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] + [B\ cj>'\ = [B © B\ cf> © <j>'\(b) [B, (/>] = 0 if (B, 4) is traceless; (c) [B, C} = C [By 0] if C € Qi satisfies Cn = 1. This definition is similar to that of a Grothendieck group. 11.1.6. Now let C' be another linear category with a given functor a* : C' —> C' such that a*n is the identity functor from C' to C'. Let b : C C' be a linear functor. Assume that we are given an isomorphism of functors ba* — a*b : C —> C'. Then b induces a linear functor b : C —• C' by b(A,(f>) = (bA,<j>') where ' : a*bA —> bA is the composition a*bA = ba*A bA. It is clear that [A, 4] ^ [bA, (j)'] respects the relations of JC(C), JC(C') and hence defines an 0-linear map fC(C) —> K,(C'). 11.1.7. Assume now that C is, in addition, an abelian category in which any object is a direct sum of finitely many simple objects. Let B be a simple object of C. Let ts be the smallest integer > 1 such that ( a * ) t B B is isomorphic to B; let f s (a*)tBB —> B be an isomorphism. We have n = n'tB where n' is an integer > 1.
11.1. Periodic Functors
91
The composition (a)
B = (a*)n'tBB
-> • • • a*2tBfB> a*2tB B
a
*'B/B> a*tB B
B
is a non-zero scalar times identity (since B is simple); hence by changing f b by a non-zero multiple of we can assume that the composition (a) is the identity. Consider the isomorphism 4>b : a*(B © a*B © • • • ( a ^ ^ B ) = a*B © a*2B © • • • (a*)tBB which maps the summand a*^B onto the summand a*i B by the identity map (for 1 < j s) is an object of C. Let 5 be a set of simple objects of C with the following property: any simple object in C is isomorphic to a*^B for a unique B in S and some j > 0. For each B in S we choose 0B as above. It is easy to see that any object of C is isomorphic to (b)
©BGS((£ © a*B © • • • (a*)tB~1B)
® EB, 4>B ® 1>B)
where for each B, EB is a finite dimensional Qj-vector space with a given automorphism tps Eb —• Eb such that ipB = 1 a n ( ^ = 0 for all but finitely many B. Note that the summands corresponding to B such that t s > 2 are traceless. Let B be a simple object of C such that a* B = B. The isomorphisms (f> : a*B = B such that (B,(f>) G C, generate a free (9-submodule of rank 1 of Home( a *B, B)\ w e denote this (9-submodule by OB- It is easy to see that OB depends only on the isomorphism class of B. 11.1.8. From 11.1.7 it follows easily that (a)
K(C) =
®bOb
as O-modules (the sum is taken over the isomorphism classes of simple objects B such that a*B = B); to the element (j) G Ob such that (B, ) G C corresponds the element [B,(p] G JC(C).
Quivers with Automorphisms
12.1.1. An admissible automorphism of the graph (I, H,h > [h]) consists, by definition, of a permutation a : I —> I and a permutation a : H —> H such that for any h G H, we have [a(/i)] — a[h] as subsets of I and such that there is no edge joining two vertices in the same a-orbit. In this chapter we assume that we are given an integer n > 1 and an i > [ft]) in 9.1.1. We assume admissible automorphism a of the graph (I, H,h — that a n = 1 both on I and on H. From the definition it follows that (a) any a-orbit on I is a discrete subset of I, in the sense of 9.1.3. An orientation h —> h\h —> h" (see 9.1.1) of our graph is said to be compatible with a if, for any h G H, we have (a(h))' = a(h') and {a{h))n — a{hn). From property (a) we can deduce that there is at least one orientation of our graph which is compatible with a. This is seen as follows. Choose a set of representatives Ho for the a-orbits on H. For each h G HQ, choose one element h! of [h]\ let h" be the other element of [h]. Now let h G H. We can find n G Z such that h = an ho where ho G Ho is uniquely determined. We set h' — an(ho) and h" = an(hQ). We must prove that h h " are independent of the choice of n. We are reduced to verifying the following statement: if ra G Z satisfies a m /io = ho, then we have a m (/iQ) = h'0 and a711 (h^) = /IQ. If this is not the case, we have
12.1.2. Let Va be the category of finite dimensional I-graded A;-vector spaces V = 0 i e i V | with a given linear map a : V —> V such that a(V\) = V a (j) for all i G I and a 7 |vi = lvi f° r all i G I and all j > 1 such that a J (i) = i. The morphisms in V a are isomorphisms of vector spaces compatible with the grading and with a. Note that a : V —• V automatically satisfies
For each v G N[I] a , we denote by V® the full subcategory of Va whose objects are those V such that dimVi = v\ for all i G I. Then each object
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_13, © Springer Springer Science+Business Science+Business Media, 2011 Media, LLC LLC 2010
92
12.1. The Group fC(Qv)
93
of Va belongs to VJ for a unique v G N[I]° and any two objects of VJ are isomorphic to each other. Moreover VJ is non-empty for any v G N [ I ] A . We choose an orientation of our graph, compatible with a. If V G V a , then we may regard V as an object of V (by forgetting a); in particular, G v and E v are defined. There is a natural automorphism a : G v —>' Gv given by a(gz) = a(g)(a(z)) for all g G Gv and z G V; moreover, there is a natural automorphism a : E v —• E v (x •—• a(x)), such that for any x = (Xh) G E v and any h G H, the compositions VH' ^
V h „ A V A ( F C „) and V h . A V a{h>)
va{h,t)
coincide. Both of these automorphisms satisfy a n = 1. Note that a(gx) = a(g)(a(x)) for all g G G v and x G Ev- Taking the inverse image under a : E v —'• E v gives us a functor a* : £>(E V ) —>> ^ ( E v ) . v f o r a11 1 Let v = (i/1, i / 2 , . . . , i/ m ) G X be such that dim Vi = \ € here, vl G N[I] for each I. Let i / = (i/ 1 , i / 2 , . . . , i / m ) G * be defined by v'\ = ^(i)- We have natural isomorphisms a : Tv> —> Tv and a : fw — given by a ( f ) = (V = a(V°) D a ^ 1 ) D - •• D a ( V M ) = 0) for / = ( V = V ° D V 1 D • — D VTH = 0) and a(x,f)
= (ax, a / ) .
It
follows that a*Lu = LU'. From this we see that a* takes Qv into itself and TV into itself. Applying the definitions in 11.1.2, 11.1.5 to the linear categories Q v , ? V and to the periodic functor a* on them, we obtain the linear categories Q v , ^ V and the O-modules /C(Qv), JC(Pv). 12.1.3. We shall use the notation O' = 0{v,v~l] (v is an indeterminate). Now K(Qv) is naturally an O'-module by vn[B, ] = [B', '], where (B', ') is obtained from (B,) by applying the shift [n]. We define O'-linear maps fC(Qv)
O'
K(Vw)
IC(Qv)
as follows: p' sends a symbol [B,] (where B G Vv) to [2?,$]; p sends v~n[HnB, Hn()]. It is easy to a symbol [B,] (where B G Qv) to verify that p, p' respect the defining relations; hence they are well-defined. It is also clear that pp' = 1. / If K G Qv and we are given an isomorphism <j> : a* K —*• K, then induces isomorphisms (j)n : a*(HnK[—n]) = HnK[—n] for each n; moreover we have (K,4>) = ®n(HnK[—n\,(f)n) as objects of Qv- This follows from
94
12. Quivers
with
Automorphisms
8.1.3(a) by repeated applications of 11.1.3. This shows that p'p — 1. Thus, p' is an isomorphism G OB such that (B,]EK{Qv)).
12.1.4. Now let T , W be two objects of V a . We can form E T , E W and their product ET X EW- This has an action of GT X GW (product of actions as in 9.1.2) and an automorphism a (product of the automorphisms a of the factors) such that an = 1. The functor a* : £ > ( E T X E W ) —*> T>(ET X E W ) takes the subcategories VT,W and QT,W into themselves. Applying the definitions in 11.1.2, 11.1.5 to these linear categories and to a*, we obtain linear categories VT,W, QT,W and (9-modules JC(VT,W)I)C(QT,W)These are special cases of the definitions of Vv, Qv and K(Vv), £(Qv) in 12.1.2; indeed, T x W and E T x E W are special cases of V and E v where our graph has been replaced by the disjoint union of two copies of itself. Thus, JC(QT,W) is naturally a O'-module and we have CT o £(7>t,w) = K(Qt,W) and ®bO'
®O
Ob=JC(QT,W)
where the sum is over a set of representatives B for the isomorphism classes of simple perverse sheaves in VT,W such that a*B = B, OB is a free Omodule of rank 1 defined as in 11.1.7 (to G OB such that (B,(f>) G VT,W corresponds [B,<j>] G £ ( Q T , W ) ) From the definitions it is clear that any simple object B G "PT.W IS the external tensor product B' B" of two simple objects B' G VT and B" G Vw (and conversely); we have a*B = B if and only if a*B' ^ B' and a*B" ^ B" and then Horn(a*B,B) = Hom(a + B / , B') ® Hom(a*£", B") and OB = OB' ®O OBIt follows that £ ( Q T , W ) = K ( Q T ) ®O> / C ( Q W ) .
12.2. Inner Product
95
1 2 . 1 . 5 . Assume now that W is an I-graded, a-stable subspace of V and that T = V / W with the induced grading and a-action. The functors Indip W : QT,W —5- Qv and Res^ w : Qv —'• QT,W are linear and compatible with a*; hence by 11.1.6, we have induced linear functors QT,W —» Qv, Qv —• QT,W and CMinear maps
ind^W
:
£(QT,W)
fc(Qv)
and
res^W
:
£(Qv)
—>' £(QT,W)-
These last two maps are in fact O'-linear since Ind^ w commute with shifts.
and R e s ^ w
1 2 . 2 . INNER PRODUCT
12.2.1. Let (#', B" induce isomorphisms 4>' : a*B' B' and 4>" : a*B" B" where B' = t^B'.B" = t\,s*B" are semisimple complexes in G v \ ( T x E v ) defined as in 8.1.9. Now ft j>" : a*(B' B") -> {& B") induces an isomorphism of Hj+2dim^r)(rn(B'®B")) onto itself, or equivalently, an isomorphism of D j ( E v , Gv; B\
B")
onto itself, denoted again by a. Here u denotes the map of G v \ ( r x E v ) into the point. It is clear that a : D j ( E v , Gw\B', B") -»• D j ( E v , G v ; B', B") satisfies a n = 1. We define {(B',4>'),(B",")}
= £ j
tr
( a - D j ( E v , G v ; B ' , B " ) ) f - J ' € 0((i;)).
12. Quivers with
96
Automorphisms
The last inclusion follows from 8.1.10(e). (In general, given a ring R, a n € R such that an — 0 for the ring of power series E n e z n 0) is denoted R((v)) (resp. /2[[v]]; or R((v~i)); or R[[v~'}]).) If (B', (f>f) is traceless, then we can write B' = C ® a*C © • • • © (where t > 2) and (pf acts like a matrix without diagonal terms. We have a corresponding decomposition Di(Ev,Gv;B/,B") = Dj(Ev,
C, B") © Dj(Ev, GV; a*C, B") © • • •
©Dj(E w . G w ' ^ - ^ C . B " ) Then a : r>j(EVjGv; B', B") D j ( E v , Gv; B") acts with respect to the above decomposition like a matrix without diagonal terms; hence it has trace zero. Thus, {(£', '), (B", ")} = 0 if either (£',') or {B",(f>") is traceless. It follows immediately that {[B',4>% \B\4,"}} =
(B",f)}
is a well-defined symmetric CX-bilinear pairing JC(Qv) x/C(Qv)x - 0 ( ( v ) ) . We can define in the same way a symmetric O'-bilinear pairing {, } : AC(QT,W) X £ ( Q T , W ) X
0({v));
indeed, as we have seen, /C(QT,W) is a special case of /C(Qv) in the case where our graph has been replaced by the disjoint union of two copies of itself. Using the definitions and 8.1.10(f), we see that {^i (8) A3, A2 ® A4} = for any A\,A2
6 JC(QT)
Ai
and AZ,Aa
e
£(Qw)- We then have
A3, A2®A4e
We have the following result.
Aa)
/C(QT,W)-
12.3. Properties of Lv
L e m m a 12.2.2. Let a G
£(QT)w),/? G K(QV).
97
We have
{a, re$T )W (/?)} = {«™J^ w (a),/3}. Let (A, 0) be an object of QT,W and let (£?, 0') be an object of Q Y We take the trace of the automorphism a in both sides of 9.2.10(a); the lemma follows. 1 2 . 3 . P R O P E R T I E S OF LU
12.3.1. Let XA be the set of all V = (i/ 1 , z/ 2 ,..., VM) G X such that vi e N [j]a for all I L e t u = (j/^ I/ 2 ) . . . ^ " i j e r be such that dim Vi = ^i for all i G I. We have natural automorphisms a : Tv —> Tv and a : Tv —> Tv defined as in 12.1.2. The obvious isomorphism a*l = 1 on T v induces an isomorphism 00 : a*{nv)\l = (7^)1 (o* 1) = (nu)\l in P ( E V ) . It is easy to see that (L„,0o) is an object of Qv- Applying to 0o a shift by / ( f ) (see 9.1.3(c)), we obtain an isomorphism a*Lu = Lu which is denoted again by 0o- Note that (Z/^,00) is an object of QvLet W C V and T = V / W be as in 12.1.5. Let i / = (i/ 1 , i / 2 , . . . , i / m ) G a A* A' and i/" = ( i / / ; 1 , i / / 2 , . . . , i / , / n ) G A"1 be such that d i m T j = dim W i = for all i G I. L e m m a 12.3.2. The following equality holds in /C(Qv)-' ind% W([LU>,(j>0] 0 [!/„",0O]) = [Lu*„»,<J. This follows from 9.2.7. L e m m a 12.3.3. Let v = (i/ 1 ,!/ 2 ,... ,i/ m ) G A"1 be such that dimVi = I// for all i G I. The following equality holds in J C ( Q T , W ) - ' re%iV,[LuM =
<S>
where the sum is taken over all r = ( t ^ t 2 , . . . , r m ) , u ; = (a;1, a ; 2 , . . . ,cj m ) in XA such that dimTi = dim W i = J^i for all i and T1 +ul = vl for
all I; M'(T,U)
is as in
9.2.11.
This follows from the decomposition 9.2.11. The terms in that decomposition corresponding to T,W with some coordinate not in N[I] a can be grouped according to non-trivial a-orbits and contribute traceless objects, hence they disappear from the final result.
98
12. Quivers with Automorphisms
L e m m a 12.3.4. Let i be an a-orbit on I and let 7 = ]T\ 6 i i € N[I]. Let V e Va be such that dimVi = n for all i € i and d i m V j = 0 for all i £ i. Let d be the number of elements of i. The following equality holds in n-1
(a)
[ Z 7 ) ( n _ l h , fa] = J 2 v ~ 2 8 d [ ^ 5=0
M
Using the definitions and the known structure of the cohomology of a product of d copies of a projective (n — l)-space, we see that the left hand side of (a) is represented by ( 0 1 [—2si — 2s2 — • • • — 2Sd],<j>), a sum over all sequences (s 1,32,... ,3d) with 0 < Sj < n — 1; here (f> maps the summand corresponding to («i, s2,..., Sd) to the summand corresponding to (S2, S 3 , . . . , Sd, s\). The summands corresponding to sequences whose terms are not all equal to each other, form traceless objects. The remaining terms give the right hand side of (a). 12.3.5. R e m a r k . In the previous lemma we have [L 7 ) ( n _i) 7 , o] = d n 1 hence v ( - )[Z 7 > ( n _ 1 ) 7 ,^ 0 ] and [Lni,0] = [ L n 7 > M n-1
(a)
[i-r,(n-i)7. -Ao] =
£
v~2sd[Lny,
0O],
s=0
L e m m a 12.3.6. We preserve the assumptions of Lemma 12.3.4, take n — 1. We have
an
d we
00
(a)
{[L 7 ,0o], [£-,,(>]} = Y , v 2 d S = 0- - " V s=0
Since i is discrete, we have E v = 0 and L 7 = 1. Using the definitions we see that the left hand side of (a) is computed as follows: we consider the product of d copies of an infinite projective space with the automorphism given by cyclically permuting the factors; we must find the trace of the induced automorphism on each cohomology space. This is given by the same computation as in the previous lemma. L e m m a 12.3.7. Let v = (1/1, i / 2 , . . . , i/ m ) and v' = (i/ 1 , i / 2 , . . . , v'n) be two elements of Xa such that dimVi = v\ = Yli v'\ for aM i € I- We have {[Lu,(j>o],[L^,o}} G Z((v)) n Q(v). Using 9.1.4, we see that we may assume that each vl and each v'1 is some a-orbit i in I. Using 12.3.5, we can further of the form N E i e i * assume that we always have N = 1.
