point. Let us look again at the number of different curves in the same direction up to reparametrization. As computed above, the freedom in 8u'(0) is
2(ZX)X - (YX)W'
+ 2(YW)Y -
(YX)Z' = 2(ZX)X + 2(YW)Y.
Now, every common multiple of X and Y is available from the reparametrized curves and the two-jet in a point determines the curve. Thus, for Z X = WY we can only get the reparametrizations. However, if ZX =f. YW, then we arrive at a new curve in the same direction (~). This proves the claim about the number of parameters of the family. Curves of type (c). We have to repeat the computations for type (b) with the assumption YX =f. O. If we multiply the two vector equations (5.34) by Y and X, then they imply that 4( Z X) (Y X) = (Z X) (Y X) so Z X = O. But then the original equations imply W = 0 and then Z = O. Therefore, the curves Cl and C2 of type (c) have the same two-jet in zero if and only if W = 0 and Z = O. Then, 8u'(0) is equal to [X +Y,')'X' +'YY'] = ')'[X, Y'] +,),[Y, X'], up to elements in p+. Finally, if the second jets coincide, then 8u"(0) = -,),(YX)(X + Y), up to elements in p. Thus, the only possibility to share the three-jet as well is = 0 and there clearly are different curves with the same second jet (because they cannot allow more then projective freedom in the reparametrization). We have seen on the way, that gl parametrizes all possible values of 8u'(0) and, moreover, the different values of 8u"(0) are parametrized by g2. Thus, we get a
5.3. CANONICAL CURVES
579
2n-parameter family of different unparametrized curves of type (c) in each given direction. This completes the discussion of curves tangent to H. Curves of type (d). Following the same track as above would be tricky, since we do not have a general result on the freedom in the reparametrization. We only know that the parametrized curves must be determined by their 4-jets according to Corollary 5.3.2. Fortunately, it turns out that the general procedure for treating the unparametrized homogeneous curves in Lemma 5.3.5 provides a nice way to completely sort out this case. Thus, our next goal will be to compute iteratively the vector spaces ao = p, ai+l = {
{I-i+1 E9 ... E9 £1-1 is onto for any i = 2, ... , k and any nonzero element X E £I-i, For example, consider quaternionic contact structures as discussed in 4.3.3. These correspond to a 121-grading, so we only have to check ad(X) : {II ---> {I-1 for a nonzero element X E £1-2. But here £1-2 ~ im(lHl) and g±1 ~ IHln and the bracket is given by (quaternionic) scalar multiplication, so it is surjective for nonzero X. This also shows that there are no distinguished transverse directions in this case. There are also examples of longer gradings with this nice behavior. Generic rank two distributions in dimension five as discussed in 4.3.2 correspond to a three grading with dim({I±2) = 1 and dim({I±I) = dim(g±3) = 2, and it is easy to see that the bracket condition is satisfied. It may also happen, however, that there are directions in which no canonical curves corresponding to a subset of one grading component emanate. For these geometries, one has to deal with canonical curves analogous to those of type (d) in Proposition 5.3.8. To give an example of such a structure, consider split quaternionic contact structures as discussed in 4.3.4. These are similar to quaternionic contact structures, but based on the algebra IHls of split quaternions, which can be viewed as the algebra M2(JR) of two times two matrices. This also corresponds to a 121-grading with £1-2 ~ im(lHls ) = 5[(2, JR) and {l1 ~ 1Hl~, and the bracket is given by (split quaternionic) scalar multiplication. Now for a direction transverse to the sub bundle H, one may look at the image in the quotient T M / H. In there, there is a natural cone corresponding to the rank one elements in im(lHls ). If the image lies outside of this cone, then there will be a canonical curve corresponding to some subset Ao C £1-2 in the given direction. However, among the directions whose image is in the cone, there are two geometrically distinct subclasses. Choosing a representative vector on the Cartan bundle and looking at its image under the Cartan
582
5. DISTINGUISHED CONNECTIONS AND CURVES
connection, the £I-2-component, let us call it 'IjJ, will be nonzero but non-invertible. The question of whether the £I_l-component of our element lies in the image of ad( 'IjJ) does not depend on the choice of lift, so it expresses a geometric property of the original direction. Evidently, there will be a canonical curve corresponding to some subset Ao C £1-2 in the given direction if and only if the £I_l-component does indeed lie in the image of ad('IjJ).
5.3.10. The ambient description of contact projective structures. In 4.5.6 we have seen that there is a generalized Fefferman construction, which associates to a contact projective structure (M, H, ['\7]) a projective structure on M. Our next aim is to interpret this construction in terms of canonical curves, in particular, using the chains of the contact projective structure. Before doing this, we briefly discuss the ambient description (or cone description) of contact projective structures, which was introduced by D.J.F. Fox in [Fox05a]. This article also contained the first description of a projective structure associated to a contact projective structure, and we shall prove that this structure coincides with the one constructed in 4.5.6. The cone description of contact projective structures is similar to the description of projective structures discussed in 5.2.6. As in 4.5.6, we use the group e := Sp(2n + 2, JR) and the stabilizer Pee of an oriented line in JR2n+2 to describe contact projective structures, so we deal with structures admitting global contact forms. We have described the Lie algebra £I in 4.2.6 already, namely
with blocks of size 1, n, n, and 1, £1-2 corresponding to the entry x and £1-1 corresponding to the entries X and Y. On the group level, P consists of matrices which are block upper triangular, and the block diagonal part has the form (c, tl>, c- 1 ) with c > 0 and tl> E Sp(2n, JR) corresponding to the central 2 x 2-block. The elements with c = 1 form a closed normal subgroup Q C P. Given a contact projective structure (M, H, ['\7]) with corresponding parabolic geometry (p: g -+ M,w) of type (e,p), we now define M#:= g/Q. The obvious projection 7r : M# -+ M is a principal bundle with structure group P/Q ~ JR+. On the other hand, g -+ M# clearly is a principal Q-bundle and w defines a Cartan connection on this bundle. Via this Cartan connection, we get an isomorphism TM# ~ g xQ (£I/q), where q egis the Lie algebra of Q. As in 5.2.6 one easily verifies directly that, as a representation of Q, £I/q is isomorphic to the restriction of the standard representation jR2n+2. Since Q is contained in the symplectic group of jR2n+2, this shows that we obtain a natural almost symplectic structure (Le. a nondegenerate two-form T E 02(M#)) on M#. Further, viewing TM# as a tractor bundle, we obtain a natural linear connection '\7# on T M# induced by w. From the construction of this connection one immediately deduces (using again that 9 is the symplectic algebra of jR2n+2) that '\7# is compatible with the natural almost symplectic structure on M#. The connection '\7# is called the cone connection or the ambient connection associated to the contact projective structure.
5.3.
CANONICAL CURVES
583
The manifold M# can be identified either with the frame bundle or with the subsets of positive elements in an oriented natural line bundle over M. The appropriate line bundle by construction is the natural line subbundle contained in the standard tractor bundle of the contact projective structure. From the Lie algebra description, it follows immediately that this is a square root of the bundle of contact forms. Hence, we can use it as a bundle of scales. The vertical bundle of M# -+ M is canonically trivialized by the Euler vector field. As in 5.2.6 one shows that a (local) smooth section s of M# -+ M can be used to pull back \l# to a (locally defined) linear connection \l S on T M. PROPOSITION 5.3.10. Let (M, H) be an oriented smooth contact manifold of dimension 2n + 1 ? 3, and let 7r : M# -+ M be the principallR+ -bundle described above. For t E lR+ let pt : M# -+ M# be the principal right action of t. Let X E X(M#) be the Euler vector field, and let [\l] be a projective equivalence class of partial connections on H with vanishing contact torsion. Then the cone connection \l# on T M# -+ M# has the following properties: (i) For each t E lR+, pt preserves the connection \l#. (ii) \l# is compatible with the almost symplectic structure on M#. (iii) \If X = ~ for all ~ E X(M#). (iv) \l# is torsion free, so the natural almost symplectic structure on M# is symplectic. (v) The Ricci type contraction of the curvature R# of\l# vanishes identically. (vi) For any smooth section s : M -+ M# of 7r : M# -+ M, the restriction of linear connection \lB to a partial connection on H lies in the projective class [\l]. PROOF. We have already observed (ii). Having proved that \l# is torsion free, we can write the exterior derivative of any differential form on M# as the alternation of the covariant derivative with respect to \l#. Then the fact that natural almost symplectic structure T is actually symplectic follows directly from (ii). The proof that \l# is torsion free, as well as the proofs of all other parts is closely parallel to the proof of Proposition 5.2.6. Notice that in the construction of a Weyl structure from a section s for the proof of part (vi), one only has to deal with the underlying partial connection (Le. with derivatives in contact directions). Therefore, even though we are dealing with a 121-grading here, the proof remains similar to the one of Proposition 5.2.6. 0 In [Fox05a] the author obtains a cone description without assuming vanishing of the contact torsion. While it turns out that the natural almost symplectic structure on M# is always symplectic (see below), torsion freeness of the cone connection \l# is equivalent to vanishing of the contact torsion of the contact projective structure (and to torsion freeness of the associated canonical Cartan connection). Also, the passage from the ambient picture to the Cartan picture can be done for projective contact structures in a way closely parallel to the case of projective structures discussed in 5.2.7. First, M# is available as a square root of the bundle of contact forms without any reference to a projective contact structure. It turns out that the same is true for the canonical almost symplectic structure. Namely, by construction for z E M# with 7r( z) = x EM, we can interpret Z2 as the restriction of a contact form to TxM. Hence, there is a tautological one-form a E nl(M#) given by projecting down tangent vectors and then applying the contact form determined
584
5. DISTINGUISHED CONNECTIONS AND CURVES
by the foot point. By construction, (pt)*a = t 2 a which implies that for the Euler vector field X we get ixda = .cxa = 2a. This easily implies that T = da is a nondegenerate two form and that the pullback of the contact distribution in T M# is exactly the annihilator of X under the symplectic form. One easily verifies directly that this coincides with the canonical almost symplectic form constructed above, which also shows that the latter is always symplectic. As in 5.2.7, one then defines a right action of l~+ on T M# by ~ . t := t- 1 T pt . ~. This is free and the orbit space is a vector bundle 7 of rank 2n + 2 over M#/IR+ = M. In particular, sections of 7 -+ M can be identified with vector fields on M#, which are homogeneous of degree -1. The vertical sub bundle descends to a line subbundle 7 1 C 7 which is a square root of the bundle of contact forms on M. The natural symplectic form descends to a section of A27* , which is nondegenerate in each point. Next, one defines 7° to be the annihilator of 7 1 with respect to this form to obtain a filtration 7 = 7- 1 :::) 7° :::) 7 1 . Next suppose that \7# is a linear connection on T M#, which satisfies conditions (i)-(iv) from the proposition above. Then this descends to a linear connection on 7 for which the section of A 2 7* constructed above is parallel. As in the proof of Proposition 5.2.7 one verifies that this actually is a torsion-free tractor connection and hence induces a contact projective structure on M with vanishing contact torsion. Finally, normality of the induced tractor connection is equivalent to vanishing of the Ricci type contraction of the curvature R# of \7#. Again, this construction can also be done without assuming torsion freeness of \7#. The construction of a canonical Cartan connection in [Fox05a] is done in that way, starting from a general version of the cone connection. 5.3.11. The induced projective structure and its relation to chains. In the setting of the cone connection \7# on T M# as discussed above, the construction of a projective structure induced by a contact projective structure is very easy. Recall that M # can be viewed as the space of positive elements in a square root of the bundle of contact forms. Hence, for a point Z E M# with 7r(z) = x E M, we can interpret Z2 as the value of a contact form at x. Now for a contact form () on M, () 1\ d()n defines a volume form. Replacing () by P() for a positive smooth function f, we get d(P()) = Pd() + 2fdf 1\ (), which shows that the associated volume form changes by multiplication by pn+2. But this implies that a positive square root of the bundle of contact forms can be identified with the bundle of 2n~2 -densities. Since dim(M) = 2n + 1, this is exactly the bundle to consider for the ambient description of a projective structure. Hence, given a contact projective structure on M, we can form the ambient connection \7# on TA1#, then forget about the contact and symplectic structures, and just interpret \7# as a linear connection on the tangent bundle of the "right" density bundle. Proposition 5.2.7 then shows that we obtain an induced projective structure on M. This is the subordinate projective structure as defined by Fox. This also works in the presence of contact torsion, but in that case, one uses the symmetrization of the cone connection \7# rather than the cone connection itself. In the case of vanishing contact torsion, there is a nice description of the induced projective structure, which was first obtained in [CZ08]: THEOREM 5.3.11. Let (M, H, [\7]) be a contact projective structure with vanishing contact torsion.
5.3. CANONICAL CURVES
585
(1) The associated projective structure obtained from the ambient connection coincides with the projective structure obtained from the generalized Fefferman construction described in 4.5.6. (2) The paths of the associated projective structure are exactly the paths of the contact projective structure (in contact directions) and the chains of the contact projective structure (in directions transverse to the contact distribution). PROOF. (1) Consider the inclusion G = Sp(2n+2,lRt) '---+ SL(2n+2,lRt) = G which maps the subgroups Q c PeG to their counterparts Q C PeG described in 5.2.6. On the level of Lie algebras, we have the inclusion i : 9 '---+ g, which maps p to p and q to q and induces linear isomorphisms g/p --+ 9/p and g/q --+ 9/q. We shall also denote the two latter isomorphisms by i. Let (Q --+ M, w) be the regular normal parabolic geometry corresponding to the projective contact structure, and put M# = 9/Q. To obtain the Fefferman construction, one defines 9 := 9 Xp P, so there is a natural map j : 9 --+ 9. On the extended bundle, there is a unique Cartan connection w with values in 9 such that j*w = i 0 w. From the construction it is clear that we may naturally identify 9/Q = M# with 9/Q. To complete the proof of (1), it suffices to show that the ambient connection on TM# induced by (9 --+ M,w) as in 5.3.10 coincides with the one induced by (9 --+ M, w) as in 5.2.6. To see this, consider vector fields { and '!1 on M#. Let f : 9 --+ g/q be the Q-equivariant function and let j : 9 --+ 9/q be the Q-equivariant function representing 'fJ. By definition, for u E 9 and lifts 'fJ E Tu9 and ij E Tj(u)9 of'fJ we have f(u) = w('fJ) + q and j(j(u)) = w(ij) + q. Using ij = Tuj . 'fJ, we see that j(j(u)) = i(f(u)). Hence, we get j 0 j = i 0 f and this completely determines j by equivariancy. Likewise, let us choose a local lift ~ E X(Q) of ~, then consider Tj 0 ~ along j(Q) C 9, and extend this to a lift E X(9) of~. Now by construction, the covariant derivatives of'fJ in direction ~ with respect to the two ambient connections in question are represented by ~ . f + w(~) 0 f, respectively, t· j + w(t) 0 f. But by construction, composing the second function with j, we obtain i composed with the first function, which completes the proof of (1). (2) Consider first the ambient connection '\1# associated to a projective structure. For a point y E M# consider a geodesic c for '\1# with c(O) = y such that c'(O) does not lie in the vertical subbundle of M# --+ M. Then locally around y, the curve c projects to a smooth curve f in M, and we can choose a local smooth section s of M# --+ M such that so f = c holds locally around zero. But then from the definition of the connection '\18 in formula (5.11) in 5.2.6 it is evident that f is a geodesic for the connection '\18, which by construction lies in the projective class. Since we can obtain curves through each point in each direction in that way, we see that the paths determined by the projective structure locally are exactly the projections of geodesics of '\1#, which are transverse to the vertical bundle. Now assume that '\1# comes from a contact projective structure, and let r be the canonical symplectic form on M#. For a geodesic c of '\1# and the Euler vector field X, compatibility of '\1# with r gives
t
ftr( c' (t), X (c(t))) = r('\I~(t)c' (t), X (c(t)))
+ r( c' (t), '\I~(t)X (c(t)))
= O.
Using '\It X = ~, we conclude that if r(c'(O), X) = 0, the r(c'(t), X) = 0 for all t. This exactly means that if the initial direction of the projection f of c is tangent
586
5.