99
12.4• Verdier Duality
We argue by induction on m + n. The case where m or n is zero, is trivial. The case where m = n = 1 follows from 12.3.6. Hence we may assume that m or n is > 2. Assume first that n > 2. Let W be an I-graded a-stable subspace of V such that T = V / W satisfies dimTi = v\ for all i € I. Let i/ 1 = (i/ 1 ), i/ 2 = (i/2, i / 3 , . . . , i/ m ). We have
= {[Lvi, 0O]
, 0O], res^wI-^'> <M}
and this is in Z((v)) fl Q(v), by 12.3.3 and the induction hypothesis. We can treat similarly the case where ra > 2. The lemma is proved. 1 2 . 4 . V E R D I E R DUALITY
12.4.1. Let V € V a . Let {B,) be an object of Qv. Recall that (f> : a*B ^ B. Applying Verdier duality, we obtain D() : D(B) ^ D(a*B) = a*(D(B)). The inverse of this isomorphism is an isomorphism D{(f>)~1 : a*(D(B)) D{B). It is clear that (D(B), D(<j))~l) is an object of Qv. It is easy to see that we have a well-defined homomorphism of abelian groups D : K(Qv) £ ( Q v ) given by D[B,] = [D(B), D()~1]. This homomorphism has square equal to 1. Moreover, from the definitions, we see that it is semi-linear with respect to the involution : O' —• O' given n n 1 n and C -»• C" for C such that C = 1. by v v~ Let W C V and T = V / W be as in 12.1.5. We have a homomor£ ( Q T , W ) defined in the same phism of abelian groups D : )C(QT,W) way as D : K,(Qv) —• JC(Qv) (and in fact is a special case of it). From the definition we see that the following diagram is commutative: 12.4.2.
JC(QT)
® £(QW)
I
£(QT,W)
JC(QT)®JC(QW)
1
d •
£(QT,W)
where the tensor products are over O' and the vertical maps are the isomorphisms in 12.1.4. Lemma 1 2 . 4 . 3 . For any a mcQpW(Z)Q) in K,(Qv). This follows from 9.2.5.
G £(QT,W);
we have D(incE£w(a))
=
100
12. Quivers with Automorphisms
1 2 . 5 . SELF-DUAL ELEMENTS
L e m m a 12.5.1. Let B G Vv be a simple object such that a*B = B. (a) If i G I and n € N are such that B G Vv,{i},ni> then B G Vv,i\ni where i is the a-orbit of i and 7 = X^'ei*' £ N[I]. (b) Assume that B G Vv-,i-,y where i is as in (a) and 7' G N[I] has support contained in i. Then 7' = 717 /or some n > 0, where 7 is as in (a). (c) / / V / 0 , then there exists an a-orbit i in I such that B G w/iere 7 is as in (a,), /or some n > 1.
Vv-,i-,n7
We prove (a). Our statement is independent of the orientation, as in the proof of 10.3.2. Hence we may assume that the orientation is such that h' £ i for any h G H. Such an orientation exists since i is discrete. By our assumption and by Lemma 9.3.5 we have
dim Vi/( Y ,
MVh'))>n
heH:h"=i
for all x in the support of B and dim V | / (
Y
*h(Vh,))
= n
heH:h"=i
for some x in the support of B. Since the support of B is a-invariant, it follows that the same holds when i is replaced by any i' in i. Using Lemma 9.3.5 again, we see that the conclusion of (a) holds. We prove (b). Let i G i. We can find n G N such that B G 7V;{i);ni- By (a), we have B G Vv,i;n-y By 9.3.5, the inclusions B G Vv,i-,n-y,B G Vv,iyy' imply 7' = 717 and (b) follows. (c) follows immediately from (a) and 9.3.1(b). Proposition 12.5.2. Let B be a simple object of Vv such that a*B is isomorphic to B. (a) There exists an isomorphism <j> : a*B = B such that (B,(j>) G Vv and such that (D(B), D(0)_1) is isomorphic to (B,) as objects ofVv Moreover, (f> is unique, if n is odd, and unique up to multiplication by ±1, if n is even. (b) We have D{B) = B as objects ofVv (b) clearly follows from (a). We prove the existence of <j> in (a). This is trivial when V = 0. Hence we may assume that V ^ 0. By 12.5.1, there
12.5. Self-Dual Elements
101
exists an a-orbit i on I and an integer n > 1 such that B G Pv^rry where 7 = i. Hence it is enough to prove the following statement for any fixed a-orbit i in I. For any n > 1, and any simple object B G Vv,i-,n-y such that a*B = B, there exists 0 : a*B = B such that (B,<J>) G Vv and such that (.D(B),D(<j>)-1) is isomorphic to (B,). We argue by descending induction on n. (We have n < dimV|, for any i G i). Let S be the set of all simple objects C' G Vv;i-,>n-y (up to isomorphism) such that a*C' = C; for each such C", we denote by fc> some isomorphism a*C' ^ C' such that ( C ' , f c ' ) e Vv- By 12.5.1(b), for each C' G <S, we have C' G "Pv;i;n'7 for some n' > n, hence the induction hypothesis (on n) is applicable. Hence we may assume that f c is such that (.D{C'),D(fc')~l) is isomorphic to (C'Jc) for each C' G S. Let W be an I-graded a-stable subspace of V such that T = V / W satisfies dimTi — n for all i G i and Ti = 0 for all i G I — i. Such W exists since i is an a-orbit. By Proposition 10.3.2 (for I' = i) we can find a simple object A of V\v,i,o such that Ind^f = B © (©jCjbl) where Cj G Vv ;i;>n7 for all j. Since Ind^ w commutes with a* and a*B — B, we have also Ind^f w a M = B(B(®ja*Cj\j]). By the uniqueness of A in 10.3.2, we must then have a* A = A. By the induction hypothesis (on dim V) we can find an isomorphism 0' : a* A == A such that (A, (j)') G Vw and such that ( J D(^), J D(0 / )~ 1 ) is isomorphic to {A,'). Applying to [A, (f)'} the homomorphism i n d £ w : /C(Qw) = /C(QT>w) -> £(Qv), we necessarily obtain an element of the form [£, 0] + S e e s Pc[C\ fc] where <j> : a*B = B is some isomorphism such that (B,(J>) G Vv- The coefficients P c are in O'. Since D[A,'] = [A,'\, we have D(ind%wA) = i n d ^ w , 4 (see 12.4.3). Hence we have £>[£, 0] + ^
Pc>D[C', fc] = [B, 0] + Y i
(Pc as in 12.4.1). Substituting here D[C', fc>] = [C'Jc], D[B,4>] = \B10] + Y(pc> C'
~
we obtain
Pc)[C'Jc]-
Now the elements [B,0] and [C, fc] form a subset of a basis of the (9'-module JC(Qv) and D[B, 0] is a scalar multiple of some element in that
12. Quivers with Automorphisms
102
basis. Hence the previous equality forces D[B, ] to be equal to [B, 4>}. This proves the existence of . Assume that has the same property as <j>, where ( satisfies Cn = 1. We have C[B,4>] = [B,C4>] = D[B,C4>] = D(C[B,(f>}) = ClD[B,] = Cl[B,Y, hence C = C _ 1 and £ — ±1. If n is odd, it follows that £ = 1. The proposition is proved. Lemma 12.5.3. Let (B,), (£',') be objects ofVv simple objects of Vv.
such that B,B'
are
(a) If B, B' are not isomorphic, then {[B,], [#',']} G vO[[v\\. (b) If(B',') = (D(B),D((p)~1),
then {{B, <J>], [Bf, ')} el + vO[[v]].
(c) If (B, (p) is as in 12.5.2, then {[B, ], [B, o] for various u = (i/1, v2,..., i/ m ) G Xa such that dimVi = A f o r a11 i € I. 12.6.1.
Lemma 12.6.2. For any a, a' € Mv we have {a, a'} G Z((v)) D Q(v). This follows immediately from 12.3.7. Proposition 12.6.3. The following two A-submodules ofJC(Qv) coincide: (a) M v ; (b) the A-submodule generated by the elements [B, (/>] where (B, (f>) is as in 12.5.2. We show by induction on dim V that (c) if (B,) is as in 12.5.2, then [B,(f>] G Mv. This is trivial when V = 0. Hence we may assume that V ^ 0. By 12.5.1, there exists an a-orbit i on I and an integer n > 1 such that B G ? v ; t w where 7 = J 2 i e i i. Hence it is enough to prove the following statement for any fixed a-orbit i in I. For any n > 1, and any (B,<j>) as in 12.5.2 such that B G TV;*;n7> w e have [B, 4>] G Mv-
12.6. Lv as Additive Generators
103
We argue by descending induction on n. We have n < dimVi, for any i 6 i. Let W be an I-graded a-stable subspace of V such that T = V / W satisfies dim Ti = n for all i G i and T | = 0 for all i G I — i. (Such W exists since i is an a-orbit.) As in the proof of 12.5.2, we can find a self-dual element (A,') G such that A G Vw,i,o and such that (with the notation there) ind¥ f W [i4,^] = [£,0i] + £ c , e 5 P c 4 < ? ' , / c ' ] where 0! = ±. By 12.5.1(b), for each C' G <S, we have C' G Vv,i-,n'-y for some n' > n; hence (d)
[C",/HGMV,
by the induction hypothesis. Replacing, if necessary $ by —(f)', we can assume that (f>\ = (f>. We now show that, (e) for any integer r < 0, the coefficient of vr in Pc is an integer, for any C' G Assume that this is not so. Then we can find an integer r < 0 which is as small as possible with the following property: there exists C' G S such that the coefficient c(C') of vr in Pc is in O — Z. For such C", we have
{ind¥iW[i4, '], \C', fc)} = {[B, 4>l [C, fc]} (f)
+ £ C"es
Pc"{[C",fc»],[C,fc-\}.
By the induction hypothesis, we have [A, <j>'] G M w and by 12.3.2, ind^p W carries M w into My; hence, i n d ^ w [ ^ 4 , G M y . Using now 12.6.2 and (d), we see that the left hand side of (f) is in Z((v)). In particular, the coefficient of vr in the left hand side of (f) is an integer. By 12.5.3(a), we have {[£,0], [C, fc]} G vO[[v]] since B,C' are not isomorphic. This implies that the coefficient of vr in {[B, 0], [C7, fc]} is zero since r < 0. If C" G S is not C' then, by 12.5.3(a) and 12.6.2, we have {[C", f c ] , [C, fc]} e vO[[v]] D Z((v)) = vZ[[v]], hence Pc"{[C", fc"], [Cr,fc]} £ .Pc"^Z[[v]]. Using the minimality of r, we see from this that the coefficient of vr in Pc»{[C", /c"], [C, fc]} is an integer. Similarly, using 12.5.3(c) and 12.6.2, we have {[C, fc], [C, fc]}
e (l + vO[Ml) n Z((v)) = l + vZ[[v]]-
hence Pc>{[C', fc], [C', fc]} G Pc>{l+vZ[[v]]). Using the minimality of r, we see from this that the coefficient of vr in Pc{[C', fc], [C',fc]} is c(C') plus an integer.
12. Quivers with Automorphisms
104
Combining these results, we see that the coefficient of vr in the right hand side of (f) is c(C') plus an integer; on the other hand, we have seen that the coefficient of vr in the left hand side of (f) is an integer. It follows that c{C') is an integer; this is a contradiction. Thus, (e) holds. Next we note that D[A, '] = [A, ft]. Using 12.4.3, we deduce that jD(ind^ w [i4, ']) = ind^ w [;4, ] + Y
C"es
Pc»[C",fc»]
= [B,]+ ]T
C"es
Pc»[C",fc>]
where P c is as in 12.4.1. Comparing coefficients on the two sides, we obtain Pc» = Pc" for all C". Thus, the coefficients of vr and of v~r in P c coincide. Since either r or —r is < 0, we see from (e) that the coefficient of vr in P c is an integer, for any r and any C" G S. In other words, we have Pc G A. In the identity =
Y.
C"es
p
c>[C",fc»],
all terms except [B, \ are in Mv, as we have seen. It follows that so is [B,]. Thus (c) is proved. The remainder of the proof will be similar to that of (c). Consider the set T of all simple objects C G Vv (up to isomorphism) such that a*C = C; for each such C we select fc : a*C = C such that (C,fc) G Vv and D[CJc] = [CJc]- Let a G M v . Using 12.1.3 and 12.5.2, we can write uniquely a = YtCerPclC, fc] with pc G O'. To complete the proof, it remains to show that pc G A for all C. Assume that this is not so. Then we can find an integer ro which is as small as possible with the following property: there exists Co G T such that the coefficient 6(Co) of vr° in pc0 is in O — Z. For such Co we have (g)
{«, [Co, fCo]} = J2 Pc{\C, fc], [Co, fc.]}. cer
Using (c), 12.5.3, and the definition of ro, we see that the coefficient of vr° in the right hand side of (g) is equal to b(Co) plus an integer. Using (c) and 12.6.2, we see that the coefficient of vr° in the left hand side of (g) is
12.6. Lu as Additive Generators
105
an integer. It follows that b(Co) is an integer. This is a contradiction. The proposition is proved. 12.6.4. Let M be a module over a ring R. A subset B of M is said to be a signed basis of M if there exists a basis B' of M such that B = B' U (—B'). Let By/ be the subset of K(Q\j) consisting of all elements [B, 4>] with (B,4>) as in 12.5.2 (if n is even) and of all elements ±[i?, ] with (B, ) as in 12.5.2 (if n is odd). Prom 12.1.3 and 12.5.2 we see that (a) Bv is a signed basis of the O'-module
£(Qv),
and from 12.6.3 we see that (b) By/ is a signed basis of the A-module My/.
13.1.1. We preserve the setup of the previous chapter. Given v E N[I] a , we may regard V /C(Qv) as a functor on the category V® with values in the category of ©'-modules. An isomorphism V = V ' in V® induces an isomorphism E v — E v ' compatible with the a-actions; this induces an isomorphism Qv — Q v which induces an isomorphism /C(Qv) — £ ( Q v ) that is actually independent of the choice of the isomorphism V = V ' by the equivariance properties of the complexes considered. Hence we may take the direct limit lim^ /C(Qv) over the category V®. This direct limit is denoted by o'^w By the previous discussion, the natural homomorphism The signed basis Bv of /C(Q V ) (see 12.6.4) (where V E V®) can be regarded as a signed basis of the ©'-module o'k^, independent of V; we 13.1.2. Let 0 / k = 0,y{o'K) {y runs over N[I] a ). Let B = V1UBU, a signed basis of the ©'-module Q> k. An element x E o'k is said to be homogeneous if it belongs to o'k„ for some v\ we then write |x| = v. The homomorphisms i n d ^ w can be regarded as ©'-linear maps o'krc>' (o'kw) o*ki/, defined whenever E N[I] a satisfy r -f- u = v. They define a multiplication operation, hence they define a structure of ©'-algebra on c>/k. For any u = (i/ 1 ,..., i/m) E Xa, we may regard Lu as Lemma 12.3.3 can be now restated as follows:
Since the elements Lu generate Q>k as a ©'-module (see 12.6.3), it follows that the algebra structure on eyk is associative. One can also see this more 13.1.3. The homomorphisms resip w can be regarded as ©'-linear maps O'k^ - c>'kr®c>/ (o'ka>), defined whenever r,u;, v E N[I] a satisfy r+u = v. By taking direct sums, we obtain an ©'-linear map f : o ' k —> (o'k)-
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_13, © © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010
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13.1. The Algebra o'k
107
13.1.4. We have a symmetric bilinear pairing Z[I] x Z[I] —• Z given by v • v' =
+ Vh"Vh')i h This bilinear form is independent of orientation. Let o'k(8>c>'(c>'k) be the O'-module c>'k®c>' (c>'k) with the (9'-algebra structure given by (x 0 y){x' y') = for x,x',y,y'
® yy'
homogeneous.
L e m m a 13.1.5. f : Q l = ^ l - m for m + 1 < / < ra + n. Hence where the sum is taken over all r ' = ( r ' 1 , . . . , r ' m ) , CJ' = (a;' 1 ,... , a / m ) , r//
=
(r//(m+l)?
1
that r " + uj' = v 771 + 72. We have
)T//(m+n))?
n
^/z
=
(a;//(m+l)j . . .
for 1 < / < ra and r
f(Lu,Lv»)
nl
1
ja
/'(m+»))
l m
+ u" = v" -
i n
^a
s u c h
for ra + 1 < I
u») = J 2 v M ' ( t , " ) l T ® ^u,
where the sum is taken over all r = ( r 1 , . . . , rm+n) and w = (u1,..., cjrn+n) = j/"-™ for in ; t a such that rl + J = vn for 1 < / < m and rl + ra-flk„ —• 0((v)), which is independent of V. This extends to an O'-bilinear pairing {, } : cvk x o>k —> 0((v)) such that for homogeneous x, y, {x,y} is given by the previous pairing if |x| = |y|, and is zero if |x| ^ \y\. 13.1.7. We define a (^'-bilinear pairing {,} on e>'k®o' (o'k) by {x'®x",y'®y"}
=
{x\y,}{x",y"}.