DISTINGUISHED CONNECTIONS AND CURVES
to the contact distribution, the f remains to be tangent to the contact distribution for all times. But then the same argument as above can be applied, showing that the paths associated to the contact projective structure are exactly the projections of such geodesics. In particular, this shows that the paths of the contact projective structure are also paths for the associated projective structure, so it remains to prove the claim about chains. From the explicit description of the Lie algebra 9 in 5.3.10 we see that (in contrast to 9-1) the subspace 9-2 egis contained in 9-1. Hence, for a nonzero element 'ljJ E 9 _ 2 and a point u E g, the flow line of w -1 ( 'Ij;) starting in u is mapped by j to the flow line of w- 1 ('ljJ) starting in j(u). By definition, the former flow line projects to a chain of the contact projective structure, while the latter projects to a path of the associated projective structure. Hence, the chains of the contact projective structure are among the paths of the associated projective structure, and since together with the paths in contact directions they exhaust all 0 directions, the proof is complete. Notice, in particular, that this implies that, viewed as unparametrized curves, the chains of a contact projective structure can be locally realized as geodesics of a linear connection. As we shall see below, this is in sharp contrast to the situation for other parabolic contact structures. As a consequence, we can also prove that a contact projective structure is completely determined by its chains. COROLLARY 5.3.11. Let (Mi,H i , ['\7 i ]) for i = 1,2 be two contact projective structures of the same dimension, and assume that f : M1 ...... M2 is a contactomorphism, which maps chains to chains (as unparametrized curves). Then f is an isomorphism of projective contact structures. PROOF. In 5.3.10 we have seen that the ambient space and its canonical symplectic structure depends only on the contact structure. Hence, f lifts to a symplectomorphism f# : M~ ...... M~, which is also compatible with the naturallR+actions and the Euler vector fields on the two ambient manifolds. In 5.2.7 we have constructed the standard tractor bundle from the tangent bundle of the ambient manifold using only these data, and the Cartan bundle of the induced projective structure can then be obtained as a frame bundle of the tractor bundle. The upshot of this is that denoting by gi the Cartan bundle of the induced projective structure on M i , the contactomorphism f naturally lifts to an isomorphism F : g1 . . . g2 of principal bundles. Now we claim that F is compatible with the Cartan connections, i.e. that F*w 2 = w1 • To prove this, recall from 4.4.3 that given a projective structure, we obtain the projectivized tangent bundle as a correspondence space, which then naturally carries a path geometry. Applying this in our setting, we can view gi as a bundle over P(T Mi) for i = 1,2. In the projectivized tangent bundle of a contact manifold, we can consider lines which are transverse to the contact distribution, and of course they form a dense open subset. Let us denote these subsets by Po(TMi), and consider the restrictions of the principal bundles gi to these subsets. Since f is a contactomorphism, the induced map P(T M1) ...... P(T M2) of course maps Po(TM1) to Po(TM2), so F restricts to these subsets. But on Po(TM i ), the paths determined by the path geometry are the paths of the projective structure which are transverse to the contact distribution, so by the theorem these are exactly the
5.3. CANONICAL CURVES
587
chains. Hence, F*c'P = i;} holds over Po(TM1) and since this is a dense open subset, it holds everywhere. This shows that f maps all the paths of the associated projective structure on Ml to paths of the associated projective structure on M2. In contact directions, these are just the paths of the contact projective structures by the theorem, and the result follows from 4.2.7. 0 Notice that, as promised in 1.1.4, this shows that morphisms of contact projective structures can be characterized as diffeomorphisms, which preserve both a contact structure and a projective structure.
5.3.12. The path geometry of chains. We conclude our study of chains with results on Lagrangean contact structures and partially integrable almost CRstructures, which were first obtained in [(:2;05] using many of the tools developed in this book. We start the discussion in the setting of general parabolic contact structure. For any parabolic contact structure on M, the chains introduced in 5.3.7 give rise to a canonical family of unparametrized curves in all directions transverse to the contact distribution. Looking at the projectivized tangent bundle P(TM), these directions form a dense open subset Po(TM). Over this subset, the chains clearly give rise to a generalized path geometry as defined in 4.4.3. Now it is easy to describe this in the language of (non-parabolic) correspondence spaces. PROPOSITION 5.3.12. Let g = g-2 61···61 g2 be a contact grading on a simple Lie algebra and let G be a Lie group with Lie algebra g. Let PeG be a parabolic subgroup corresponding to the contact grading and let Go c P be the Levi subgroup. Let Q C P be the stabilizer of the line in g/p corresponding to g-2 C g_ under the obvious linear isomorphism. (1) The subgroup Q contains Go and has Lie algebra go 61 g2. Moreover, the subspaces g-2 and gl of g descend to Q-invariant subspaces of g/q. (2) If (p : 9 -+ M, w) is a regular normal parabolic geometry, then the quotient g/Q is naturally isomorphic to M := Po(TM). Viewing w as a Cartan connection on 9 -+ M, we get T M = 9 x Q (g/ q). In this identification, the line bundle defining the path geometry of chains corresponds to the Q-invariant subspace of g/q induced by g-2. PROOF. (1) We have described the action of P on the line in g/p determined by g-2 in the proof of Theorem 5.3.7. From the computations made there, we see that an element g E P, which we write as go exp(Zl) exp(Z2) with go E go and Zi E gi for i = 1,2, lies in Q if and only if Zl = O. From this, all statements in (1) follow immediately. (2) We have also seen in the proof of Theorem 5.3.7 that the P-orbit of the line g-2 consists of all lines in g_ which are not contained in g-l. Hence, the homogeneous space P/Q can be identified with the lines in g_ which are transversal to g-l. Since TM ~ 9 Xp (g/p), we conclude that g/Q = 9 Xp (P/Q) can be naturally identified with Po(TM) = M. Since w clearly is a Cartan connection on 9 -+ M, we get the claimed description of T M. In this picture, the tangent map of the natural projection M -+ M corresponds to the natural quotient map g/q -+ g/p. Hence, the vertical subbundle of this projection is represented by the Q-invariant subspace in g/q induced by gl. From the construction it also follows immediately
588
5. DISTINGUISHED CONNECTIONS AND CURVES
that the tautological subbundle corresponds to the Q-submodule induced by g-2 EB gl. Hence, g-2 gives rise to a line bundle which is complementary to the vertical subbundle inside the tautological subbundle, and hence defines a generalized path geometry; see 4.4.3. Finally, since the chains are by definition the projections of the flow lines of constant vector fields corresponding to elements of g-2, we obtain the path geometry defined by the chains in this way. 0 From 4.4.3 we know that a generalized path geometry will give rise to a regular normal parabolic geometry. If our original contact manifold has dimension 2n + 1, then this geometry is associated to the 12 I-grading of 9 = sl(2n+2, JR) corresponding to the first two simple roots. The appropriate group is 6 = PGL(2n + 2, JR) with the maximal parabolic subgroup P c 6 corresponding to this grading. Our aim is to obtain a direct description of this parabolic geometry. Let us first consider the homogeneous model G / P of our parabolic contact structure. Then the proposition simply says that Po (T( G/ P)) = G / Q, so our path geometry is actually homogeneous under the group G. Hence, also the associated regular normal parabolic geometry (9 --> G/Q,w) will be homogeneous under G, and we can apply the theory of homogeneous Cartan geometries from 1.5.15. Any homogeneous Cartan geometry (9 --> G / Q, w) of type (6, P) is given by a pair (i, a) consisting of a homomorphism i : Q --> P and a linear map a : 9 --> g, which satisfy certain compatibility conditions, as follows. The bundle 9 = G Xi P is the associated bundle with respect to the left action of Q on P defined by q. h = i(q)h. Denoting by j : G --> 9 the obvious map, the Cartan connection is then uniquely determined by the fact that j*w = aow. The curvature of this geometry is described in Proposition 1.5.16. The compatibility conditions on i and a imply that (X, Y)
t--t
[a(X), a(Y)]- a([X, Y])
factors to a Q-equivariant bilinear map g/q X g/q --> g. Identifying g/q with gjr;, we obtain the bilinear map gjp. (2) The map 9 from 5.3.12 has values in the semisimple part of 90 and is only nonzero if one of its entries lies in 9~1 and the other entry lies in 9-2. For a nonzero element /30 E g~l' the trilinear map (9~1)3 ----> 9~1 defined by (R, B, T) t-+ [<poe(R, [B, /30]), T] is, up to a nonzero factor, the complete symmetrization of the map ((~~), (~~), (~~)) t-+ (Rl,B2 )CT}). Here we have split
590
5.