The identity {x,y'y"} for all x,y',y"
= {r{x),y'
®y"}
G o'k, follows immediately from 12.2.2.
13.1.8. The homomorphism D : K(Qv) ~> £ ( Q v ) (where V G VJ) can be regarded as a group homomorphism D : o'ku —> o'ku that has square 1 and is semi-linear with respect to the ring involution : O' —• O' given by vn t—> v~n and £ ^ C - 1 f° r C £ ^ C " = 1- By taking direct sums we obtain D : 'k) as an O'-algebra with
(x 0 y)(x' y') = v^'^xx'
yy'
for x,x',y,y' homogeneous. This should be distinguished from the algebra (c>'k) be the ring isomorc>/kc>'(c>'k). Let D : o'kf&oKo'k) —> phism given by D(x y) = D{x) <S> D(y) for all x, y. Let r : o> k —• o' k o' (c?'k) be the (9'-algebra homomorphism defined as the composition O'k
o'k
o'kc>'(o'k)
o'k®c>' (c>'k).
13.1.10. Let (,) : 0 >k x 0 / k -> 0 ( ( v - 1 ) ) be the O'-bilinear pairing given by (x,y) = {D(x),D(y)}. Here, ~ : 0((v)) —• 0((v~1)) is given b y E n a n V n ^ E n a n V - n (an G O). From 13.1.7, we deduce the identity (a) for all x, y' ,y" G o' k.
{x,y'y")
= {r{x),y'
®y")
13.1. The Algebra o'k
109
13.1.11. From the definition we have (a) D(b) = b for all b G B. We have (b) {&, b'} G vZ[[v\] n Q(v) for any b, V G B such that b' ^ ±b. Indeed, {6,6'} is in vO[[v]] by 12.5.3 and in Z((v)) by 12.6.2 and 12.6.3, and hence in vZ[[v]]. We have (c) {6, b}e 1 + vZ[[v]] n Q(v) for all b G B. Indeed, {b,b} is in 1 + vO[{v\) by 12.5.3 and in Z((v)) by 12.6.2 and 12.6.3, and hence in 1 + vZ[[v]]. From (a),(b),(c) we deduce: (b') (6, b') G v^ZKt;- 1 ]] D Q(v) for any b, b' G B such that b' ^ ±b. (c') (b,b) G 1 + v-lZ[[v-1}}
fl Q(v) for all b G B.
13.1.12. Let i be an a-orbit on I and let 7 = i. For any n > 0, c k n 7 has a distinguished element denoted l n j ; it corresponds to [1,1] G JC(Qv) where V G V" 7 . This element forms a basis of the (9 ; -module o>\i n i . When n = 0, this is independent of i and is denoted simply by 1 G o' ko- Note that (a) 1 is the unit element of the algebra o' k. (b) The elements Lu G c>'k are precisely the elements of o'k which are products of elements of form 1ni for various i, n. Hence the elements 1 ni generate o ' k as an O'-algebra. (for n > 1), where d is (c) We have l i l ( n _ i ) i = ^ " ^ ( E ^ o the number of elements in the orbit i. (See 12.3.4.) From 12.3.6, we have (d) {li, 1 i} = (1 — i> 2 d ) - 1 (where d is as above) or equivalently, since £>(1*) = 1»: (do ( i i I i 4 ) = ( i - « - " ) - » . From (b) and (c) we see that (e) f(li)
= r(l A. 13.2.3.
From 13.2.3 and 13.2.2, it follows that the O'-algebra homomorphism r : o ' k —> o ' k ® o ' (c?'k) (see 13.1.9) restricts to an «4-algebra homomorphism ^ k —> A^QA (^k), denoted again by r. This is an .4-algebra homomorphism if Ak®A (^k) (which is naturally imbedded in o'k(g>o' (o'k)) is given the induced ,4-algebra structure (see 13.1.9). 13.2.4.
13.2.5. The pairing (,) : o ' k x o ' k C>((v -1 )) (see 13.1.10) restricts to an ^-bilinear pairing (,) : ^ k x ^ k Z ( ( v - 1 ) ) fl Q(v) (see 13.1.11(b / ),( c/ ))The equation analogous to 13.1.10(a) continues of course to hold over A. 1 3 . 2 . 6 . Let k be the Q(v)-algebra Q(v) (^k). Note that B is a signed basis of the Q(v)-vector space k. We have a direct sum decomposition k = where \au is the subspace spanned by Bv. From 13.2.1 and 13.1.12(c), we see by induction on n that k is generated as a Q(i>)-algebra by the elements 1{ for the various a-orbits on I.
111 13.
The Algebracvkandk
13.2.7. The homomorphism r in 13.2.4 extends to a Q(u)-algebra homomorphism k —• kQ(v) k (denoted again by r) where k®Q( v ) k is regarded as a Q(v)-algebra by the same rule as in 13.1.9. 13.2.8. The pairing (,) on ^ k extends to a Q(v)-bilinear pairing (,) : k x k —• Q(i>). Prom 13.1.11(b/),(c/), we see that the restriction of this pairing to k u is non-degenerate, for any v. (Its determinant with respect to a basis contained in B u belongs to 1 -I- v - 1 Z[[v - 1 ]] D Q(v) and hence is non-zero.) 13.2.9. Let I be the set of a-orbits on I. We identify Z[/] with the subgroup Z\l]a = {ueZ[l]\u^va{i)
Vi G 1}
of Z[I] by associating to each v G Z[J] the element of Z[I] (denoted again i/), in which the coefficient of i is v^ where i is the a-orbit of i. For i/, v' G Z [I] we define v • v' G Z by regarding z/, v' as elements of Z[I] as above and then computing v-v' according to 13.1.4. (This is a symmetric bilinear form). According to this rule, we have, for i,j G I: i • j = minus the number of h G H such that [h] consists of a point in i and a point in j, if i j and i • i = twice the number of elements in the orbit i. Note that, Hi then —244 G N; indeed, this is the number of h G H such that [/i] consists of a given point in i and some point in j. Hence we have obtained a Cartan datum (/,•)• 13.2.10. Let f be the Q(v)-algebra, defined as in 1.2.5, in terms of the Cartan datum (/, •) just described. Recall that f = ' f / J where 'f is the free associative Q(i>)-algebra on the generators 9i(i G I) and J is a two-sided ideal defined as the radical of a certain symmetric bilinear form (,) on 'f. Let x 'f —^ k be the unique homomorphism of Q(v)-algebras with 1 such that x{Qi) = l i f° r e a c h ® £ IT h e o r e m 13.2.11. x induces an algebra isomorphism f = ' f / J = k. The homomorphism x is surjective, since k is generated by the l i as a Q(v)-algebra (see 13.2.6.) The homomorphism r : 'f —> 'i 0 'f (see 1.2.2) and the homomorphism r : k —• k k (see 13.2.7) make the following diagram commutative: r
k
T
* k k
112
13. The Algebra cvk and k
Indeed, first we note that \ ® X 1S a n algebra homomorphism, since v • v' on Z [I] has been defined in terms of the pairing on Z[I]. Hence the two possible compositions in the diagram are algebra homomorphisms; to check that they are equal, it suffices to do this on the generators But they both take 0; to U 1 + 1 (see 13.1.12(e)). (the right hand side is as in For x,y G 'f, we set ((x,y)) = (x(x),x(y)) 13.2.8). We have (a) ((ft, = (li, li) = (1 - v-"'2)-1 (1i 5 1 , ) = 0, if i ^ j (trivially);
(b)
(see 13.1.12(d')); ((ft,*;))
-
((*,vV')) = ( x ( * ) , x ( y W ) ) = (r(x(x)),x(y')®x(y")) r x = (X ® x)( ( )> x(y') ® x{y")) = («*), 0, there is a unique 1-1 correspondence b b' between an d B' such that O^b = b' plus B{. vn an .4-linear combination of elements in 0Un'>nBi. ,. n)
n
113. The Subsets
of B
121
an Let Mn = En':n'>n ^ d let M'n be the .4-submodule of f generated by Un':n'>nB Bi ; n and a TTitn : ^Bijo —^ a B | . n be the bijections analogous to 7Ti>n : B i ; 0 —> B i ; n and a7ri?n : CTBi;o —> i;n in 14.4.6. Now a : I —• I induces an algebra automorphism a : f —> f which restricts to a bijection a : B —> B whose fixed point set is denoted by B a . The I x {1,2,... }-colored left graph structure on B (as in 14.4.7) defines a I x {1,2,... }-colored graph structure on the subset B a as follows. We say that 6, b' G B a are joined by an edge of color (i, n) if they can be joined in the left graph on B by a sequence of edges of colors (ii, n), (i 2 , n ) , . . . , (i s , n) where ii, i 2 , . . . , i s is an enumeration of the elements of i in some order. This is called the left graph on B a . By replacing "left" by "right" we obtain a I x {1,2,... }-colored graph structure on B a , called the right graph on B a . Theorem 14.4.9. There is a unique bijection \ B —*• B a compatible with the structures of I x {1,2,... }-colored left (resp. right) graphs and such that x(l) = 1. The two bijection corresponding to "left" and "right" coincide.
14- The Signed Basis of f
124
The inverse bijection is obtained geometrically by attaching to a pair (B,) where B is a simple object of a suitable Vv and (j> is an isomorphism a*B = B, the simple object B without specifying . The fact that this bijection is compatible with the colored graph structures is also clear geometrically (using, in particular, 12.5.1(a)). 14.4.10. Remark. This theorem shows that to describe the left or right graph structure on B it is enough to do the same in the case where the Cartan datum is symmetric. Theorem 14.4.11. Assume that the root datum is Y-regular. Let A G X+ and let Ax = f / E i f0 +1 be the U-module defined in 3.5.6. As in 3.5.7, let T) G Aa be the image of 1. Let B(A) = niG/(Un;0,nV~n n€ N
where
fb,b',n
€ N.
The theorem asserts the positivity of certain integers; in our definition in the framework of perverse sheaves, these integers are the dimensions of certain Qj-vector spaces. The theorem follows.
14-5. Examples
125
14.4.14. R e m a r k . For non-symmetric (7,-), the integers in question are not dimensions, but traces of automorphisms of finite order of certain Qjvector spaces and it is not clear whether they are positive or not. 14.4.15. In the case where (7, •) is symmetric, the set B„ is (conjecturally) in natural 1 — 1 correspondence with the set Xu of irreducible components of a certain Lagrangian variety naturally attached to (I,-) and to v (see [9, 13.7]). The union UuXu has a natural colored graph structure (defined as in [8]) and one can hope that the previous bijection respects the colored graph structures. 14.5.
EXAMPLES
14.5.1. Assume that (7, •) is a simply laced Cartan datum of finite type. Let (I, i f , . . . ) be the graph of (7, •) (see 14.1.3); note that 1 = 7. We choose an orientation of this graph. Let V G V and let G v , E v be as in 9.1.2. From the results in [9], it follows that there is a 1-1 correspondence between the set of orbits of G v on E v (a finite set, by Gabriel's theorem) and the set of isomorphism classes of objects of Vv (see 9.1.3): to an orbit of G v corresponds the GV-equivariant simple perverse sheaf whose support is the closure of that orbit. This is well-defined since the action of G v has connected isotropy groups. Assume that (7, •) is such that 7 = {z, j} and i-i = j -j = = j • i = — 2. Then (7, •) is a symmetric Cartan datum of affine type. Let (1,77,...) be the graph of (7, •) (see 14.1.3); note that 1 = 7 and 77 has two elements. We orient this graph so that h! = i for both h G 77. Let V G V and let G V , E V be as in 9.1.2. Note that E v consists of all pairs T, T' of linear maps V; —• V j . Assume that both V* and V j are ndimensional and n > 2. Then Gv acts on E v with infinitely many orbits. Let v = (2n terms). Then ir„ : —> E v (see 9.1.3) is a principal covering with group Sn (the symmetric group) over an open dense subset of Ev- This gives rise to irreducible local systems over an open dense subset of E v , and hence to simple perverse sheaves on E v , indexed by the irreducible representations of Sn. These simple perverse sheaves belong to Vv14.5.2.
14.5.3. Assume that (7, •) is such that 7 = {i} and i-i = 2. The canonical basis B of f consists of the elements (a G N).
126
14- The Signed Basis of f
14.5.4. Assume that (I, •) is such that I = {i,j} and i • i = j • j = 2 and i • j = j • i = — 1. The canonical basis B of f consists of the elements O^O^O^ (a,b,c € N , 6 > a + c) and of the elements c) a) (a,b,c € N , 6 > a + c) with the identification 0i, Ei, F{ are defined in Chapter 16. Chapter 17 contains a proof of a crucial result of Kashiwara on the behaviour of Ei,Fi in a tensor product. Chapters 18 and 19 are concerned with various properties of the canonical basis of AA, in particular with the fact that this basis is almost orthonormal for the natural inner product. Chapter 20 deals with bases at oo (or crystal bases in Kashiwara's terminology). Chapter 21 deals with the special features which hold in the case where the Cartan datum is of finite type. Chapter 22 contains some new positivity results. In the remainder of this book we assume that, unless otherwise specified, a Cartan datum ( / , •) and a root datum (Y, X,...) of type ( / , •) have been fixed. The notation f, U, etc. will refer to these fixed data.
L e m m a 15.1.1. The algebra homomorphism x : 'f —• U given by Oi »—• E[ = (Vi — v~l)K-iEi (i G / ) factors through an algebra homomorphism Let x' ' ; f U be the algebra homomorphism given by 0i »-» Ei /). A simple computation shows that, if / G then
(i G
where TV depends only on v and not on / . Hence if / is a homogeneous element in 1 (so that x ' ( f ) ~ 0), then x ( / ) = 0- The lemma is proved. 15.1.2. Let U+ be the image of x- Using the previous lemma we see that Let U ° be the subalgebra of U generated by the elements for % G I. From the triangular decomposition of U, we can deduce that multiplication U. These defines injective maps U_U°U+ -> U and U+ ® U ° ® U " maps have the same image, which is a subalgebra U of U; this follows from the identity E^Fj = v^FjEt + S{J( 1 - K^) for all i j . Note that the elements K - i which are invertible in U are not invertible in U. The left ideal generated by them in U coincides with the right ideal generated by them in U. The quotient of U by this ideal will be denoted by il. We have obvious algebra homomorphisms U~ —> il and U ^ —> il and it is clear that (a) multiplication defines isomorphisms of vector spaces U~ U"1" = il (b) the algebra il is defined by the generators ei,i (i G I) and the relations e ^ j = vl'i(/>j€i + 6 i j for all together with the relations f(ei) = f{(j>i) = 0 for any homogeneous / = f(9i) G I. Here, e^,^ are the images 15.1.3. There is a unique Q-algebra homomorphism u : il —» il such that u(ei) = Vi(pi, iv(cf)i) = —Vi€i, u(v) = v~l. We will not use it. Note that
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © © Springer Springer Science+Business Science+Business Media, LLC 2010 2011
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15. The Algebra U
131
Lemma 15.1.4. For each i G I we define <j>i : f —• f to be left multiplication by Qi and e^ : f —> f to be the linear map ^r in 1.2.13. (a)
make f into a left ii-module.
(b) Ei : f —> f is locally nilpotent for any i € / . The identity e ^ = v l 'i(frjCi+Sij (as maps f —• f) follows from ir(6jy) = ir(9j)y-\-vjl0j(ir(y)). Let / = / ( f t ) be a homogeneous element of J . Prom the definition we have f(i) = 0 as a linear map f —• f. We must show that f(ei) = 0. Let / ' = a ( f ) G J . From the definition we have (x,€i(y»
= (1 - v~2)(iX,y)
for all x,y G f. It follows that ( x 1 f ( e i ) ( y ) ) = c(f((f>i)(x)1y) where c G Q(v). From f'( f. Thus the relations 15.1.2(b) of il are verified; (a) is proved. If x G then Ci(x) G iu~i if Vi > 1 and €i(x) = 0 if Vi = 0. It follows that €i : f —• f is locally nilpotent. The lemma is proved. L e m m a 15.1.5. There is a unique algebra homomorphism d : It —> It U (vi-Vil)l®K-iEi such that d((f>i) = i be the category whose objects are Q(v)-vector spaces P provided with two Q(f)-linear maps e;,i : P —> P such that €i is locally nilpotent
the morphisms in the category are Q(t>)-linear maps commuting with P be defined as For P G Vi and s G Z, let \a) : P and as 0, if s < 0. From (a) we deduce by induction on N:
if 5 > 0
This is well-defined, since e^ is locally nilpotent. For N > 0, we define a subspace P(N) of P by P(0) = {x G P\ei(x) = 0} and P(N) = [N)P(0). L e m m a 16.1.2. (a) We have e;IIt = 0 for all t> 0. (b) We have Ylt>o
— 1- The sum is well-defined since,
(c) We have a direct sum decomposition P = ®n>oP(N) as a vector space. Moreover, for any N > 0, the map restricts to an isomorphism
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011
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16.2. Admissible Forms
133
and (a) is proved. Now (b) follows immediately from 1.3.4. We prove (c). If x G P, we have by (b): x = fi^x^ where xn = N ( N ~ I Y 2 H N ^ X Y By V ~ w e h a v e XN € -P(O). It remains to show the uniqueness of the x^; it is enough to prove the following statement. If 0 = Y2N>o ifri^BNi where XN G P(0) are zero for all N > Nq (for some N0 > 0),~then xNo = 0. We argue by induction on No. For No = 0 there is nothing to prove. Assume that No > 1. Applying and using 16.1.1(b), we obtain 0 = v N 1 ( ) N 1 x 2AT>O \ ~ ^ t \ ~ ^ N- The induction hypothesis is applicable to this equation and gives XN0 = 0 . This proves (c). (d) follows immediately from (c). 16.1.3. We define linear maps fa, e^ : P —> P by *MN)y)
AND
= 4>iN+1)v
FOR A11
*i(iN)v) =
v E P(O).
L e m m a 16.1.4. Let M G C[ and let x G Ml. (a) We can write uniquely x — X)s s > o s + t > o ' w h e r e xs G ker(£'i : fy[t+2s m) and xs = 0 for large enough s; we can write uniquely x = £s;s>0;s-t>0
E S) x
\
's
wher
e
X
s €
ker(F;
enough s. (b) We have £a;s>0;a+(>0 note either of these sums by Fi(x).
: M t _ 2 s -> M) and x's = 0 for large = £
a ; s
>
0 ; s
_
(
>
0
W e
(c) We have E . , . > o i . + t > o ^ ' " 1 ) « . = £ s ; s > 0 ; 3 - t > 0 ^ + 1 ) < note either of these sums by Ei(x).
We
dede
~
This follows from 5.1.5 (we are reduced by 5.1.4 to the case considered there.) The operators Q(v) is said to be admissible if (a) (x,c -1 )) H Q(t>) of Q((u - 1 )). 16.2.4. Let B be a basis of the Q(v)-vector space P (resp. M). We say that B is integral if (a) the ^4-submodule of P generated by B is stable under Ci,<j)^ : P —> P for all t > 0 (resp. the ,4-submodule ^Af of M generated by B is stable under E^.F^ : M -> M for all* > 0); in the case of M, it is further assumed that B fl Mn is a basis of Mn for all n. Assume that we are given an admissible form (,) and an integral basis B for P (resp. M) which is almost orthonormal (see 14.2.1). Let C{P) = {x e AP\{x,x)
e A}
C(M) = {x e AM\(x,x)
e A}.
and
Lemma 16.2.5. (a) C(P) is a Z[v~ l ]-submodule of AP and B is a basis of it. (b) Let x G aF be such that (x, x) G 1 + v~x A. Then there exists b G B such that x = ±b mod v~lC(P). (c) Let x G aP be such that that (x,x) G v~lA.
Then x G
v~lC(P).
(d) C(M) is an A(Z)-submodule of ^M and B is a basis of it. (e) Let x G be such that (x,x) G 1 + u _ 1 A . Then there exists b G B such that x = ±b mod v~1£(M). (f) Let x G
be such that that (x,x) G v _ 1 A . Then x G
This follows from Lemma 14.2.2.
v~lC(M).
16.2. Admissible Forms
135
Lemma 16.2.6. Let y G aP (resp. y G jM fl Ml with t > 0) be such that Ciy = 0 (resp. Eiy = 0,); let n > 0 (resp. 0 < n < t). We have = *n{y,v) (resp. {F^y.F^y) = Kfav)) where 7rn G 1 + v- 1 Z[[v- 1 ]] (resp. < e 1+ v~1Z[[v~1]]). It suffices to show that
(resp. (F^n+1)y,F^n+1)y) = ^'(F^y, F^y)) where f € l + tr'ZUtT 1 ]] _1 _1 (resp. 7r' e 1 + i> Z[[u ]]) and n > 0 (resp. 0 < n < t). We have Wl n + l ) ?/,0l n + 1 ) !/) = «n + i ] " V i t P y J ^ y ) = (1 - v-'r'ln = (i - v^)-1
+1 l - H ^ e ^ y ) [»+i]r
V(*1B)».
4n)y)-
Similarly, ( F ^ y , F ^ y ) = ([„ + l]"1^"
V
F?+»y)
= v 4 - t + 2 B + , [ - n + t] < [n+ l l r ' ^ t f , ^ ) . It remains to observe that [n + i l r V e i + ^ ' z i b " 1 ] ] for 0 < n and
for 0 < n < t. The lemma follows. Lemma 16.2.7. (a) Let x G aP; write x = J2n>O Vn where y^ = and XN G P(0) are zero for large N (see 16.1.2(c)). Then XN G aP for all N. (b) If x G C(P), then each xn and yx above is in C(P). I f , in addition, (x, x) G 1 + v - 1 A, then there exists No > 0 and b G B such that XN0 = ±b mod v~1C(P) and XN = 0 mod v~1C(P), YN = 0 mod v~lC(P) for all N^N0.
16. Kashiwara's Operators in Rank 1
136
(c) Let x G M* f l ^ A f . We write x = J2s-,s>0;s+t>0 V* where Vs = F^xs and xs G ker(Ei : Mt+2s —> M) are zero for large enough s; then xs G A M for all s. (d) If x G Ml fl C(M), then each xs and ys above is in C(M). I f , in addition, (x, x) G 1 + v~*A, then there exists sq > 0 and b G B such that xSo = ±b mod v~1C(M) and xs = 0 mod v~xC(M), ys = 0 mod v~1C(M) for all s so. ls We prove (a). We have XN = V^N('N~1^2IIN(X). Since stable under €{ and for all t, we see that A P is stable under IIAT : P • P. Hence xn G .4P. This proves (a). Next we show that the subspaces (N^P(0), < ^ ^ ' ^ ( 0 ) are orthogonal to each other under (,), if N ^ N'. We argue by induction on TV + N'. If N > 1, we have for z,z' G P(0) that z,(f>\N ^z') is equal to a scalar N times Ci(j>\ V ) , hence to a scalar times ^N so that it is zero by the induction hypothesis. We treat similarly the case where N' > 1. If N < 0 and Nf < 0, the result is trivial; our assertion follows. Now let x G AP be non-zero. We have (x, x) = (2/iv, 2/iv)- We can l t+1 find t G Z such that v- yN G C(P) for all N and v~ yN (£ C(P) for some N. Then there exist integers a ^ > 0, not all equal to 0 such that v 2t ~ (VN,yN) ~ &N G V - 1 A for all N. Hence
(e) v~2t(x,x)
~J2NaN
£ v~lA
and ^2NaN
> 0.
If x G £ ( P ) , then (e) shows that t < 0; hence yN G C(P) for all N and, using the previous lemma, we see that xn G C(P) for all N. If now x G £ ( P ) satisfies (x,x) G 1 + v _ 1 A , then (e) shows that t = 0 and a^o = 1 for some No and a^ = 0 for all N ^ No. In other words, we have {yNoiVNo) G 1 + v~1A and (vn^Vn) £ v _ 1 A for all N ± No. Using 16.2.6, we deduce that ( x n 0 , x n 0 ) G 1 + v _ 1 A and (x;v,x;v) G v _ 1 A for all N ^ No and the second assertion of (b) follows from 16.2.5. We prove (c). If xs = 0 for all s, then there is nothing to prove. Hence we may assume that xs 0 for some s and we denote by N the largest index such that XN ± 0. We have N > 0, N + t > 0. We argue by induction on N. If N = 0, there is nothing to prove; hence we may assume that Since N > 0. We have E^x = = E\N)aM C AM, we have p " * ' ] .sat G AM. We have [ ^ J " 1 G A, hence xn G a M . Then x' = x — F^xn G AM. The induction hypothesis is applicable to x' and (c) follows.
16.2. Admissible Forms
137
Next we show that (F$N)z, F ^ z ' ) = 0 if N + N' and z,z' are homogeneous elements in the kernel of Ei. We argue by induction on N + N'. If N > 1, we have that (F^N) z,F^N'] z') is equal to a scalar hence to a scalar times (F^'^z, F$N'~l)z) times (F^N~1)z1EiF^N,)zf)i so that it is zero, by the induction hypothesis. We treat similarly the case where N' > 1. If N < 0 and N' < 0, the result is trivial; our assertion follows. The remainder of the proof is entirely similar to that of (b). Lemma 16.2.8. (a) fa,€i : P —• P map C(P) into itself. (b) Fi,Ei : M
M map C(M) into itself.
Let x G C(P). We must show that fax G C(P),€IX G C(P). By 16.2.7, we may assume that x = for some xn as in that lemma. But then G C(P), by 16.2.6. We argue fax = 0, we denote by TN(P) the set of all x G aP such that x = [N)xf for some x' G P(0) D AP with (x',x') = 1 mod v~xA. For any 5, t such that 5 > 0, s + 1 > 0, we denote by T S)t (M) the set of all x G A M such that x = F^x' for some x' G ker (Ei : Mt+2s M)n^M l with (x',x') = 1 mod v~ A. From the definitions we see that (a)
fa(TN(P))cTN+1(P);
( b ) ii(TN(P))
C T
N
^ ( P ) f o r N > 1, Ci(T0(P))
(c) if N > 0, then fa : TN(P) -> TN+1(P) are inverse bijections;
= 0;
and u : TN+1(P)
TN(P)
(d) Fi(T8tt(M)) C T a + i, t _ 2 (M) if s > 0,5 + 1 > 1, and Fi(Ts,t(M)) if 5 > 0,5 + t = 0;
= 0
C rs_1)t+2(M) if s > 1,5 + t > 0, and Ei(Ts>t(M))
=0
(e) Ei(Ta,t(M)) i f s = 0,« > 0 ;
(f) if 5 > 0,s + t > 1, then Fi : TSyt(M) —i• TSit(M) are inverse bijections.
T s + i, t _ 2 (M) and Et :
Lemma 16.2.10. (a) Case of P. We have ±B + v~lC(P)
= UN>0TN(P)
+
Moreover, the sets BN — BC\(TN(P)-\-v~1 C(P)) of B.
v~lC(P). (N > 0) form a partition
16. Kashiwara's Operators in Rank 1
138
(b) Case of M. We have ±B + v~1C(M)
= Us,t-s>o,s+t>oTSjt{M) +
Moreover, the sets BSft = B D (T A , T (M) + v~lC(M)) form a partition of B.
v^CiM). (s > 0,S + t > 0)
By 16.2.6 and 16.2.5, we have TN(P) C ±B + v~lC(P). Conversely, let x G ±B. We have (x,x) G l + v - 1 A . Hence, by 16.2.7, we have x = y' + y" where y" G v~lC(P) and y' = <j)\N)x'' for some x' G P(0) D AP such that x' G ±B 4- v~lC(P) and some N > 0. Thus x G TJV(P) + v~xC{P) and the first assertion of (a) follows. To prove the second assertion of (a), it is enough to show that TNl(P) fl (TN2(P) + v~lC{P)) is empty for NX ^ N2. Assume that (f>\Nl)xi = x2+v~lz where z G C(P) and xx,x2 G P ( 0 ) n ( a P ) satisfy (x\,xi) = 1 mod v _ 1 A and (x2,x 2 ) = 1 mod v - 1 A . By 16.2.7, we can write 2? = £at>o \N)zN where zN G C(P) D P(0). We have \Nl)Xl = + V"1 Eiv>0 4N)*N. This implies, by 16.1.2(c), that x X ZN = 0 for N ^ N I , N 2 , V~ ZNX = X\ and V~ ZN2 = — x2. From the last 2 equality we deduce that ( x 2 , x 2 ) = V~ (ZN2,ZN2) £ v~2A, a contradiction. Thus (a) is proved. The proof of (b) is entirely similar. 16.2.11. Using the previous lemma and the results in 16.2.9, we deduce the following. In the case of P we have: (a)
fc(±BN
+ v~lC(P))
C ±Bn+1
+
v~1£(P);
(b) €i(±BN + v - 1 £ ( P ) ) C ± £ j v - i + v _ 1 £ ( P ) for N > 1, and ii(±B0 v~ C(P)) C v~1C(P);
+
1
±BN+1 + v~lC(P) (c) if N > 0, then fc : ±BN + v~lC(P) -1 _1 €{ : ± B N + 1 + v £ ( P ) —• + v £ ( P ) are inverse bijections.
and
In the case of M we have: (d) Fi(±Ba,t + v~1C(M)) C ± B a + i f t _ 2 + v - 1 £ ( M ) ) if s > 0,5 + t > 1, and Fi(±Ba,t + v~lC(M)) = 0 if s > 0,5 + t = 0; (e) £i(±B s ,t + f _ 1 £ ( M ) ) C 2 + v _ 1 /:(M)) if s > 1, s + t > 0, 1 and Ei(±Ba,t + v~ C(M)) = 0 if 5 = 0, t > 0; ±BS+M_2 (f) if s > 0,s + t > 1, then Ft : ±Ba,t + v~1C(M)) v~ C(M)) and Ei : ± P S + M _ 2 + t r ^ M ) ) ±BSjt + v~1C(M)) inverse bijections. l
+ are
16.3. Adapted Bases
139
1 6 . 3 . ADAPTED BASES
16.3.1. P, M, (,) are as in the previous section. We say that a basis B of P is adapted if there exists a partition B = Un>o-B(n) and bijections ?rn : B(0) B(n) for all n > 0 such that (a) for any N > 0, B(N)UB(N
+ 1)U£(AT + 2 ) U . . . is a basis of (f>\N)P;
(b) for any b G B(0) and any N >0 we have (f>[N)b - <jrN(b) G ^N+1)P. We say that a basis B of M is adapted if there exists a partition B = Us,t;s>o,s+t>oB(s,t) and bijections
for all s, t as above, such that (a) B(s, t ) U B ( s + M ) U B ( a + 2 , t ) U . . . is a basis of Ml n f\s) (b) for any b G 5 ( 0 , 2 s + t), we have
M\
- 7rs>t(b) G i ^ ( s + 1 ) M .
In this section it is assumed that B is integral, almost orthonormal (with respect to (,)) and adapted. Lemma 16.3.2. Let be
B.
(a) Case of P. We have b G Bo if and only ifb£
fa(P).
(b) Case of M. We have b G Ut>o#o,t V and only if b £ FiM. We prove (a). Assume first that b G Bn with N > 0. Then b = <j>^x' + v~lz where z G C(P) and x' G P. Since B is adapted, we can write 4>\N^x' = J2 c b'b' where b' runs over B fl <j>iN^P and Cb> G Q(v). We can also write z = J2b" db"b" where b" runs over B and dy> G Z[v - 1 ]. If b £ (f>[N^P, then by comparing the coefficients of 6, we obtain 1 = v~xdb, Since N a contradiction. Thus, we have b G 4N)p> 0, we have b G fa P. Conversely, assume that b G faP and b G Bo. Then b = x' + v~lz where z G £{P),x' G P(0), and b G <j>mP(0); using the equation N) (P(0), (f>\ P(0)) = 0 for N > 0, we deduce that (x\ b) = 0, hence (6, b) = v~1(z,b) G v~1A, a contradiction. This proves (a). The proof of (b) is entirely similar.
16. Kashiwara's Operators in Rank 1
140
Lemma 16.3.3. (a) Case of P. Let b G Bo,N > 0 and let b' be the unique element of ±B such that f{b) = b' mod v~lC(P). Then b' = npjb. (b) Case of M. Let b G Po,«+2* where s > 0. If s +1 > 0, then there is a unique element b' G ±B such that F?(b) = b' mod v~1C(M) and b' = tr 3ft b. If s + t < 0, then F?(b) = 0 mod v~lC(M). We prove (a). We write b = x + v~xz where z £ C(P) and x G P(0)C\aP satisfies (x,x) = 1 mod v - 1 A . Using 16.2.7, we write z = YIN' Z n ' w ^ e r e zN> G £ { P ) H (T>?z G C(P) and \ z G 4>?b = 4>\N)b mod v~1C(P) + 1 and e^b) = 0 mod v~ C(P) if N = 0. In particular, we have B^ = B(N) for all N. (b) Case of M. Let b E B(s,t). Let b0 E £ ( 0 , 2 s + t) be the unique element such that 7rSjt60 = b. We have Fi(b) = 7r s+ i )t _ 2 6 0 mod v~1C(M) if s 4-1 > 1 and Fi(b) = 0 mod v~lC(M) if s + t = 0. We have Ei(b) = 7r s _ M+2 fco mod v~lC(M) if s> 1 and Ei(b) = 0 mod v~lC(M) if s = 0. In particular, we have BS}t = B(s,t) for all s,t. This follows from 16.3.3.