DISTINGUISHED CONNECTIONS AND CURVES
each of the elements of 9~1 ~ ]R2n into two elements of]Rn and ( , ) denotes the standard inner product on ]Rn. PROOF. (1) Evidently, the value of i remains unchanged if we replace the matrix by its negative, so we obtain a well-defined map Q ---7 P. An easy direct computation then shows that this induced map is a homomorphism. The map a evidently maps q to p, and it injects 9_ EB91 into 9_, so the map 9/q ---7 9/p will be injective and thus a linear isomorphism by dimensional reasons. Another easy direct computation shows that 0: restricts to i' on q. The verifications of equivariancy of 0: is a slightly tedious but straightforward computation. (2) To obtain cI>a it suffices to compute [o:( ), a( )]- 0:([, ]) on two elements of 9- EB 91, which we denote by (f3i, Xi, Ii, Zi, Wi) for i = 1,2. A direct computation shows that all components of the result vanish except the lower right (2n) x (2n)block, which equals ( -Un - tr(Un)id U21
+ U~l
U 12 + UI2 ) UI1 + tr(Uu)id '
where the n x n matrices Uij are given by Un
= 1(X1Z2 + W 1Y2 U12 = 1(Xl wi U21 =
X 2Z 1 - W 2Y1),
W1X~),
1(Y1t Z2 - ZiY2).
Since the block matrix is evidently tracefree, we see that cI>a has values in the semisimple part 90s. To get cI>a as a map defined on A2(9/p) one has to interpret the original elements as consisting of f3i E 9~\, (:it) E 9~1 and (;1) E 9-2. This shows that the only nonzero component of cI>a is the' one taking one ~ntry from 9~1 and one entry from 9-2, To compute the trilinear map (9~1)3 ---7 9~1 in the last claim, we have to take the above matrix for (..:-'if) = (~~), (;}) = (~~), and Xl = Y 1 = Z2 = W 2 = 0, and then apply this matrix to the vector (~~). Specializing the matrix, we get Un = R1S~ + SlR~, U12 = -R1Sf, and U21 = R2S~, and from this the last claim 0 follows easily. This allows us to determine the path geometry of chains for the homogeneous model of Lagrangean contact structures, which has some unexpected consequences. THEOREM 5.3.13. Let i : Q ---7 P and 0: : 9 ---7 9 be the maps from the lemma. Then the invariant Cartan connection on G Xi P ---7 G / Q determined by a is torsion free (and hence regular) and normal. Thus (G x i P, wa ) is the regular normal Cartan geometry describing the path geometry of chains on G/ P. This path geometry is non-fiat, so there is no local coordinate system on G / P in which all chains are straight lines. There is not even a local linear connection on T( G/ P) which has the chains among its geodesics. PROOF. Since the curvature of the homogeneous Cartan geometry determined by (i,o:) is induced by the map cI>a computed in the lemma, torsion freeness follows immediately from the fact that cI>a has values in 90 c p. To prove normality, we have to show that 8*cI>a = O. This can be verified by a direct computation, but it is more easily done by using a bit of representation theory. As it stands, cI>a
5.3. CANONICAL CURVES
591
is an element of 9:':..2 (8) (9:::-1)* (8) 90 s , which as a representation of 90 s ~ s[(2n, JR) is isomorphic to JR2n* (8)JR2n* (8) s[(2n,JR). The last statement in the lemma says that cl>a actually lies in the intersection of that space with S3JR2n * (8)JR 2n. From the latter representation, there is a unique s[(2n, JR)-equivariant trace with values in s2JR 2n*, whose kernel is irreducible. From the explicit description of cl>a in the lemma, one immediately verifies that it lies in the kernel of this trace. Now, on the other hand, 8* is 90s-equivariant and has values in 9:':.. (8) 9. Since cl>a is homogeneous of degree three, its image under 8* must be contained in 9:':..2 (8) 91 EEl 9-1 (8) 92. Now, evidently, this contains at most tensor products of two copies of JR 2n* , so it certainly cannot contain an irreducible component isomorphic to the tracefree part of s3JR 2n* (8)JR 2n. Hence, 8*cl>a = and normality follows. Having found the canonical parabolic geometry associated to the path geometry of chains on G / P, we can apply the results from 4.4.3-4.4.5. Since cl>a i=- 0, this path geometry is non-flat, and in particular, the paths cannot be realized as straight lines in any coordinate system. We have described the harmonic curvature components of path geometries in 4.4.3. There are three such components for n = 1 and two for n > 1, but in any case all but one of these components are homogeneous of degree less than three. Since cl>a is homogeneous of degree three, it must already coincide with the unique harmonic curvature component of homogeneity 3, which was called p in 4.4.3. Now the last claim follows from Corollary 4.4.4. D
°
This result is remarkable in several ways. On the one hand, we see that even in the homogeneous model, the chains form a pretty complicated family of curves. On the other hand, since the automorphism group of the path geometry of chains contains G, we obtain an example of a non-flat path geometry with a large automorphism group. In view of the connection between path geometries and systems of second order ODE's discussed in 4.4.5, we also obtain an example of a nontrivial system of ODE's with large automorphism group. This is the context in which this example was first considered (in the case n = 1 and without mentioning the connection to chains) in [GroOOj. It also provides an example for the class of torsion-free path geometries which was studied in that reference. In this context, one may go one step further and pass to the space of chains, which carries a canonical Grassmannian structure; see 4.4.5. Applied in this special case, we obtain an example of a non-flat homogeneous Grassmannian structure with large automorphism group (containing G). This example is particularly interesting, since it is even a Grassmannian symmetric space, i.e. for each point x there is an automorphism 0" x of the structure which has x as a fixed point and satisfies TxO"x = -id; see [ZZ08j.
5.3.14. Chains in Lagrangean contact structures. According to Theorem 1.5.15, the pair (i,o:) from Lemma 5.3.13 gives rise to an extension functor mapping Cartan geometries of type (G, Q) to parabolic geometries of type (G, p). One simply extends the structure group via the action of Q on P coming from i, i.e. 9 = g Xi P. Denoting by j : g -+ 9 the canonical inclusion, there is a unique Cartan connection wa on 9 for which j*wa = 0: 0 w. The results in 5.3.13 show that for the homogeneous model (and hence for locally flat Lagrangean contact structures), this extension functor maps the geometry coming from the canonical Cartan connection for the Lagrangean contact structure to the canonical parabolic
592
5. DISTINGUISHED CONNECTIONS AND CURVES
geometry associated to the path geometry of chains. Hence, it is natural to ask, whether the same holds for a larger class of Lagrangean contact structures. It is clear how to attack this question, since the curvature of the geometries coming from the extension functor are described in Proposition 1.5.16. To obtain the result, however, we need a rather careful analysis of the Cart an curvature. THEOREM 5.3.14. Let (p : Q -+ M, w) be the regular normal parabolic geometry associated to a Lagrangean contact structure on M. Put M = Po(TM) = Q/Q and let (9 -+ M, wo ) be the value of the extension functor induced by the pair (i, 0:) from Lemma 5.3.13. Then the geometry (9 -+ M,w o ) is regular and normal if and only if w is torsion free. PROOF. By Proposition 1.5.16, the curvature function determined by (5.39)
K, 0 j
K,
of
(9
-+
M,w o )
is
= 0: 0 K 0 g,-1 + 0'
where g, : g/q -+ gjp is the linear isomorphism induced by 0: and 0 is the map computed in Lemma 5.3.13. We start by proving the necessity of torsion freeness of w. This is vacuous if n = 1, since in that case any regular normal Cartan connection of type (G, P) is torsion free, so we may assume n > 1. If w is normal, then the lowest nonzero homogeneous component of K, must be harmonic by Theorem 3.1.12. From the description of the harmonic curvature of the parabolic geometries associated to generalized path geometries in 4.4.3, we then conclude that K, must be homogeneous of degree at least two, and the homogeneous component of degree two must be contained in the space of maps g~l x g-2 -+ g~l' In particular, the restriction of K, to A2g_2 has to be homogeneous of degree at least three, which implies that K, must have values in g-1. Now from the definition (5.38) of 0: in Lemma 5.3.13, we see that 0: restricts to a linear isomorphism g-l -+ g-2. By regularity, K maps A2g_1 to g-l, and from Lemma 5.3.13 we know that 0 vanishes on A2 g_ 2. Hence, we conclude from (5.39) that K has to map A2g_1 to p. From the discussion of harmonic curvature components of Lagrangean contact structures in 4.2.3, we conclude that this implies torsion freeness of w. Let us conversely assume that w is torsion free. Since we already know that 0 has values in ggs, (5.39) together with torsion freeness of w immediately implies that K, 0 j has values in g-1, which implies regularity of wo' To proceed, we need a bit of input from BGG-sequences. By torsion freeness, the curvature function K of w has values in A2p+®p. Now the Lie bracket defines a P-equivariant map A2p+ -+ g2, so its kernel is a P-submodule A~p+ C A2p+. Likewise, the projection I' -+ 1'/1'+ ~ go is P-equivariant, and the semisimple part ggs C go is a P-submodule. Hence, the elements whose projections to go lie in ggs form a P-submodule Po c p. Now for n > 1, the harmonic part of K has values in the highest weight submodule of A2g1 ®go, which certainly is contained in the P-submodule A~p+ ®Po C A2p+ ®p. For n = 1, this is also verified easily, so by Corollary 3.2 of [Cap05], torsion freeness implies that all of K has values in A~p+ ® Po. Since we have already proved that 8*0 = 0 in Lemma 5.3.13, it remains to show that for each u E Q, the map F(u) = 0: 0 K(U) 0 g,-l lies in ker(8*) c L(A2 g,g). To verify this, it is better to interpret F(u) as an element of A2p+ ® g, and interpreting K(U) in a similar way, we get F(u) = (0: ® A2¢)(K(U)), where ¢ : p+ -+ p+ is dual to the composition of the natural projection g/q -+ gil' with
5.3. CANONICAL CURVES
593
9/p --4 g/q. To compute ¢ we can use the trace form on both algebras, which is just a multiple of the Killing form. But then expanding the defining equation, one immediately verifies that
Q-l :
wt)
0 'Y Z OZ'Y 0000 ( ¢(oow)=OOO O·
000
0000
Now suppose that I'i:(u) = LUi A Vi ® Bi for Ui , Vi E 1'+ and Bi E 1'. Then F(u) = L ¢(Ui) A ¢(Vi) ® a(Bi ), and from 3.1.11 we know that 8*(F(u» = Li ( - ¢(Vi) ® [¢(Ui ), a(Bi)]
(5.40)
+ ¢(Ui ) ® [¢(Vi) , a(Bi)]
- [¢(Ui ), ¢(Vi)] ® a(Bi »).