17.1.
FIRST APPLICATION TO TENSOR PRODUCTS
17.1.1.
In this chapter we shall give three applications of Proposition
17.1.2. Let M, M G C[. Then MM is an object in C[ (see 5.3.1). Now let P eVt and M G C-. We define Q(v)-linear maps : P (g> M P ®M
where Ki : M —> M is the linear map given by Kiy — v™y for y G Mn. It is easy to check that (P(g)M, 6*) is an object of T>i. (This also follows from 15.1.5.) Hence the linear maps (j>i,€i : P M —* P 0 M are well-defined. From the definitions we deduce (using the quantum binomial formula)
for all x G P, y G M and t > 0; the sum is taken over all tr ,t" G N such L e m m a 17.1.3. ( a ) / / ( , ) : P x P Q(v) and (,) : M x M Q(v) are admissible symmetric bilinear forms in the sense of 16.2.2, then the symmetric bilinear form on P M given by (x (g> y, x' y') = (x, x')(y, y') (b) / / ( , ) : M x M -> Q(^) and (,) : M x M -> Q(v) are admissible symmetric bilinear forms in the sense of 16.2.2, then the symmetric bilinear form on M ® M ^wen by (x y, x' ® y') = (x, x')(y, y') zs admissible. We prove (a). Let x,x' € P,y € Mn,y'
G M n \ We have
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011
142
17.2.SecondApplication to Tensor Products
143
On the other hand, we have ()-vector We define Q(v)-linear maps fa, ei : Po —s• Po space with basis /?0, by = [a + l]iA+i for s > 0 and c 0 (with the convention = 0). It is easy to check that this makes Po into an object of Vi. We have # ( & ) = for s > 0,* > 0, and cf(ft) = v ~ t ( t + 1 ) / 2 + s < & _ t , for s > 0,t > 0, (with the convention /3_i = /?_2 = • • = 0). Let (,) be the symmetric bilinear form on Po given by s
t=l It is easy to check that this bilinear form is admissible. 17.1.5. We fix an integer n > 0. Let Mn be the Q(v)-vector space with basis bo,bi,... ,bn with Z-grading such that bm has degree n — 2m. It will be convenient to define bm = 0 for ra > n and for ra < 0. Let Ei, Fi : Mn —> Mn be the linear maps given by Ei(bs) = [ra — s + l]i&s_i and Fi(bs) = for all s. It is easy to check that in this way Mn is an object of C-. Note that for t > 0, we have E^fa) = [n~i+t]fi8-t s and F^fis) = P t T i ^ + t f° r • Let (,) be the bilinear form on Mn given 9{n s) by (bs, ba>) = 6s^v; ~ for 0 < s < n and 0 < s' < n. It is easy to check that (,) is an admissible form on Mn. 17.1.6. Let P 0 be as in 17.1.4 and let Mn be as in 17.1.5 (ra > 0). Then, as in 17.1.2, P = P 0 Mn is a well-defined object of Vi. Note that P has a basis {bSjSt = /3S 6 s /|s > 0,0 < s; < ra}. For any t > 0 we have t' + s t'
r
t"
Vs+t',s'+t"
where the sum is taken over all t', t" 6 N such that t' +1" = t, and U&.,, = «rB+a''+*"1fc.-1,3'
+ vrn+2s'~2{vi
- Ol"
-
+ 1 }ib,,s'-i-
17. Applications
144
By convention, we set bStS> = 0 if either s < 0 or s' < 0 or s' > n. For any m e Z, we define P m to be the Q(v)-subspace of P spanned by the vectors with S + S' = M. We have P = © m P m , P (resp. LI : P —• P ) is an (infinite) linear combination of operators (resp. it follows that (a)
fc{PM)
C P m + 1 and EI(Pm) C P™"1.
17.1.7. Let a*'= XXt(n+t~s'} t=0
5+ t t
Vs+t,s'-t
for s > 0, s' > 0, s + s' < n, n -f t - sr t
-t(s+t) t=0
t>s+t,s'-t
for s > 0, 0 < s' < n, s + s' > n; the two definitions agree if s + s' = n. Note that C*,*' E PS+S'. For s + s' > n, we have (a)
= J2 » R t'=o
, W
-1 - n + s'
Indeed, the right hand side of this equality is, by definition, s' Y^
s'-t' ^
-+"
-t (where 0 < t < s') is n+ t-
E t'+t"=t
t"
[ " - 1 - n + s'" \A
.
We replace the exponent -t"(s + t' + - t' by (n + t - s')t' ( - 1 - n + s')t" + / where / = t(-n - t - s + s' - 1) depends on t', t" only through their sum. Hence the coefficient of bs+ttS'-t is s')t'-(-l-n+s')t"
=
t'+t"=t t- 1 yf = vf8t, o = (with s > 0 and 0 < s' < n) is a basis of P. For ra G Z, let £ m be the Z[t; -1 ]-submodule of P generated by the vectors ba,s> with s + s' = ra. Lemma 17.1.8. (a) For any s > 0 and 0 < s' -bSy8
-t in _t(n + t - s ' ) + st = + Here t > 0; hence 2 t(s + s' — n) — t < 0 ; the inequality becomes an equality only for t = 0. Assume next that s + s' > n. The coefficient of bs+t,S'-t in (s,s' is in
Here t > 0; hence — t(s + t) + t(n — s') = t(n — s — s') — t2 < 0; the inequality becomes an equality only for t = 0. This proves (a). The previous proof also shows that the matrix expressing the vectors Cs,s' in terms of the vectors bs>s' (with s + s' = ra fixed) is upper triangular, with diagonal entries equal to 1 and with off-diagonal entries in v~1Z[v~1]. This implies (b). The lemma is proved. Lemma 17.1.9. The A-submodule j^P of P generated by B is stable under ^ : P —> P for all t > 0. (In other words, the basis B of P is integral) The formulas in 17.1.6 show that Ci(6S)S/) 6 j^P and 0. The lemma follows. Lemma 17.1.10. Assume that 0 < s < n and t > 0. We have ^ K f i = CS,t if s + t < n and i%,0
t-\- S — n
£ u;u>0;u>t—n;u<s+1—n
U
Cs+u,t—u
17. Applications
146
if s + t > n. Assume first that s + t < n. We have 4
Pi Ko =
'if + 5
2^vi
t'
t'= 0
bs+t',t-t' — Ca,t'
Assume now that s + t > n. We have r
d*)h
tf'b.,0
_
V^
=
n-t'(n-t+t')
£
f
t'-,t >0;t't—n
'if + 5' Cs+u,t—u)• i t' t" Since the index t' satisfies t' > t—n and u>t', the index u must satisfy u > t-n. We substitute i " 1 " ? , = ( - i f ft"]* = (-^["t'^i and V7*"('+*')-*"-*'(»-*+*') = v7(n-t+u)t'+(-s-i)t\ — 1 — 71
t — t'
The condition on u implies n — t + u> 0; hence [n~5,+u] . is automatically zero unless n — t + u > t", i.e., if t' > t — n. Thus the condition t' >t — n can be omitted in the summation and we obtain
E E
0 t - n - , u < s + t - n ['"Tl The lemma is proved.
17.2.SecondApplication to Tensor Products
147
Lemma 17.1.11. We consider the partition of B into the subsets B(t)
where t =
= {Ca,t|* + t < n} U { < t + 2 * - » , n - * | « + t >
n}
0,1,2,....
(a) For any t> 0, the set B(t) U 5 ( f + l ) U 5 ( t + 2 ) U - - is a basis of (b) The basis B of P is adapted. Prom 17.1.10, we see that CM G
if s + t ft+"n)P
if s 4-1 > n. (The last inclusion is seen by induction on t.) It follows that (c) B(t) U 5 ( t + 1 ) U 5 ( H 2 ) U - C
4t)p-
Hence X(t) C (frf*P where X(t) is the subspace of P spanned by B(t) U B ( (d) H l ) ^Ub5c( fX + ( t2)) U - - - . We now prove the inclusion for any b G B fl Pm, by induction on ra > 0. Note that B(0) = {Cs,o|0 < s < n) = {fes,o|0 < s < n}. If b G B(0), then (d) follows from 17.1.10. If m = 0, then b = b0y0 G #(0), hence (d) holds. Assume now that ra > 1. If b G B(0), then (d) holds; hence we may assume that b G B — B(0). Then b G X(l) and by (c) we have b = fay where y G Pm~l. By the induction hypothesis we have (f>^+1)y G X(t + 1) C X(t), G X(t). This hence proves (d). Thus (a) is proved. Prom (a) we see that {6 G B\b $ faP} = B{0). Let Trt : B0 -»- Bt be the bijection given by (e) 7TtCs,o = Cs,t if s + t n. From 17.1.10, we see that ,o - ntCs,o G X(t + 1), hence t+1) P. (f) ^ C - , 0 = 7rtC*,o mod 4 The lemma is proved. Lemma 17.1.12. Consider the admissible form (,) on P defined as in 17.1.3, in terms of the admissible forms 17.1.4, 17.1.5, on Mn and Po. Then B is almost orthonormal with respect to (,). From the definition it is clear that the basis (bStS>) of P is almost orthonormal (actually different elements in this basis are orthogonal to each other). Since B is related to this basis as described in 17.1.8, it follows that B is also almost orthonormal.
148
17.
Applications
We now see that the hypotheses of 16.3.5 are verified in our case. Applying Proposition 16.3.5 to B, and taking into account 17.1.11(e),(f), we obtain the following result. Proposition 17.1.13. We have 4>i(Cs,s')
= Cs,s'+1
mod v~1C(P) if s + s' < n,
4>i(Cs,s')
= Cs+i,s'
mod v~1C(P) if s + s' > n,
uiCsy)
= Cs,s'~i
mod v~1C(P) if s + s' 1,
e»(C«,«') =
Cs-i,s'
mod v~xC(P) l
€i(Ca,o) = 0 mod v~ C(P)
if
if s + s' > n, s+x mod v ~ 1 £ ( P ) n P H V + 1 i f s + s' < n, 4>i(bStS') = bs+i>s/ mod v~lC(P) fl PS+S'+L u(bs,s>)
= bs,s>-1
l
m o d v~ C{P)
D P ^ ' "
l
h(ba,s.) = b8-its> mod v~ C(P) fl PS+S'~L
if s + s' > n, 1
i f s +
s ' < n ,
if s + s' > n.
What we actually get are the statements of the corollary with C(P) fl ps+s'±i replaced by C(P). But ba,a* G Ps+s'; hence from 17.1.6(a), 4>i(bs,s>) E PS+A'+L and €I(BS^) G P*+*'-\ The corollary follows. Corollary 17.1.15. Let P eVt, M € C[ and let (P ® M,<j)i,€i) G Vi be defined as in 17.1.2. Let x G P and y G Mn be such that CiX = 0, Eiy = 0. (Then n > 0.) For any m >0, let Cm be the Z[v'^-submodule of P ® M generated by the vectors Vs)x (8) Fls )y with s + s' = m. We set C-\ = 0. We have M(f>[s)x F^y) = (f>[9)x ® Ft(3'+1)y mod t r *£,+.'+1 if s + s' < n; tiiti^x ® F ^ y ) = ii(48)x
® Ft{s,)y) = 4s)x
e i l ^ W ^ ^ ^ "
® F^y Fls'~l)y 1
^ ^
mod v~lCs+s,+1 if s + a' > n; mod v^C^-x 1
mod v~ £s+s'-i
if s + s' < n; i f s + s'> n.
We may identify Po ® Mn with the subspace of P ® M spanned by the vectors (j) \s)x ® F^s )y with s > 0 and 0 < s' < n. It is in fact a subobject in T>i. Therefore the result follows from the previous corollary. 17.2.
SECOND APPLICATION TO TENSOR
PRODUCTS
17.2.1. We consider two integers p > 0 and n > 0 and form the tensor product M = Mp Mn. This is again an object of C[\ hence the operators
17.2. Second Application to Tensor Products
149
Ei,Fi : Mp Mn —> Mp Mn are well-defined. Now Mn has a basis {bs'\0 < sf < n} as in 17.1.5; similarly, Mp has a basis {bs|0 < s < p} as in 17.1.5; Then {bs,s> = bs®bs> |0 < 5 < p , 0 < < n) is a basis of M. As in 17.1.7, we define s+t t
t=o
for 0 < s < p, s' > 0, 5 + s' < n, "ra +
C
=
t=o
t - sr Ds+ty-t t
for 0 < 5 < p, 0 < s' < n, 5 + s' > n; the two definitions agree if 5 + s' = n. The vectors just described form a basis B of the vector space M, which is related to the basis (bSfS>) by a matrix with entries in Z[v - 1 ] whose constant terms form the identity matrix. (This is seen as in 17.1.8 or can be deduced from that lemma, using the natural surjective map P —> M which takes bSiS' to bSfS> if s < p and to zero if s > p; that map also takes Cs,8' to if s < p and to zero if s > p.) Hence the A(Z)-submodule of M, generated by the elements (6S,S')> coincides with the A(Z)-submodule generated by the elements (Ca,s')» w e denote it by C(M). As in 17.1.9, we see that the ^4-submodule of M generated by B is stable under hence B is an integral basis. As in 17.1.12, we see that B is almost orthonormal with respect to the form (,) on M defined as in 17.1.3 in terms of the forms (,) on Mp,Mn (see 17.1.5). As in 17.1.11, we see that the basis B of M is adapted. (Again, this could be deduced from the corresponding result for P.) We now see that the hypotheses of 16.3.5 are verified in our case. Applying Proposition 16.3.5 to J3, we obtain the following result, analogous to 17.1.13. Proposition 17.2.2. We have Fi(ts,sf) = Cs.s'+I
m o d
Pi(Cs,s') = Cs+its'
m
V~1C(M) if s + s' < n;
° d v~1C(M) 1
Fi(Cp,s') = 0 mod v~ C(M)
if s
Ei((s,s') = Ca,s'~i m°d v~lC(M) f
Ei(Cs,s ) = G - m ' mod v 1
_1
Eiits,o) = 0 mod v~ C(M)
if s < p and s + s' > n; sf > n; if s + s' n; if
s mod v~lC(M) l
Ei(ba,o) = 0 mod v~ C(M)
if s < p and s + sf > n;
if
if s + s' 1; if s + s' > n; s 0,n > 0.) For any m > 0, let C be the A(Z)-submodule of M M generated by the vectors We have Fi(F^s)x ® F^y)
= F$a)x ® F^+1)y
mod v~lC if s + s' < n;
Fi(F^s)x ® F^y)
= Ft(3+1)x ® F^y
mod v~lC if s + s'> n;
Ei{Fls)x
F^y)
= F^s)x F^'^y
mod v~lC if s + a' < n;
= F^~l)x ® F^y mod v~lC if s + s' >n. Ei{Fls)x F^y) We may identify Mp Mn with the subspace of M <S> M spanned by ^y with 0 < s < p and 0 < s' < n. It is in fact a the vectors F^x subobject in C[. Therefore the result follows from the previous corollary. 1 7 . 3 . T H E OPERATORS
: f —• f
17.3.1. We shall regard f as a il-module as in 15.1.4. Thus, for each i € / , <j>i : f —• f acts as left multiplication by 0i and €i : f —• f is the linear map ir in 1.2.13. For any i G I, f with the operators i,ei : f —> f is then an object of T>i. (See 16.1.1.) Hence the operators : f —• f (see 16.1.3) are well-defined. Note that the form (,) on f is admissible in the sense of 16.2.2 for any i. 17.3.2. For a fixed i G / , we define a Q(v)-basis B% of f as follows. By definition, B% = U t >oB*(t) where B%(0) is any subset of B^q such that Bi-o = B'(0) U (-£*(())) and, for t > 0, B*(t) is the image of £ f ( 0 ) under TIM : Bi-o ^ Bi-t (see 14.3.2(c)). By definition, we have B = B{U (-B1) and Bl is adapted (in the sense of 16.3.1) to f € Vi. By definition of B, we see that Bl is almost orthonormal for (,) and the ^4-module it generates is ^ f .
17.3. The Operators $iy u : f -»> f
151
17.3.3. Let £ ( f ) = {x e Af\(x>x) G A}. From Theorem 14.2.3 and Lemma 14.2.2, it follows that £ ( f ) is the Z[v -1 ]-submodule of f generated by B\ Lemma 17.3.4. (a) ^f is stable under the operators any i G I.
: f —> f, for
(b) £ ( f ) is stable under the operators fa,ii : f —• f, for any i G I. The stability under is clear from definitions. The stability under e^ follows from 13.2.4. This gives (a). Now (b) follows from Lemma 16.2.8(a) applied to f, (,) and Bl. Applying Proposition 16.3.5 to our case, we see that the following holds. Proposition 17.3.5. Let b G Bl(t). Let bo G Bl(0) be the unique element such that 7riftbo = b. We have fa(b) = 7ri)t+i&o mod v - 1 £ ( f ) . We have ii(b) = 7Tij-ibo mod v _ 1 £ ( f ) if t > 1 and c,(6) = 0 mod v~1C(f) if t = 0. 17.3.6. The following result shows that the endomorphisms of the Zmodule £ ( f ) / v _ 1 £ ( f ) induced by fa, ii act, with respect to the signed basis given by the image of B, in a very simple way, described in terms of the bijections 7Ti>n in 14.3.2(c). Corollary 17.3.7. Let i G I and let b G 0i(b) = 7r i)t+ i6 0 mod
v~lC(f);
(b) €i(b) = 7rM_i&o mod v~lC(f) i f t = 0.
if t > 1 and ii(b) = 0 mod v _ 1 £ ( f )
(c) If i G / and b G B, then we have i(b) = b' mod v - 1 £ ( f ) for a unique b' G B. Moreover, lib' — b mod v~1C(f). (d) If i G / and b G Bi;n for some n > 0, then we have ii(b) = b" mod v~lC(f) for a unique b" G B. Moreover, fab" = b mod v~lC(f). We apply 17.3.5 to b if b G B\ or to -b if -b G B\. This gives (a) and (b). Let b' = 7Ti>n+ibo G B j ; n + W e have fa(b) = b' mod v XC(f) by (a) and e i (6 / ) = b mod v _ 1 £ ( f ) by (b). This proves (c). Assume now that b G Bi]n with n > 0. Let b" = 7TI>N_I6O G Bi-n-1. We have ii(b) = b" mod v~lC{i) by (b) and fa(b") = b mod v~lC{f) by (a). This proves (d).
Study of the Operators Fu Ei on AA
18.1.1. In this chapter we assume that the root datum is ^-regular. Let A E X + . As in 3.5.6, we set A a = f f 0 t < < , A > + 1 . S i n c e A w i l 1 b e fixed in this chapter, we shall write A instead of A\. As in 3.5.7, we denote the Recall that there is a unique U-module structure on A such that EiJ] = 0 for all i E / , K^rj = for all /i E Y, and Fi acts by the map obtained from left multiplication by di on f. From the triangular decomposition for U, we see that A can be naturally identified with the U-module
by the unique isomorphism which makes rj correspond to the image of For any v E N[/], we denote by (A)^ the image of under the canonical map f —> A. We have a direct sum decomposition A = 0i/(A)i/. Note that (A)„ is contained in the (A — v)-weight space A A _ l / (the containment may 18.1.2. By Theorem 14.3.2(b), the subset U i>n;n >(i |A )+i a S»;n of B is a signed basis of the Q(t;)-subspace of f. Hence the natural projection f —> A maps this subset to zero and maps its complement B(A) = ^2G/(un;0 f A which clearly satisfies (a) and (b). We show that it satisfies (c). For x = 1, (c) is trivial. Since the algebra f is generated by the various Oj, it is enough to show that (c) holds for x = Ojx', assuming that it holds for x'. We have E(eix) = E(ei(l)jx') = E(v1^jEix' 4- Sijx') = vimjj€iE(xr) +
6itjE(x')
and
e,(S(x)) = tiiStoj*)) = hence (c) holds for x. This proves the existence of S. The uniqueness of E (assuming only (a),(b)) is clear since f is generated by the Oj as an algebra. 18.1.5. Let £(f) be as in 17.3.3. We have £ ( f ) = ©„£(f)„ (sum over all v G N [ I ] ) where £(f)„ is the Z[v _1 ]-submodule of f generated by Bu. Lemma 18.1.6. (a) IfbeBis not equal to ±1, then there exist i G / and b" G B such that b - fab" G v " 1 ^ ) . (b) If v G N [I] is non-zero, then c(f)„ =
Yl i;ui>0
We prove (a). According to 14.3.3, if 6 is as in (a), then there exist i G / and n > 0 such that b G By 17.3.7, we then have fab" - be v _ 1 £ ( f ) for some b" G B. We prove (b). The sum j r ^ x ) faWfyv-i) 1S a Z[v _1 ]-submodule of C(f)u (by 17.3.4) and the corresponding quotient module is annihilated by v~l (by (a)). By Nakayama's lemma, this quotient is zero; therefore (b) holds. The lemma is proved.
154
18. Study of the Operators Fi, Ei on A \
Proposition 18.1.7. Let v G N [ I ] . (a) £(f)u coincides with the 7i[v~l\-submodule of f generated by the el• * * itl for various sequences ti,Z2,... iw / in which i ements appears exactly Vi times for each i E / . consisting of the images of the el(b) The subset of C(f)u/v~1£(f)l/ 05 ements (f>i2 • • • 1 for various ii,^--above, coincides with l the image of Bu in C{i)u/v~ C{i)u. This follows immediately from the previous lemma. 18.1.8. We will denote by L(A) the A-submodule of A generated by the signed basis {b~r]\b G #(A)} of A. We have a direct sum decomposition L(A) = ©„L(A)„ (v runs over N[/]) where L(A)l/ is the A-submodule of A generated by the elements {b~7]\b G 5(A) D Bu}. We have L{A)U C (A)„. Since A is integrable (see 3.5.6), A belongs to the category C[ for any i G /; hence the operators Ei,Fi : A —> A (see 16.1.4) are well-defined. For any v G N[/], we will denote by L'(A)U the A-submodule of A generated by the elements F ^ F ^ • • • Fi t rj for various sequences . . . , it in I in which i appears exactly Vi times for each i G I. Let L'{A) = L'(A)v C A. We have L'(A)U C (A)^. 1 8 . 2 . A GENERAL HYPOTHESIS AND SOME CONSEQUENCES
Until the end of 18.3.6, we shall make the following General hypothesis 18.2.1. N is a fixed integer > 1 such that, for any v G N[/] with tr v < N, we have (a) L{A)U = L'{A)„; (b) if i is such that Vi > 0, then Fi(x~rj) = (ix)~r) mod v~1L(A), all x G C(f)u-i; (c) if i is such that Vi > 0, then Ei(b~rj) = (€ib)~rj mod v~1L(A), all b G Bu such that b~r) ^ 0; in particular, Ei(L{A)u) c L{A)v-i.
for for
In this section and the next we will derive various consequences of the general hypothesis; we will eventually show that this is not only a hypothesis, but a theorem (see 18.3.8). Lemma 18.2.2. Let v be such that tr v < N, let i G I and let x G L'{A)u. x r w/iere the xr G (A)^-^ satisfy EiXr = 0 for all r Write x = and xr = 0 unless r + (i, A — v) > 0 (see 16.1.4)- Then (a) xr G L'(A)u-ri
for all r.
18.2. A General Hypothesis and Some Consequences
155
(b) I f , in addition, x = b~rj mod v~lL(A) for some b G Bv fl B(X), then there exist ro G [0,1/*] and bo G Bv-rQ% fl B(A) such that xro = b^rj and xr G v~1L'(A)l/-ri for all r ^ romod v~lL'(A)u-roi, Let t be an integer such that 0 < t < Ui and xr = 0 for r > t. We prove (a) by induction on t. If t = 0, then (a) is obvious. Assume now that t > 1. We have E{X = 2 r = o - F ^ W + i a n d xr+1 = 0 unless r + (i, A — v + i') > 0. (If we had simultaneously xr+i 0 and r + (i, A — v + i') < 0 then r + 1 + (i, A — v) < 0, a contradiction.) By the general hypothesis, we have EiX G L(A)„_;. By the induction hypothesis applied to EiX, we have xr G L'(A)„-ri for all r > 0. Hence F^r)xr = F[xr G L'(A)U for all r > 0. Since x G L'(K)U, it follows that x0 G L'(K)U. This proves (a). We prove (b) by induction on t as above. If t = 0, then (b) is obvious. Assume now that t > 1. By the general hypothesis, we have EiX = (iib)~r] to mod v~1L(A). By 17.3.7, we have that e ^ is equal modulo either 0 or to b' for some b' G Bu-i. If the first alternative occurs, or if the second alternative occurs with (ib')~r) = 0, then Ei(b~r)) G v _ 1 L(A); applying (a) to vEi(b~rj) we see that xr G v~1L(A) for all r > 0. We then have xo = b~rj mod v - 1 L(A), as required. Hence we may assume that (c) iib = b' mod v~lC{f),
where b' G Bv-i
fl B{A).
We have, by assumption, (d) EiX = Ei(b~rj) mod v^EiCtf),,
.
By the general hypothesis, we have EiC(i)u C £(f)„_i and Ei(b~rj) = (we have b~r) / 0, by assumption). Introducing (eib)~T] mod v~lL(K)u-i this in (d), and using (c), we obtain EiX = bf~rj
mod
v~1C(f)l/-i.
By the induction hypothesis applied to EiX, we see that there exist v~1L'(A)„-roi, r 0 G [1, Vi] and b0 G Bu-roiC\B(A) such that xro = b^r) mod l for all r such that r > 0 and r ^ ro- It follows that and xr G v~ L'(k)u-ri ^ x = Fl°~lxro
mod
v~lL{A).
By the general hypothesis, we have FiEiX = FiEi{b~r,) = Fi{{eib)~n) = (^€i(C(f) © L'(A)V) c £(f) © L'{A) and Ci^flOL'lAJJcAflOi'tA). By Lemmas 18.2.2(a) and 18.2.3, the A-module £ ( f ) © L'(A)U is generated by elements cjy^x x' where x G £ ( f ) and x' G L'{A)v-.a'i satisfy €i{x) = 0 and Ei(x') — 0. The image of such elements under fc or li is contained in £ ( f ) © L'{A) by Corollary 17.1.15. The lemma follows. Lemma 18.2.6. Let x G L'{A)U where tr v < N. Assume that there exists b G Bu fl B{A) such that x = b~r] mod v~lL'{A). Assume also that Fix £v~lL'(A). Then fc(l ® x) = 1 ® Fix mod v~lC(f) © L'(A). By 18.2.2, we may assume that x = F^xf where x' G L'{A)U-Si satisfies Eix' = 0. Since x' / 0 and Eix' = 0, we have n = (i, A - i/ + si') G N; moreover, F^n+1)xf = 0. By Corollary 17.1.15, n). If the second alternative occurs, then Fix = F*+1x' = 0, contradicting our assumptions. Thus the first alternative occurs and the lemma is proved.
18.2. A General Hypothesis and Some Consequences
157
Lemma 18.2.7. For any v such that tr v < N, we have H(£(f)„) C £ ( f ) © L'(A). We argue by induction on tr v. If v = 0, the result is obvious. Assume that v ^ 0 and that the result is known when v is replaced by v' with tr v' < tr v. By 18.1.6(b), the Z[v -1 ]-module C(f)v is spanned by vectors fax with i e I and x G £(f)„_i, so it suffices to show that for such i, x, we have E(fax) G £(f)©Z/(A). Since E is a morphism in Vi, we have E(fax) = fa(E(x)). By the induction hypothesis, we have E(x) G £ ( f ) © L'(A). Prom the definition of E we see immediately that E(f„')c
I/"; tr u"< tr u'
f ® tr„77.
Combining this with the previous inclusion, we see that E(x)e v"\
tr
£(f)©£'(A)„". u"f*z © V where z G £ ( f ) (homogeneous) and z' G L'(A)U< satisfy €i(z) = 0, Ei(z') = 0 and a, a' G N. We may assume that z' ^ 0. Let n be the smallest integer > 0 such that F™+lz' = 0. By Corollary 17.1.15, we have that 4>i{y) is equal to (c) 4a+1)z
modulo t r 1 / ^ ) © L'{A), if a + a' > n, and to
<S> F^z'
(d) \a)z © F^a'+1)z'
modulo v^Ctf)
© L'(A), if a + a' < n.
If a > 0 or ^ ^ fo, then y and both vectors (c),(d) are in the kernel of pr, by the definition of pr; on the other hand, by (a), we have pr{v~lC{f) © Z/(A)) C v~lL'{A). Hence in this case the lemma holds for y. Hence we may assume that a = 0 and z = 1. We then h a v e p r ( y ) = F±a moreover, by the previous argument: MMv)) = F^'+l\z')
mod v ^ L ' i A )
if a! i(y)) = 0
mod
v~1L,(A)
if a' > n. On the other hand, by the definition of Fi, we have Fi{pr(y)) = f l ( j t f a V ) ) = = 0 if a! > n (by the definition of It remains to observe that F^a'+1)(zf) n). The lemma is proved. L e m m a 18.2.10. Let x G C(i)u Fi(x~rj) mod v~lL'(A).
with
trvix)~rj =
Using 18.2.8(a) and the commutation of H with
we have
(•4>ix)~r] = pr(E(4>ix)) = pr(i(E(x))). Using again 18.2.8(a), we have
Fi(x~n) = #4(pr(S(s))). It remains to show that pr(ME(x)))
= Fi(pr(E(x)))
mod
v~lL'(A).
This follows from Lemma 1 8 . 2 . 9 ( b ) applied to y = E(x). (We have E(x) G f C(i) © L'{A) b y 18.2.7, and E(x) G tr „"< tr „ ® hence E(x)e
Y1 v"; tr
C(f)Q>L'(A)^ u" 0 and any x G £(f)»/-». By 18.2.10, we have (fax)~r] = Fi(x~rj) mod v~1L'(A)u. Hence it suffices to show that Fi(x~r)) G 2/(A)„. By the definition of L(A)„_j, we have x~r) G L(A)u„i. Using our general hypothesis, we deduce that x~rj G L'(A)u-i. It remains to observe that Fi(L'(A)u-i) C L'(A)U (from the definitions). The lemma is proved. Lemma 18.2.12. If tr v = N, we have L'(A)„ C L{A)u +
v~lL'{A)u.
We have L\A)u
= J2
=
i\Vi>0
=
E i\Vi>0
Afi(£(f)^). i;j/»>0
The first and third equalities are by definition; the second one follows from our general hypothesis. Hence it suffices to show that Fi(x-r))eL(A)u+v-lL'{A)„ for all x G £ ( f ) „ _ ; (where Vi > 0 ) .
By 18.2.10, we have Fi{x~r]) = (4>ix)~r) mod v~lL'{A)u. On the other hand, we have fax G C(f)u (see 17.3.4); hence we have ( f a x ) ~ r j G L(A)u. The lemma is proved. Lemma 18.2.13. If tr v = N, we have L(A)U = L'{A)„. By 18.2.11, L(A)U is an A-submodule of L'(A)^. The corresponding quotient module is annihilated by v~l, see Lemma 18.2.12. This quotient is then zero by Nakayama's lemma. The lemma is proved. 1 8 . 3 . FURTHER CONSEQUENCES OF THE GENERAL HYPOTHESIS
Lemma 18.3.1. Let y = F^F^ • • • Fitr) G A where i\ + + • • • + it = v and t = tr v < N. Let x = fatfa2 • • • fat(l rj) G f A. Assume that A). Thenx = l®y mod v~l£(f) © L'(A). yiv~lL'{ We argue by induction on t. If t = 0, there is nothing to prove. Assume now that t > 0 and that the result is known for t — 1.
18. Study of the Operators Fi, Ei on A \
160
Let y' = Fi2 - - • Fitrj G A and let x' = • • • it(l © rj) G f © A. Since F ^ t T 1 ! / ^ ) ) C u~ 1 L / (A), and y ix(\ © y') = 1 © F ^ ' = 1 © 2/ mod v _ 1 £ ( f ) © Z/(A). (That lemma is applicable since F^y' = y £ v~1L'(A).) Hence x = 1 © y mod v~~lC(f) © L'(A). The lemma is proved. L e m m a 18.3.2. If tr v < AT, then Ei{L(A)u)
C Z/(A).