From above we see that ¢ has values in 9f EB92, which implies that any two elements in the image of ¢ have vanishing bracket, so for all i we have [¢(Ui ), ¢(Vi)] = 0 in (5.40). From our explicit formulae, one next immediately verifies that for R E 1'+ and T E 1'0 we have [¢(R), a(T)] E 9f EB 92, and the first component equals the 9fcomponent of a([R, T]), while the 92-component equals twice the 92-component of a([R, T]). Knowing that I'i:(u) E A~I'+ ® 1'0, we conclude that 8*K(U) = 0 implies 8* F(u) = 0, which completes the proof. 0 Hence we see that the extension functor associated to the pair (i, a) produces the canonical parabolic geometry associated to the path geometry of chains for a large class of Lagrangean contact structures, which corresponds to the class of CRstructures among partially integrable CR-structures. Hence, for this subclass the geometry of chains is intimately related to the original parabolic contact structure. Analyzing the Cartan curvature, we can draw some first basic consequences. COROLLARY 5.3.14. (1) The chains of a torsion-free Lagrangean contact structure on a manifold M cannot be obtained as geodesics of a linear connection on T M. (2) The path geometry of chains associated to a torsion-free Lagrangean contact structure is torsion free if and only if the original structure is locally flat. PROOF. The curvature K, of the path geometry of chains is described by equation (5.39) in the proof of the theorem. Now p+ contains the P-invariant sub= £12. This induces a P-invariant filtration on A2p+ which has the form space A292 C p+ A 92 C A2p+. From Lemma 5.3.13 we know that the function ~o: has values in (p+ A 92) ® p. On the other hand, the fact that the map ¢ : 1'+ --4 p+ constructed in the proof of the theorem has values in 9f EB 92 shows that A2¢ has values in p+ A92. Hence, from (5.39) we conclude that K,oj has values in (p+A92)®9 and this then has to hold for all of K, by equivariancy. The quotient (p+ A92)/ A292 is clearly isomorphic to 91 A92 as a representation of P, so this decomposes into a direct sum according to 91 = 9f EB 9f. Hence, we obtain P-equivariant projections
Pt
irE : (p+ A 92) ® 9
ir V
:
--4
(9f A 92) ® 9,
(p+ A 92) ® 9 --4 (9f A 92) ® 9·
Now the description of ~o: in Lemma 5.3.13 shows that irE(~o:) = 0 and that ir v (~ 0:) equals ~ 0:, so in particular, is nonzero. On the other hand, since the map ¢ has values in 9f EB 92, we conclude that ir v ((A2¢ ® a)(I'i:(u») = 0 for all u.
594
5. DISTINGUISHED CONNECTIONS AND CURVES
(1) The last considerations show that 1i"v (~) i- O. But if the chains of the Lagrangean contact structure were geodesics of a linear connection on T M, then the associated path geometry would locally drop to a projective geometry on a twistor space; see 4.4.4. In particular, the distribution corresponding to 9f would be integrable and vectors from this distribution would insert trivially into the Cartan curvature of the path geometry. But this would contradict the fact that 1i"v (~) i- O. (2) Lemma 5.3.13 implies that locally flat Lagrangean contact structures give rise to torsion-free path geometries, so let us conversely assume that w is torsion free. By Theorem 3.1.11, the lowest nonzero homogeneous component of ~ has to be harmonic. In view of the description of harmonic curvature quantities for path geometries in 4.4.3, this implies that ~ is of homogeneity at least three, and the homogeneous component of degree three is a map 9~1 x 9-2 ---; 90. Moreover, the whole harmonic curvature then has to have values in this space, which is evidently contained in ker(1i"E)' By Corollary 3.2 of [Cap05], the curvature function ~ has values in this P-submodule. Since '1 + >'e
ei-ej, i#ji,j=1, ... ,e+1 Ae
= ei - ei+l = el + ... + ei
ai
(e ~ 1) >'i
Bl (e ~ 2)
(3
±ei, ±ei ± ej, i
#j
al
= el
i, j
= 1, ... , e
= el + e2 (3 = al + 2a2 + ... + 2ae (3 = 2>'2, e = 2 (3 = >'2, e > 2 (3
= ei - ei+ 1, (i < e) >'i = el + ... + ei, (i < e) ai
>'l=~(el+ .. ·+e()
±2ei, ±ei±ej, i#ji,j=1, ... ,e ae
Cl (e ~ 2)
ai
= ei - ei+1, (i < e) >'i = el + ... + ei
±ei ±ej, i #j i,j a(
De (e ~ 3)
= 1, ... ,e
= e(-1 + el
= ei - ei+1. (i < e) >'i = el + ... + ei, (i<e-1) >'e-l = ~(el + ... + ei-l - ee) >'i = ~(el + ... + ee-l + ei) ai
= 2el + ... + 2al-l + a( (3 = 2>'1 (3
= 2el
(3
= 2al
= el + e2 (3 = al + 2a2 + ... + 2al-2 (3
+ae-l +al (3
= >'2 + >'3, e = 3 (3 = >'2, e> 3
(to be continued)
B.TABLES
609
Table B.2 (continued) Roots Type
Highest root
Simple roots Fundamental weights
±2e7, ei - ej, ei + ej + ek ± e7 i,j,k= 1, ... ,6 (3 = 2e7
0i = ei - ei+l! (i < 6)
E6
06 = e4 + e5 + e6 + e7
(3 = 01 + 202 + 303 + 204 + 05 + 206
Ai= e1 + ... + ei +min{i,6 - i}e7, (i < 6)
(3 = A6
A6 = 2e7 ei - ej, ei + ej + ek + el i,j, k, l = 1, ... ,8 (3 = -e7 + es
0i = ei - ei+l, (i < 7)
(3 = 01 + 202 + 303 + 404 + 305 + 206 + 207
07 = es + e6 + e7 + es
E7
Ai =e1 + ... + ei +min{i,8 - i}es, (i < 7) A7
(3
= 2es
ei - ej, ±(ei + ej + ek) i, j, k
= 1, ... ,9 (3
= ei - ei+l! (i < 8) Os = e6 + e7 + es
0i Es Ai
(3
= e1 -
eg
= 201 + 302 + 403 + 504 + 605 + 406 + 207 + 30s
= e1 + ... + ei - min{i, 15 - 2i}eg, (i < 8) As
= A6
(3
= -3eg
= A1
±ei, ±ei ± ej, i =j:. j i, j = 1, ... , 4 !(±e1 ± e2 ± e3 ± e4) 01 = ~(e1 - e2 - e3 - e4)
F4
02
= e4, 03 = e3 - e4 04 = e2 - e3
(3 (3
= e1 + e2
= 20 1 + 40 2 + 303 + 204 (3 = >'4
= ~(3e1 + e2 + e3 + e4) A3 = 2e1 + e2 + e3, A4 = e1 + e2
A1
= ell
>'2
±ei, ei-ej, i=j:.ji,j=1,2,3
G2
= -e2, 02 = e2 - e3 A1 = e1, A2 = e1 - e3
01
= e1 - e3 (3 = 30 1 + 202 (3 = A2 (3
B.TABLES
610
Table B.3 Inverses of the Cart an matrices of the complex simple Lie algebras.