We will prove, for any n > 0, that Ei(L(A)v) C vnL'(A), by descending induction on n. This is obvious for n large since L(A)„ is a finitely generated A-module. Hence it is enough to prove the following statement. (a) Assume that n > 1 and Ei(L(A)„) C vnL'{A); then F»(L(A)„) C v L'(A). n_1
We first show that (b) e~i(£(fV © L(A)„») C v n £ ( f ) © Z/(A) provided that 1/ -+ v" = 1/. In the case where tr v" < AT, this follows from 18.2.5. Assume now that tr v" = N; then 1/ = 0. It suffices to show that €i( 1 © x) G v n £ ( f ) © Z/(A) for any a; G L{A)U. We write z = Ylr>0where xr = 0 unless r + (z,A — f ) > 0 and EiXr = 0. By the assumption of (a), we have £ r > i i^ ( r _ 1 ) rc r G vnL'{A) and by 18.2.2, we deduce that G vnL'(A) for r > 1, or equivalently, F[~1xr G vnL'(A) for r > 1. Using the general hypothesis (r — 1) times, we have E l ~ \ F r l x r ) C v n L'(A); hence xr G vnL'(A) for all r > 1. We have e/(l © ar) = J2r>i ® since e ^ l © x0) = 0. By 17.1.15, this -1 belongs to the Z[v ]-submodule generated by the elements ®F^xr with r > 1 and ri + 7*2 = r — 1 and these elements belong to v n £(f)©Z/(A). Thus (b) is proved. To prove (a), it suffices to show that Ei(y) G vn~1L'(A) for all y of the form y = FixFi2 ... Fitr) where ii + i2 H h it = v. If y G v - 1 Z/(A), then our inductive assumption shows that Ei(vy) G vnL'{A); hence Fj(?/) G w n_1 Z/(A), as desired. Thus we may assume that y £ v~1L'(A). Using now Lemma 18.3.1, we see that (c) 4>h4>i2 • • • 4>it (1 © 77) = (1 © y) + v_1z
18.3. Further Consequences of the General Hypothesis
161
where z G £ ( f ) © L'{A). From the definition of the operators fa on f © A we see that we have necessarily
Hence, by (b), we have ii(z) G vnC(f) 0 L'(A). Thus applying ii to (c), we obtain (d) iifa,fa2 • • • fat (1 © vi) = €i{ 1 © y) mod ^ " ^ ( f ) © L'( A). We have (using 18.2.7) hfaxfa2
• • • fat(l ® T]) = hfaxfa2 ••• l7 We have c f (l © ?/) = 2 r > i e i ( l © F / r V ) , since 6^(1 ©2/0) = 0. By 17.1.15, we have for any r > 1, yr plus a linear combination with coefficients in v~ Z[v~ ] of terms © F$ V2) y r where n + r 2 = r - 1. The last linear n l combination is in v ~ C{i) © L'(A) since yr G vnL(A) for r > 1. Taking sum over r > 1 we obtain 1
1
cix(j>i2 • "4>iNl) = 1
m w )
modulo v~ C(f) © Z/(A) (using Lemma 18.2.7). Using Lemma 18.3.4, we have €i( 1 © b~rj) = 1 © Ei{b~iij) mod v _ 1 £ ( f ) © L'(A). We deduce that E(ei(b)) = 1 0, then Ei(b~rj) — (eib)~rj mod v~1L(A), all b G Bu such that b~rj / 0; in particular, Ei(L(K)u) C Prom now on, we shall not distinguish between L(A) and L'(A).
for
CHAPTER
19
Inner Product on A
19.1.
FIRST PROPERTIES OF THE INNER P R O D U C T
In this chapter, we preserve the setup of the previous chapter. In particular, we write A = AA where A G X+ is fixed, except in subsections 19.1.1.
Let p : U —* \Jopp be the algebra isomorphism given by the composition
Proposition
19.1.2.
There is a unique bilinear form (,) : A x A —» Q ( ^ )
This bilinear form is symmetric. If x
G (A)^,?/ G ( A ) ^
with v ^ v', then
For any u G U, we consider the linear map of the dual space A* = Hom(A, Q(f)) into itself, given by £ »—• u(£) where u(£)(x) = £(p(u)x) for all x G A. This defines a U-module structure on A*, since p : U —» \Jopp is an algebra homomorphism. Let £o £ A* be the unique linear form such that £ 0 iv) — 1 a n d £o is zero on (A)^ for v ^ 0. It is clear that E^q = 0 for all t G / and = for all /x G Y. We show that 0 = 0. It is enough to show that, for any x in a weight space of A, the vector E\1'X^1X cannot be a non-zero multiple of 77. This follows from Lemma 5.1.6, since E^r] = 0 and — 0. From the description 18.1.1(a) of
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19.1. First Properties of the Inner Product
165
A, we now see that there is a unique homomorphism of U-modules A —• A* which takes r) to Now it is clear that there is a 1-1 correspondence between homomorphisms of U-modules A —• A* which take rj to £o and bilinear forms (,) on A which satisfy (a) and (b). The existence and uniqueness of the form x, y (x, y) follow. The form x,y >—> (y, x) satisfies the defining properties of the form x,y i—• (x,y), since p2 = 1. By the uniqueness, we see that these two forms coincide; hence (,) is symmetric. P r o p o s i t i o n 19.1.3. Let v G N[/]. (a) We have (L(A)„,L(A)„) € A. (b) For any i G / such that Vi > 0, and any x G L(A)u-i,x' have (Fix,xf) = (x,Eix') modulo v_1A.
G L(A)„ we
When tr v = 0, (a) and (b) are trivial. We may therefore assume that tr v = N > 0 and that both (a),(b) are already known for v' with and Ei(L(A)„) C L(A)„-i tr i/ < N. Since L(A)„ = Zi;i,t>0FiL(A)l,-i, whenever ^ > 0, we see that (a) for v follows from the induction hypothesis. (We use (a) and (b) for tr v' = N - 1.) We now prove (b) for v. Let i,x,x' be as in (b). By 18.2.2, we may assume that x = F^y and x' = F^y' where y G L(A) l / _ i _ s i , y' G L(A) I/ _ s / i , Ety = Eiy' = 0 and s > 0, > 0, s + (i, A - v + i') > 0, s' + (z, A — v) > 0 . We must show that (c) ( F f + 1 ) 3 , , . F f V ) = ( f f ' W - V )
mod f _ 1 A .
For any r, r' we have from the definitions (if V i f V ) = = v'
(y,vfK.riE^F?y) r - r' + (i, A - u + s'i')
This is zero unless r' > r. By symmetry, it is also zero unless r >r'. (Ft)y,Fr,y')
" G 1 + v~1Z[v~1] v.- a 5+ 1 and "2(5 + l ) +
(t,A-i/>'
ei +
v-iziv-1]
and (e) follows. The proposition is proved. Lemma
19.1.4.
Let
6,6'
G
5(A).
(a) Ifb' ^ ±b then (6-77,^-77) G
(b) We have (b~7],b~rj) G 1 + v
_1
v~*A.
A.
We may assume that 6, b' G Bu for some v. We argue by induction on N = tr v. The case where TV = 0 is trivial. Assume now that N > 1. We can find i G I such that i/j > 0 and b\ G Bu—i such that (f>ib\ = b and ub = b\ (both equalities are modulo i ; - 1 £ ( f ) ) . By 18.3.8, we have Ei(b~rj) = 6]"77 and Fi(b^rj) — b~r) (both equalities are modulo v - 1 L(A)). It follows that b^rj / 0. (From 77 = 0 we could deduce by applying Fi that b~rj G v~lL(A) which contradicts b G B(A).) Thus bi G B(A). Using again 18.3.8, we have Ei(b'~rf) = (e.ib')~r) mod v _ 1 L(A).
19.2. Normalization of Signs
167
Using the previous proposition, we have (c) V ) = ( F ^ v l b ' - r , ) = (^77, E ^ - f i ) ) = equalities modulo v _ 1 A . Assume first that b = b'. Then (eib')~r] = b^r) mod v~1L(A) and (c) becomes (b~7],b~r)) = (6^?7,6f?7) mod v~1L(A); by the induction hypothesis, we have (6^77,6^77) G 1 + v _ 1 A so that (6-77,6-77) G 1 + v - 1 A , as required. Assume next that b' ^ ±b. There are two cases: we have either hb' = 62 mod v _ 1 £ ( f ) for some 62 £ B or lib' G v - 1 £ ( f ) . If the second alternative occurs, then (6^77, (eib')~rj) G v~lA by 19.1.3(a); hence (b~7],b'~r]) G v _ 1 A , by (c). If the first alternative occurs, then 62 G B(X) (by the same argument as the one showing that 61 G B(A)) and we have b2 ^ ±61 (if we had b2 = ±61, then by applying fc we would deduce b' = ±b mod v~lC(i)\ hence b' = ±6). Prom (c) we have 77, b'~rj) = (6^77,6^77) mod v~l A and from the induction hypothesis we have (6^77, b2rf) G v~x A. It follows that (b~rj, b'~rf) G f _ 1 A . The lemma is proved. 1 9 . 2 . NORMALIZATION OF SIGNS
19.2.1. Let B„ be as in 14.4.2. From 17.3.7 and 18.1.7, we see that the following two conditions for an element x G f„ are equivalent: (a) xeBv
+
v~lC{f)v
(b) x = fcxfc2 • • • fctl mod v - 1 £ ( f ) l / , for some sequence i\, i2,... , it in I such that i\ + 12 H \~H = f . For the proof of Theorem 14.4.3, we shall need the following result. We regard f A as an f-module with Oi acting as fc (see 18.1.3.) Lemma 19.2.2. (a) Let b G B„ and b' G B„'. The vector fc(b b'~rj) is equal modulo v~1C(f) © L{A) to fc(b) (g> b'~t] or to b Fi(b'~rj). (b) Let bo G The vector &o(l ® vi) is equal modulo v~1C(f) © L(A) to bi ® 77 for some b\ G BUl, b2 G B„2 with v\ + v2 = v. We prove (a). By 18.2.2, (which is now known to be valid unconditionally) there exists ro > 0 such that v[ > ro and y-n = mod v _ 1 L(A) where a;' G L(A)„>_ roi , Eix' = 0, x' ^ 0. By 16.2.7(b), there exists 7*1 > 0 such that 1^ > ri and 6 = mod v - 1 £ ( f ) where a; G £ ( f ) „ _ r i i , eiX = 0 , x ^ 0 . By 18.2.5 (which is now known to hold unconditionally), we have mod v - 1 £ ( f ) © L(A). fc(b ® b'-rj) = fc(iri)x ®
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19. Inner Product on A
By 17.1.15, 4>i{(f)\ri)x © i ^ ( r ° V ) is equal modulo i r ^ f ) © L(A) to © F[°x' or to (jf^x © F / * 0 + V or equivalently to ®b'~r] or b © Fi{b'~rj). This proves (a). We prove (b). From 19.2.1, it follows that
4>r{1+1x
60 = 4>n4>i2
mod i> - 1 £(f),
for some sequence i \ , . . . ,it in I such that i\ +
H
1-it = v. We have
6 0 (1 0r?) = 6 0 E(1) = H(60) = E(4> h fa 2 • • • j> it l) = 4>ix4>i2 • • • <ME(i)) = j>h4>i2
4>it(i ©77).
The third equality is modulo v _ 1 £ ( f ) © L(A). It remains to show that 4>ilfa2 • * • © 7?) is equal to b\ © b^rj and 62 G B„ 2 with v\ + 1/2 = f . mod v~lC{f) © L(A) for some b\ G We show that this holds for any sequence ii, i2,..., it, by induction on t. The case where t = 0 is trivial. We assume that t > 1. Using the induction hypothesis and (a), we have that il 0 such that ^ > n, and b" G B„_ n in#i;o such that b = 7Ti,n&". By 17.3.7, we have A\. The canonical basis B(AA) of AA (see 14.4.11) is clearly an ,4-basis of A^XFor any v G N[/], let (AA\)V be the image of J$ V under the canonical map f —AA- We have a direct sum decomposition A A = ©I/(^AA)I/. Proposition 1 9 . 3 . 2 . ^ A A is stable under the operators x~, x+ : A A —S• AA, for any x G a?• For this is obvious. To prove the assertion about x + , we may assume that x = 6^ for some i,n. Let y G (^AA)^. We show that E^y G by induction on JV = tr v. If N — 0, the result is obvious. Assume that N > 1. We may assume that y = F^y' where 1 < t < Vj and y' G ( ^ A A ) „ - T J . By 3.4.2(b), the operator E \ n ) o n A a is an ^-linear combination of operators F^ ^ E \ n \ By the induction hypothesis, we have y G ^AA; hence F^E^y' G ^A A SO that E\n)y = E^F^y' ^AA- This completes the proof. The following result is a strengthening of Lemma 19.1.4.
G
Proposition 1 9 . 3 . 3 . Let b,b' G B(A). (a) Ifb' / ±b then (b~rj,b'-r)) G
v-lZ[v~l).
(b) We have (6-77,6-77) G 1 + V ^ i T 1 ] . We shall prove by induction on tr v that (c) (x,y) G A for any x,y G (^AA)^. When tr v = 0, (c) is trivial. We may therefore assume that tr v — N > 0 and that (c) is already known for v' with tr v' < N. We may assume that x = F^x' where 0 < r < Vi and x' G (A^x)u-ri- From the definitions we have (d) ( f * V , y ) =
(x>,vfK-riE\r)y).
By 19.3.2, we have K - r i E ^ y G (.aAa)^-™; hence the right hand side of (d) is in A, by the induction hypothesis. This proves (c). The proposition follows by combining (c) with Lemma 19.1.4, since A f l . 4 = Z[t>-1].
19.3. Further Properties of the Inner Product
171
19.3.4. From the description 18.1.1(a) of AA, we see that there is a unique Q-linear isomorphism : A* —j• A* such that urj\ = ur)\ for all u G U. It has square equal to 1. Theorem 19.3.5. Let be A\.
We have b G B(AA) if and only if
(1) b G ^AA, 6 = 6 and
(2) there exists a sequence t'i, i2,..., mod v~1L(A\).
ip in I such that 6 = F^F^ • • • Fiprj\
We have 6 G ±B(AA) if and only if 6 satisfies (1) and v~lZ[v-1].
(3) (6,6) = 1 mod
If 6 G ±B(AA), then 6 obviously satisfies (1); it satisfies (3) by 19.3.3. Assume now that 6 satisfies (1) and (3). Since the canonical basis is almost orthonormal, from Lemma 14.2.2 it follows that there exists b' G B(A a ) such that 6 = ±6' mod v " 1 ! , ^ ) . Since 6 - (±6') = 6 - (±6'), it follows that 6 - (±6') = 0. Assume now that 6 G B(AA). We show that 6 satisfies (2). We have 6 = (3~rjA for some /3 G B. We can find a sequence ii,i2,•..,ip in I such that (3 = fcx4>i2 ••'ip 1 mod v - 1 £ ( f ) . Using 18.3.8, it follows that 6 = FhFi2 • • • Fiprj\ mod v~1L(Ax), as required. Finally, assume that 6 satisfies (1),(2). Let i\, i2,..., ip in I be as in (2). Using again 18.3.8, we see that 6 = • • • fcpl)~r]\ mod V _1L(AA). Let (3 be the unique element of B such that =
mod v - 1 £ ( f ) .
We have 6 = /3~rfx mod v~lL(Ax). Let b' = (3~r)x. Then 6 - b' G ^A a , _1 Y^V = 6 - 6' and 6 - b' G V L(A a ). It follows that 6 - b' = 0. Thus, 6 G B(AA). The theorem is proved. 19.3.6. We will now investigate the relation between the inner product (,) on f (see 1.2.5) and the inner product (,) on AA, which we now denote by (, )A since A G X+ will vary. Proposition 19.3.7. Let x,y G f. When A G X+ tends to oo (in the sense that (i, A) tends to oo for all i), then the inner product (x~r)\,y~r)\)\ G Q{v) converges in Q ( ( v - 1 ) ) to (x,y). We may assume that both x and y belong to f„ for some i/. We prove the proposition by induction on N = tr v. When N = 0, the result is
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19. Inner Product on A
trivial. Assume now that N > 1. We may assume that Vi > 0 and x = Oix' for some i and some x' G fu-i- We have (x~VXi 2/_77A)A = (Fix'~r)\,y~r)x)\ = (x , ~r)\,ViK-.iE i y~7)x)\. Using the commutation formula 3.1.6(b) and the equality EiT)\ — 0 we see that the last inner product is equal to
Using the induction hypothesis, we see that in the last expression, the converges to 0) and first term converges to 0 for A —* oo (note that 2 1 the second term converges to (1 — v~ )~ (x', ir(y)) which by 1.2.13(a), is equal to (x,y). The proposition is proved.
CHAPTER
20
Bases at oo
20.1.1. Let M be an object of C. We define a basis at oo of M to be a (a) a free A-submodule L of M such that M = Q(v) 0 A L — M and it is required that the properties (c)-(f) below are satisfied. (c) L is stable under the operators E^ Fi : M —> M for all i; thus, Ei, Fi
(e) we have L = ®LA where Lx — L fl Mx and b = UbA where b A = (f) given 6, b' E b and z E / , we have Eib =
if and only if Fib' = b.
The definition given above of bases at oo is due to Kashiwara who calls L e m m a 20.1.2. Let x E L fl Mx
and let i E / . Let t = (i, A).