I
Type
Ai (£ ~ 1)
I
Inverse Cartan matrix
1 £+1
--
£ £-1 £-2
£-1 2(£ - 1) 2(£ - 2)
£-2 2(£ - 2) 3(£ - 2)
... ... ...
2 2·2 3·2
1 2 3
2 1
2·2 2
3·2 3
... ...
(£-1)2 £-1
£-1 £
12 2 2 2 4 4 1 2 4 6 2 2 4 6 2 4 6
Bi (£ ~ 2)
12 2 2 4 2 4 1 2 2 4 ,I 2
Ci (£ ~ 2)
14 4 4
Di (£ ~ 3)
1 4
... ... ... ...
...
...
6 3
...
...
2(£ - 1) 2(£-1)
...
2 4 6
1 \ 2 3
2 4 6
\
2 4 6
2 4 6
2(£-1) £-1
2(£ - 1) £ /
... ...
£-1 £
4 8 8 8 12
... ...
8 12
2 4 6
2 4 6
8
...
4(£ - 2) 2(£ - 2) 2(£ - 2)
2(£ - 2) £ £-2
2(£ - 2) £-2 £
4
4
\
-
4 2 2
4 4
12 6 6
... ...
(to be continued)
B.TABLES
611
Table B.3 (continued)
I Type I
Inverse Cartan matrix 14
5 6 4 2 3' 5 10 12 8 4 6 1 - 46 12 18 12 6 9 8 12 10 5 6 3 2 4 6 5 4 3 ,3 6 9 6 3 6,
E6
13
E7
4 5 6 4 2 3' 4 8 10 12 8 4 6 1 5 10 15 18 12 6 9 6 12 18 24 16 8 12 2 4 8 12 16 12 8 6 2 4 6 8 6 4 4 3 6 9 12 8 4 7,
12
Es
F4
G2
4 3 6 8 4 8 12 5 10 15 6 12 18 4 8 12 2 4 6 3 6 9 3
5 10
15 20 24 16 8 12
~ (~ 2
3 6 2 4 1 2
6 12 18 24 30 20 10 15
4 8 6 3
G;)
4 2 3\ 8 4 6 12 6 9 16 8 12 20 10 15 14 7 10 7 4 5 10 5 8
i)
I
612
B.TABLES
Table B.4 Real simple Lie algebras, Satake diagrams, the automorphsims v and s of the Satake diagram mapping a weight to its dual, respectively, its conjugate, and the indices of irreducible representations. See the end of 2.3.15 for more explanations. Real form
.s((l +
Satake diagram with a weight Al
1, JR)
-
Al
AI-I
.-0-_ .
.s((m, lHI) l = 2m-1
.su(p, l + 1 1:::;p:::;4
A2
0-0-···-0--0
Al
p)
A2
A3
AI- 1
Al
·-0--.
.su(l +
1)
l even l = 2m-1 .so(p,2l + 1 - p) l:::;p:::;l p= 2k P = 2k+ 1 .so(2l +
1)
.sp(2l, JR) .sp(p,l- p) 1:::; p:::; 121 .sp(p,p) l = 2p
.sp(l)
Index
e
-:f.e
e
-:f.e
(-1)~~1 A2i -
-:f.e -:f.e -:f.e -:f.e
( _1)(m+p )A m
-:f.e -:f.e
+1
-:f.e -:f.e -:f.e -:f.e
+1 (_l)mAm
+1
1
I-C---[ --.. --l
Al
A2
AI
AI-I
Ap
Ap+ l
At-p At+l_p
l even l = 2m-1 .su(p,p) l = 2p-1 p2:2
Is Iv I
Al
A2
+1
Ap- l
l=I=:::J>oAp A2p A2p- 1 Ap+ l
.-.- -.--.
Al
Al
A2
...
AI- 1
A, - 1 At
Ap Ap+ l
0-· .. - 0 - . - . . .
.--
Al
A2
Al
A2
At
-.~.
A'-I
A,
A, - 1
A,
... -.~.
0 - 0 - · .. -O=¢=O
A2p A2p+1
Al A2 As
. - 0 - . - . . . - 0 - - . - · ..
AI_l Al
-.::::J¢:: •
Al A2 Aa A2p- 2 A2p- 1 A2p ___ o-.-···--o-----e~o Al
A2
A,-I
At
. - . - ... -.=¢=.
(-1) (k+ Htt l ) )Al
e e
e e
(-1 )(k+ 1(lt 3) )Al
e
e
( -1 )!l!±..!lA 2 I
e
e
+1
e
e
(-l)~i=1
e
e
e
e
IL¥J
A2i - 1
(_1)~;=IA2i-1 IL¥J
(-1)~i=1
A2i - 1
(to be continued)
B.TABLES
613
Table B.4 Real form
Satake diagram with a weight
50(1,1)
oAI- l A, A2 Al-V 0-0-----0 ""'0 Al
e e
1 even lodd 50(p, 21- p) 1:::;p:::;1-2
s
I
1/
e =/=e
I
(continued) Index I
+1 +1
.Al- I A, A2 Ap Ap+l /. 0-0-- - --0-_- - --. AI-2 ""'. Al
e e (_l)¥(AI- I+Azl e =/=e (_l)¥(Az- I+Azl =/=e =/=e +1 =/=e e +1
p,l even p, 1 odd p even, 1 odd p odd, 1 even 50(l- 1,1 + 1)
OAI_ I A6_~: __ - AZ-o( ] Al
1 even lodd 50(21)
.-.-----.
A,
A2
=/=e e =/=e =/=e
+1 +1
e e =/=e =/=e
( _l)~(AI-I+Azl
.AI- 1
Al-V
""'. Al
1 even lodd 50* (2l) l=2m
.Al- I A, A2 A3 AI - 3 Al-V .-0-.- - - -- . - - 0 ""'0 Az
50* (2l) 1 = 2m+ 1
~~~-~~ - - - ~t"3 Al-.(] Al
e
e
+1 (_1)L::"A2i- ,
oAI- l
=/=e =/=e
(-1 )L::'I A2i - I
(to be continued)
B.TABLES
614
Table B.4 Real form
ISatake diagram with a weight I
(continued) Index I
v
e
¥-e
+1
A5
¥-e ¥-e
+1
o~o A2 A5
¥-e ¥-e
+1
0-0-1-0- 0
Al
EI
I
s
A2
A3
A4
A5
A6
I
o~o
Ell
Al
A2
A3 A4 A6
EIII
Al A2 A3 A4 A5 e-e-e_-e
1A6 Al A2 A3 A4 A5 o - o - o - r : -o-
A6 o
Al A2 A3 A4 A5 A6 e-o-e-o-o-o
1A7
0-0--1--
Al
EVIl
e
1A6
compact form of E6
EVI
A3 A4 A6
A4 A5 Al A2 A3 o-e-e-e-o
EIV
EV
1
Al
A2
A3
A4
A5 e-
A6 o
¥-e
+1
¥-e ¥-e
+1
e
e
+1
e
e
( _1)AI+A3+ A7
e
e
+1
e
e
(_l)AI +A3+ A7
A7
compact form of E7
Al A2 A3 A4 A5 Ae e-e-e-e-e-e
1A7
(to be continued)
B.TABLES
615
Table B.4 Real form
Satake diagram with a weight Al
EVIl I
A2
A3
A4
A5
A6
(continued) 81 v 1Index 1
A7
0-0-0-0-0-0-0
1
e
e
+1
e
e
+1
e
e
+1
e
e
+1
e
e
+1
e
e
+1
e
e
+1
e
e
+1
As
o-o-o---r---o --------r-----
Al
EIX
A2
A3
A4
A5
A6
A7
As
compact form of Es
Al
A2
A3
A4
As
A6
A7
As
FI
Al
FH
Al
compact form of F4
Al
A2
A3
A4
o-o~o-o
A2
A3
A4
o-_~_-_
A2
A3
A4
---~---
G2
Al
compact form of G 2
Al
A2
O:::EO
A2
-:::E-
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Index
P-frame bundle, 274, 384, 421, 605 normal,277 WP,325 W p ,325 I::-height, 292 a-string through /3, 167 r-determined, 562 Ill-grading, 296 Ikl-grading, 238
Kostant's version, 351 Bruhat decomposition, 333 Bruhat order, 324 bundle of scales, 503 conformal, 124 canonical curves in Cartan geometries, 110 on homogeneous spaces, 69 preferred parametrization, 568 Camot group, 267 Cartan connection, 71 conformal, 128 Cartan decomposition, 202 global,206 Cartan geometry, 71 Cartan involution, 202 global,206 Cartan matrix, 169 Cartan product, 186 Cartan subalgebra 9-stable, 213 complex, 162 maximally compact, 213 maximally noncompact, 213 real, 200, 213 Cartan's criteria, 150 Casimir element, 190 Casimir operator of a representation, 152 center of a Lie algebra, 144 of the universal enveloping algebra, 192 central character, 192 centralizer, 163 chains, 574, 584, 591 character, 194 chart, 15 Christoffel symbols, 39 cocyc1e of transition functions, 27 commutator ideal, 142 complete reducibility, 152 complexification
absolute derivative, 38 action, 24 effective, 25 free, 25 transitive, 25 adjoint action, 20 admissible family of curves, 110 affine extension, 47 AHS structure, 137 algebraic bracket of adjoint tractors, 85 ambient connection for contact projective structures, 582 for projective structures, 528 ambient metric, 531 associated bundle, 28 associated graded vector space, 235 atlas, 15 automorphism, 163 infinitesimal, 97 inner, 163 Bianchi identity general,88 reductive, 90 biholomorphism, 32 Borel fixed point theorem, 307 Borel subalgebra, 291 standard, 184, 291 Borel-Moore homology, 337 Borel-Wei! theorem, 185 Bott-Borel-Weil theorem, 357 623
624
of a Lie algebra, 147 of a representation, 148, 227 cone, 471 cone structure, 471 conformal circles, 560 conformal holonomy, 527 conformal structure almost Einstein, 526 anti-self-dual, 383 generalized, 471 self-dual, 383 connection affine, 42 distinguished, 365 general,37 induced,40 invariant linear, 62 invariant principal, 57 linear, 35 on a G-structure, 46 partial affine, 48 partial linear, 47 principal, 38 projective equivalence of, 384 special symplectic, 555 contact connection, 405 contact form, 403 contact grading, 298 contact structure, 403 contact torsion, 422 coordinates normal,45 correspondence, 467 correspondence space, 99 cotangent bundle, 17 cotangent space, 17 Cotton-York tensor 111-graded case, 365 conformal, 131 of a Weyl form, 537 covariant derivative, 35 covariant exterior derivative, 37 CR-structure, 414, 552 codimension two elliptic, 445 hyperbolic, 445 higher codimension, 443 curvature harmonic, 265 of a general connection, 37 of a linear connection, 36 of a principal connection, 39 of a Weyl form, 518 curvature form, 71 curvature function, 71 curve quaternionic, 573
INDEX
densities conformal, 135, 371 derivation inner, 153 of a Lie algebra, 153 with values in a representation, 158 derived series, 142 development, 108 diffeomorphism, 15 differential form, 18 direct sum of Lie algebras, 145 distribution, 18 bracket generating, 429 horizontal, 36 integrable, 18, 19 smooth,18 dominant, 176 Dynkin diagram, 170 extended, 178 Einstein metric, 525 Engel's theorem, 143 equivariant, 145 Euclidean group, 47 exponential map, 20 extension functor, 106 exterior absolute differential, 38 exterior derivative, 18 Fefferman construction classical, 479 general, 108, 478 Fefferman space, 478 fiber bundle, 25 homogeneous, 50 fibered manifold, 25 fibered morphism, 25 filtered vector space, 235 flow, 17 foliation, 19 frame bundle, 27 adapted, 285 noncommutative, 604 orthonormal, 113 frame form, 274 Freudenthal multiplicity formula, 194 Frobenius reciprocity algebraic, 160 geometric, 54 Frobenius theorem, 19 fundamental derivative definition, 86 fundamental vector field, 25 G-structure, 45, 113 generalized flag variety complex, 302 real,314
INDEX
generalized geodesics, 560 generic distribution rank n in dim. n( n2+l), 430 rank 2 in dim. 5, 431, 493 rank 3 in dim. 6, 430, 494 rank 4 in dim. 7,434 grading element, 118, 239 grading section, 503 growth vector, 429 Harish-Chandra map, 193 Hasse diagram, 325 height of a root, 174 Heisenberg algebra, 403 quaternionic, 433 split quaternionic, 435 highest weight theorem of the, 184 highest weight vector, 182 holomorphic, 33 holonomy, 526 affine, 534 exotic, 535, 555 symplectic, 555 homogeneity of linear maps, 236 homogeneous model, 71 homogeneous space, 25 homomorphism of Lie algebras, 141 of Lie groups, 20 horizontal differential form, 29 horizontal lift, 36 horizontal projection, 37 ideal, 24 in a Lie algebra, 141 immersion, 16 induced module, 68, 159, 184 infinitesimal character, 192 infinitesimal fiag structure, 248 regular, 251 integrability for CR-structures, 414 invariant differential operator, 65 isotropy subgroup, 25 isotypical component, 189, 348 Iwasawa decomposition, 209 global, 210 jet, 30 semi-holonomic, 95 jet prolongation of a bundle, 30 Jordan decomposition, 161 kernel
625
of a Klein geometry, 49 Killing form, 149 Klein geometry, 49 effective, 49 infinitesimally effective, 49 reductive, 50 split, 50 Klimyk's formula, 197 Kostant codifferential, 261, 341 Kostant Laplacian, 263, 343 Kostant multiplicity formula, 196 Kostant partition function, 195 Kostant's version of the BBW-Theorem, 351 Laplacian conformal, 137 leaf, 19 Levi decomposition, 156 Levi factor, 156 Levi subgroup, 242 Levi bracket, 251 Lie algebra, 19 Ikl-graded, 238 abelian, 141 compact, 200 filtered, 237 nilpotent, 142 reductive, 144 semisimple, 144 simple, 144 solvable, 142 Lie algebra cohomology definition, 157 Lie algebra homology definition, 157 Lie bracket of adjoint tractors, 85 of vector fields, 17 Lie derivative, 31 Lie group, 19 complex, 32 Lie subalgebra, 141 Lie subgroup, 24 virtual, 24 Lie's theorem, 143 Liouville theorem, 133 local diffeomorphism, 15 locally fiat, 74 logarithmic derivative, 21 lower central series, 142 lowest form, 190 Mobius space, 116 Mobius structure, 439 manifold almost complex, 33 complex, 32 filtered, 251
626
Maurer-Cartan equation, 21 Maurer-Cartan form, 21 morphism of Cartan geometries, 73 of homogeneous bundles, 50 of infinitesimal flag structures, 248 of representations, 145 multiplicity of a weight, 181 of an irreducible component, 188 natural bundle, 29, 79 Newlander-Nirenberg theorem, 33 Nijenhuis tensor, 33 for almost CR-structures, 414 nilradical, 156 normal conformal case, 128 one-parameter subgroup, 20 operator conformally invariant, 135 orbit, 25 parabolic geometry complex, 280 definition, 244 normal, 265 regular, 252 parabolic subalgebra complex, 291 real, 308 parabolic subgroup, 242 opposite, 267 parallel transport, 38 partition of unity, 16 path geometry, 463, 587 Plucker embedding, 307 Poincare conformal group, 118 Poincare-Birkhoff-Witt theorem, 159 principal bundle, 26 holomorphic, 32 homogeneous, 50 morphism, 26 projective holonomy, 528 prolongation algebraic, 114 pseudo-sphere, 116 pullback of natural bundles, 31 of one-forms, 17 of vector fields, 17 quadric, 443 Racah's formula, 197 radical, 155 rank of a distribution, 18
INDEX
of complex semisirnple Lie algebra, 163 real form compact, 200 of a Lie algebra, 147, 200 split, 200 real rank, 310 real structure, 226 Reeb field, 405, 542 regular element in g, 163 representation adjoint, 20, 146 completely reducible, 147 complex, 145 conjugate, 226 constructions with, 146 contragradient, 146 derivative of a, 20 dual, 146 faithful, 146 fundamental, 186 holomorphic, 32, 197 indecomposable, 147 index, 228 induced, 54 irreducible, 146 of a Lie algebra, 145 of a Lie group, 20 quaternionic, 228 semisimple, 147 simply reducible, 147 unitary, 147 Rho tensor, 501, 518 Ill-graded case, 365 conformal, 131 Ricci identity general, 88 reductive, 90 root, 164 compact, 214 positive, 169 restricted, 207, 308 simple, 169 root decomposition, 164 root lattice, 169 root reflection, 168 root space, 164 root system, 168 Satake diagram, 216 scale, 504 conformal, 124 scaling element, 503 Schubert cell, 337 Schubert variety, 337 Schur's lemma, 146 section, 25 semisimple
INDEX
element of g, 162 Serre relations, 179 sign of a Weyl group element, 174 simple subsystem, 174 soldering form, 42, 518 of a Weyl structure, 501 stabilizer, 25 standard generators, 179 standard parabolic subalgebra complex, 291 real,308 structure affine, 44 almost complex, 33, 280 almost Grassmannian, 375, 469 almost Lagrangean, 398 almost quaternionic, 394, 473 almost spinorial, 400 conformal, 116 contact projective, 420, 492, 553, 582 Grassmannian, 380 hypercomplex, 394 Lagrangean contact, 410, 458, 547, 588 partially integrable almost CR, 412 projective, 10, 383, 458, 492, 584 ambient description, 528 cone description, 528 quaternionic contact, 433, 488 split quaternionic contact, 435 symplectic, 402 structure group, 26 reduction of, 27 submanifold, 16 embedded, 16 immersed, 16 submersion, 16 subrepresentation, 146 subspace invariant, 146 support, 16 symbol of a differential operator, 66 symbol algebra, 251 symmetric space, 224 Hermitian, 304 quaternionic, 304 tangent bundle, 16 tangent map, 16 tangent space, 16 torsion, 44 of a Cartan connection, 85 of a Weyl form, 537 torsion free Cartan geometry, 74 trace form, 149 tractor bundle, 83
627
adjoint, 83, 256 almost Grassmannian, 381 conformal standard, 523 projective standard, 527 tractor connection, 83 abstract, 287 transition function, 25 translation, 19 twistor correspondence conformal, 470 Grasmannian, 469 twistor space, 102, 457 for almost quaternionic structures, 473 for conformal structures, 477 for quaternionic contact structures, 489 unitary trick, 204 universal enveloping algebra, 159 vector bundle, 26 associated graded, 237 filtered, 237 holomorphic, 32 homogeneous, 50 homomorphism, 26 vector field, 17 complete, 17 constant, 71 left invariant, 19 right invariant, 20 Verma module, 184 generalized, 320 vertical projection, 37 vertical tangent bundle, 29 Webster scalar curvature, 548, 553 Webster-Ricci curvature, 546, 553 Webster-Tanaka connection, 544, 553 weight, 164 p-algebraically integral, 318 p-dominant, 318 algebraically integral, 180 analytically integral, 197 dominant, 180 fundamental, 180 highest, 183 weight lattice, 181 weight space, 164 Weyl chamber dominant, 176 Weyl character formula, 194,359 Weyl connection, 501, 518 conformal, 120 Weyl curvature, 383 Ill-graded case, 365 conformal, 131 Weyl denominator, 195 Weyl dimension formula, 195 Weyl form, 518
628
normal,519 Weyl group, 174 affine action, 193 . Weyl structure, 499 associated principal connection on Q, 511 closed, 125, 508 exact, 125, 508 normal,515 Yamabe operator, 137
INDEX
Titles in This Series 154 Andreas Cap and Jan Slovik, Parabolic geometries I: Background and general theory, 2009 153 Habib Ammari, Hyeonbae Kang, and Hyundae Lee, Layer potential techniques in spectral analysis, 2009 152 Janos Pach and Micha Sharir, Combinatorial geometry and its algorithmic applications: The Alcala lectures, 2009 151 Ernst Binz and Sonja Pods, The geometry of Heisenberg groups: With applications in signal theory, optics, quantization, and field quantization, 2008 150 Bangming Deng, Jie Du, Brian Parshall, and Jianpan Wang, Finite dimensional algebras and quantum groups, 2008 149 Gerald B. Folland, Quantum field theory: A tourist guide for mathematicians, 2008 148 Patrick Dehornoy with Ivan Dynnikov, Dale Rolfsen, and Bert Wiest, Ordering braids, 2008 147 David J. Benson and Stephen D. Smith, Classifying spaces of sporadic groups, 2008 146 Murray Marshall, Positive polynomials and sums of squares, 2008 145 Tuna Altinel, Alexandre V. Borovik, and Gregory Cherlin, Simple groups of finite Morley rank, 2008 144 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci How: Techniques and applications, Part II: Analytic aspects, 2008 143 Alexander Molev, Yangians and classical Lie algebras, 2007 142 Joseph A. Wolf, Harmonic analysis on commutative spaces, 2007 141 Vladimir Maz'ya and Gunther Schmidt, Approximate approximations, 2007 140 Elisabetta Barletta, Sorin Dragomir, and Krishan L. Duggal, Foliations in Cauchy-Riemann geometry, 2007 139 Michael Tsfasman, Serge Vladut, and Dmitry Nogin, Algebraic geometric codes: Basic notions, 2007 138 Kehe Zhu, Operator theory in function spaces, 2007 137 Mikhail G. Katz, Systolic geometry and topology, 2007 136 Jean-Michel Coron, Control and nonlinearity, 2007 135 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci How: Techniques and applications, Part I: Geometric aspects, 2007 134 Dana P. Williams, Crossed products of C*-algebras, 2007 133 Andrew Knightly and Charles Li, Traces of Heeke operators, 2006 132 J. P. May and J. Sigurdsson, Parametrized homotopy theory, 2006 131 Jin Feng and Thomas G. Kurtz, Large deviations for stochastic processes, 2006 130 Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, 2006 129 William M. Singer, Steenrod squares in spectral sequences, 2006 128 Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov, Painleve transcendents, 2006 127 Nikolai Chernov and Roberto Markarian, Chaotic billiards, 2006 126 Sen-Zhong Huang, Gradient inequalities, 2006 125 Joseph A. Cima, Alec L. Matheson, and William T. Ross, The Cauchy Transform, 2006 124 Ido Efrat, Editor, Valuations, orderings, and Milnor K-Theory, 2006
TITLES IN THIS SERIES
123 Barbara Fantechi, Lothar Gottsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli, Fundamental algebraic geometry: Grothendieck's FGA explained, 2005 122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities and asymptotic methods, 2005 121 Anton Zettl, Sturm-Liouville theory, 2005 120 Barry Simon, Trace ideals and their applications, 2005 119 Tian Ma and Shouhong Wang, Geometric theory of incompressible flows with applications to fluid dynamics, 2005 118 Alexandru Buium, Arithmetic differential equations, 2005 117 Volodymyr Nekrashevych, Self-similar groups, 2005 116 Alexander Koldobsky, Fourier analysis in convex geometry, 2005 115 Carlos Julio Moreno, Advanced analytic number theory: L-functions, 2005 114 Gregory F. Lawler, Conformally invariant processes in the plane, 2005 113 William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith, Homotopy limit functors on model categories and homotopical categories, 2004 112 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups II. Main theorems: The classification of simple QTKE-groups, 2004 111 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups 1. Structure of strongly quasithin K-groups, 2004 110 Bennett Chow and Dan Knopf, The Ricci flow: An introduction, 2004 109 Goro Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups, 2004 108 Michael Farber, Topology of closed one-forms, 2004 107 Jens Carsten Jantzen, Representations of algebraic groups, 2003 106 Hiroyuki Yoshida, Absolute CM-periods, 2003 105 CharaIambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, second edition, 2003 104 Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence sequences, 2003 103 Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanre, Lusternik-Schnirelmann category, 2003 102 Linda Rass and John Radcliffe. Spatial deterministic epidemics, 2003 101 Eli Glasner, Ergodic theory via joinings, 2003 100 Peter Duren and Alexander Schuster, Bergman spaces, 2004 99 Philip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, 2002 97 V. A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D. Neusel and Larry Smith, Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2: Model operators and systems, 2002
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