Write
(b) I f x mod v~lL belongs to b, then there exists so such thatxs E v~lL for s so, xSo mod v~lL belongs to b and x = F^s°^xSo mod v~lL. We prove (a) by induction on N > 0 such that xs = 0 for s > N. For N = 0, the result is clear. Assume now that N > 1. We have EiX = where xs = 0 for s > N. By definition, we have E 8 -s>i-s+t>o
hypothesis, we have xs E L for all s > 1. Since L is stable under Fi and
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Media, LLC 2010 2011
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174
Fj;s)xs = Ffxa, it follows that F$s)xs G L for all s > 1. Since x G L, we deduce that xq G L. This proves (a). We prove (b) by induction on N > 0 as above. If N = 0, there is nothing to prove. Assume that N > 1. If EiX G then v
J2s>-,s'>o-s>+t'>i FiS 1 e L and b y (a) we have vxs'+1 G ^ for all s' > for all s > 1. As before we then have F^xs G v-1L 0. Hence x8 G x then E{X mod v~lL for s > 1 and x = xq mod v~ L. If EiX £ belongs to b. By the induction hypothesis, there exists So > 1 such that F^So~^xSo xs G v~lL for s ^ so and s > 1. Therefore we have EiX = mod v~xL. Equivalently, we have EiX = F*°~1xSo mod v~lL. Applying Fi to this and using 20.1.1(f), we obtain x = F ^ x = F*°xSo = F^So)xSo mod v~lL. The lemma is proved. 20.1.3. In the next theorem we assume that the root datum is y-regular. Let A G X+. Let L be the A-submodule of AA generated by the canonical basis B(AA) and let b be the image of the canonical basis in L/v~1L. T h e o r e m 20.1.4. (L,b) is a basis at oo of A\. Property 20.1.1(c) follows from Theorem 18.3.8. We prove that property 20.1.1(d) is satisfied. Let be b. There exists (3 G B such that b is /3~rj\ mod v~lL. From Theorem 18.3.8 we see that Fib is <j>i(/3)~ri\ mod v~xL and Eib is €i((3)~r)\ mod v~lL or zero. By 17.3.7, we have i(/3) = (3' mod v~lC(F) for some (3' e B and this is necessarily in B. Then 4>i(/3)~r)\ = and (3'~rj\ mod v~lL (3'~r)\ mod v~lL so that Fib is (3'~r)\ mod is in b U { 0 } . By 17.3.7, we have either c - 1 £(f). Then ei(/3)~r}\ = j3"~r)\ mod v~xL or ei(/3)~r]\ = 0 mod v~lL so that Eib is (3"~r)\ \ mod v~lL is in b u { 0 } . This proves property mod v~lL or 0. Now 20.1.1(d). Property 20.1.1(e) is clearly satisfied. We prove that property 20.1.1(f) is satisfied. Let b,b' G b. We have b = (3~r)\ mod v~lL and b' = (3'~r]\ mod v~lL where /?,/?' G B. By 18.3.8, we have Eib = b' if and only if ( E ^ ) " ^ = /?/-T?A mod v~xL. This is equivalent to the condition that (a) u(3 = (3' mod
v~lC(f).
Similarly, the condition that Fib' = b is equivalent to the condition that (b) j>i(3' = (3 mod ^ - ^ ( f ) . Now conditions (a) and (b) are equivalent by 17.3.7. The theorem is proved.
20.2. Basis at oo in a Tensor Product
175
2 0 . 2 . BASIS AT OO IN A TENSOR P R O D U C T
20.2.1. Let M, M ' G C. Assume that M and M' have finite dimensional weight spaces. Assume that (L, b) (resp. (L', b')) is a given basis at oo of M (resp. M'). Consider the tensor product M © M' G C. T h e o r e m 20.2.2. The free A-submodule L ©A L' of M ® M' and the Q define basis bob 'of (L ©A L')/v~l{L ©A L') = {L/v~lL) ©Q (L'/v^L') a basis at oo of M © M'. Only properties (c),(d),(f) in the definition 20.1.1 need to be verified. In verifying these properties, we shall fix i G / and write Ll for the sum ©LA over all A such that (i, A) = t. The notation Ln has a similar meaning. Let Gl be the set of all z G Ll such that z mod v~xL belongs to b and such that E{Z = 0. Let Gn be the set of all z' G Ln such that z' mod v~lL' belongs to b ' and such that Eiz' = 0. From the definitions, all elements of the form F^z (z G GtJs G [0,t]) belong to b modulo v~1L and according to 20.1.2, all elements of b are obtained in this way. Similarly, all elements of the form F^s>)z' {z' G Gn\ s' G [0, t'\) belong to b ' modulo v~lL' and all elements of b ' are obtained in this way. Using Nakayama's lemma, which is applicable since the weight spaces are assumed to be finite dimensional, we deduce that the elements F^z (z G G*,s G [0, t]) generate the A-module L; similarly, the elements F^s ^z' (z' G Gn ,s' G [0, t']) generate the A-module L'. Let zeG\z' G Gn',se [0,£],s' G [0,t']. According to 17.2.4, we have FiiF^z
© Fl s > ) z') = F l s ) z © F l 8 ' + 1 ) z '
mod v~l{L ©A L')
© F\s'\')
= F^s+1)z
mod v~l{L ©A L')
© F^z')
= Ft(s)z © F^~l)z'
mod v~\L
©a L')
© F^z')
= F^s~l)z
mod v~\L
©a L')
if 5 + s' < t'\ Fi{F\s)z
© F^z'
if s + s' > t'\ Ei{Fls)z if s +
< t'\ EiiF^z
© Ft(s,)z'
lists' > t'. It follows that Ei, Fi map a set of generators of the A-module L ©A L' into L ©A L'\ hence they map L ©A L' into L © A L'. This verifies property 20.1.1(c) of a basis at oo. Properties 20.1.1(e),(f) of a basis at oo are also clear from the previous formulas. The theorem is proved.
20. Bases at oo
176
20.2.3. Assume that z G G\z' G Gn' and that s G [0,t],s' G [0,£'] are such that t + t' = 2(5 + s'). By the formulas in 20.2.2, the condition that Fi(F^ s ) z G v~l(L ®A L') is that either s' = t' and s + s' < t', or s — t and s -f s' > t'. The first case cannot occur since s > 0. Hence the condition is that s = t and s + s' > t'. But if s = t then t' = s + 2s' hence s + s' > s + 2s' so that s' — 0. Thus the condition is s = t = t', s' = 0. We can reformulate this as follows. P r o p o s i t i o n 20.2.4. Let (M, L, b), (A/7, £/, b') be as above. Let b G b, b' G b'. Assume that b G b A and b' G b , v and (i, A) + (z,A;) = 0. Then the following two conditions are equivalent: (a) Fi(b®b')
= 0 in ( L ® A L')/v~l(L®A l
(b) Fi(b) = 0 in L/v~ L
and Ei{b') = 0 in
L'); L'/v^L'.
CHAPTER
21
Cartan D a t a of Finite Type
In this chapter we assume that the Cartan datum is of finite type; then the root datum is automatically F-regular and X-regular. Let A' = —wq(\). Then A' E X+ and we may consider the U-module "AA' as in 3.5.7. Since = Av as a vector space, the canonical basis B(AV) of Ay may be regarded as a basis of "Ay. 21.1.1.
P r o p o s i t i o n 21.1.2. There is a unique isomorphism A\ ^AA' such that x maps B ( A A ) onto B ( A A ' ) -
of U-modules x
By 6.3.4, AA' is a finite dimensional simple object of C'. Hence uAy is a finite dimensional simple object of C . By definition, its (—A')-weight space is one dimensional and the (—A' — i')-weight space is zero for any i. By Weyl group invariance (5.2.7), it follows that the A-weight space is one dimensional and the (A -f z')-weight space is zero for any i (we have A = wq{—\')). Let x be the unique element of B(AA') in this A-weight space. Then EiX = 0 for all i. By Lemma 3.5.8, there is a unique morphism (in C') X : AA —• "Ay which carries rj to x. Since x is a non-zero morphism between simple objects, it is an isomorphism. We can regard x as a n isomorphism of vector spaces AA = AA' such that x(uv) — w(u)x(y) f° r all u E U and y E A A and such that x(v) € B ( A A ' ) We have (a) x U A a ) C
AAy.
Indeed, let y E ^Aa- Then y = g~r] for some g E ^ f ; hence x(y) = g^xiv)- ^ remains to use the fact that is stable under g+ (see 19.3.2). We have (b) x(x) = x(x) for all
X
E AA.
Indeed, we can write x = urj with u E U. We have x(ut]) = u(u)x( rj) = u(u)x(v)
= v(u)xW
=
=
as required.
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Media, LLC 2010 2011
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178
We have (c) ( x , x ' ) =
for
x,x' G AA.
Indeed, if we set ((£,#')) = ( x W , x(x')) w e obtain a form satisfying the defining properties of (x,x'), hence equal to it. Let 6 G B(AA). Let b' = x(&)- Using (a),(b), we see that b' G and 6' = b'. Using (c) and the fact that (6,6) and (x(r/), x(v)) a r e i n 1 + v~1Z[v~1], we see that (6', 6') G 1 + Since b' G ^Aa', we have also (b',b') G A; hence (6 / ,6 / ) G 1 + v~lZ[v~1}. Using Theorem 19.3.5, it follows that ±6' G B(AV)- This argument also shows that, if (L, b), (Lr, b') are the bases at oo of AA, AA' defined in 20.1.4, then x(L) = L'. - Fitr) We can find a sequence i\, i2, •.., it in I such that 6 = F^F^ — EiX f° r It follows mod v~lL. From the definitions we have that that b' = Ei1Ei2 • • • Eitx(rj) mod v~lL', so that b' mod v~xV belongs to b'. Since ±6' G B(AA')> it follows that b' G B(AA')- The proposition is proved. I n particular, 21.1.3. We shall identify the U-modules A A and ^AA' via U the generator £ = £_A> of A\> (see 3.5.7) is now regarded as a vector in the wo(A)-weight space of AA, which belongs to the canonical basis of AA-
CHAPTER
22
Positivity of the Action of FuEi in the Simply-Laced Case
22.1.1. In this chapter, the root datum is assumed to be F-regular. We fix A G X + and we set A = A\. The main result of this chapter is Theorem 22.1.7, which asserts, in the simply laced case, that the matrices of the linear maps Ei and Fi of A into itself, with respect to the canonical basis of A, have as entries polynomials with integer, > 0 coefficients. Using Theorem 18.3.8, we see that Lemma 18.2.7 is true unconditionally.
22.1.3. In the following corollary we shall use the notation
C o r o l l a r y 22.1.4. Let b G B. Write r(b) = Ylhb-,bub2h ® b2 and f(b) = S ® b 2 where hb.bub2 G A and gb.but>2 G A; here bi,b2 run over B .
It suffices to prove the statement about gb;bl,b2. By 3.1.5, we have A(b~) = ®
*
the definition of
By the previous theorem, if b2 G B(A), the coefficient of bi^b^rj is in A. This coefficient is clearly in A; hence it is in Z[v~x]. The corollary follows.
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011
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22. Positivity of Action of Fi,Ei
180
Corollary 22.1.5. Let i G I. Let b G B. Let us write ir(b) where b' runs over B and dbtoiti/yn are integers. Ylb'inezdbjit^nV71^
=
(a) Let b' G B(A) and let n G Z be such that db,ei,b'yn ^ 0. Then to(A-|6'|) + n > 0 . (b) We have vio(A-|6'|)+n
E
_
v-io(A-|6'|)-n
db,0i,b',n
Ti
&
f]
where b' runs over B(A). We apply the previous corollary to hb;bi,b2 with b\ = Of, we obtain 5> 0. Hence fb,b',i,n € N. Again by Theorem 14.4.13, the integers db,0i,b',n are > 0 and, by Corollary 22.1.15(a), we have io(\ — \b'\)+n > 0 for any n such that db,di,b',n / 0. Since [N] is a sum of powers of v if N > 0, we deduce that fb,b',i,n € N. The theorem is proved.
N o t e s on Part III 1. Most results in Part III are due to Kashiwara [2]. An exception is Theorem 22.1.7, which is new. 2. Although Theorem 22.1.2 does not appear explicitly in Kashiwara's papers, it is close to results which do appear; the same applies to the results in 17.1. The proofs in 17.2 are quite different from Kashiwara's. 3. The proof in 19.2.3 is an adaptation of arguments in [3]. REFERENCES 1. M. Kashiwara, Crystallizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 1 3 3 (1990), 249-260. 2. , On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 6 3 (1991), 465-516. 3. , Crystal base and Littelman's refined Demazure character formula, Duke Math. J. 7 1 (1993), 839-858. 4. G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-^98.
Part IV C A N O N I C A L B A S I S OF U
The algebra U is a modified form of U in which U ° is replaced by the direct sum of infinitely many one-dimensional algebras, one for each element of the lattice X of weights. This is an algebra without unit; it has instead many orthogonal idempotents by means of which one can approximate the unit element. The U-modules in the category C can be regarded naturally as modules for U; on the other hand, more exotic U-modules, like those without weight decomposition, cannot be regarded as U-modules. Thus, U is an algebra more appropriate than U for the study of objects of C. One of its main virtues is that it has a canonical basis B with remarkable properties. In Chapter 23, we define and study the algebra U and its .A-form ^ U . In Chapter 24, we prove an integrality property for the quasi-7£-matrix and we use it to define a canonical basis in any tensor product A v , which is fixed by an antilinear involution obtained by composing the obvious involution ~~ with the action of the quasi-7£-matrix. In Chapter 25, we show that these bases have some rather nice stability properties with respect to certain transition maps (see 25.1.5). The proof is based on results in Part III. From this we deduce (Theorem 25.2.1) that these bases can be "glued together" to form a single basis B of U . In Theorem 25.2.5, we show that B is simultaneously compatible with many subspaces of U and in Proposition 25.2.6, we show that B is a generalization of B. In Chapter 26 we show that B is characterized (up to signs) by an almost orthonormality property with respect to an inner product. This is in the same spirit as Kashiwara's idea of characterizing B by an almost orthonormality property (see Notes on Part II). As an application, we show that B is stable (up to signs) under a and u T h e same should be true without signs.
184
Part IV. Canonical Basis of U
In Chapter 27 we assume t h a t the Cartan d a t u m is of finite type; we show t h a t a tensor product of form 0 Aa 2 • • A A „ has a canonical basis and t h a t this induces a canonical basis on the space of coinvariants. This last basis has an invariance property with respect to cyclic permutations, as shown in Chapter 28. In Chapter 29 we prove a refinement of the Peter-Weyl theorem (in U instead of the coordinate algebra) and we discuss an extension of the theory of cells in Weyl groups to the case of U . In Chapter 30 we define a canonical topological basis of (a completion of) U ~ U + (in finite type) and show t h a t this basis gives rise simultaneously to the canonical bases of the various tensor products A\ "Ay.
CHAPTER
23
The Algebra U
23.1.
D E F I N I T I O N AND F I R S T P R O P E R T I E S OF
U
2 3 . 1 . 1 . The objects of C may be regarded as modules over a certain algebra U, in general without 1, which is a modified form of U. We prepare
Consider the direct sum decomposition U = J(i/)> where v runs G through Z[J], defined by the conditions U ( I / ) U ( I / " ) C U ( I / ' + U(0), Ei G U(«), Fi G U(—i) for all */,*/' G Z[J], i G I, /j, G Y. We have There is a natural associative Q(t;)-algebra structure on U inherited from that of U. It is defined by the following requirement: for any A^, A'/, \' 2 , X2 and any t <E U(Ai -• AJ), 5 e^U(A'2 - A'2'), the product is
The algebra U does not generally have 1, since the infinite sum Y^xex does not belong to U (if X ^ 0); however the family (Ia)agx is in some
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011
185
23. The Algebra U
186
23.1.2. A finer d e c o m p o s i t i o n of U . We shall need a direct sum decomposition of U which is slightly different from the one in 23.1.1(b). The images of the summands U(i/) under ir\>t\» form the direct sum decomposition A'UA" = ©^(A'UA"^))- We set U(I/) = 0A',A"(A'UA"(^)). We then have direct sums decompositions U = ©„U(i/) and U = © v , a « i / ( 1 a ' U M 1 a » ) where A', A" run through X and v runs through Z[I]. Actually, the summand 1A'U(^)1A" = A'UA"(^) is zero, unless A' — A" = v in X. 23.1.3. U - b i m o d u l e s t r u c t u r e . Note that U is naturally a U-bimodule by the following rule: if if G U ( i / ) , s G U , i " G U(i/"), we have by definition, t'-KX>,\"{s)t" = irx>+u>,\"-v"{t'st") for all A', X" G X. For a l i i G / and A G X , we have the following identities in U: = l\+i>Ei, (EiFj-FjEi)lx
Fil\ = =
lx-i'Fi;
6itj[{i,\)]ilx;
and more generally, E^ U =
F^U
Ei°)F?hx = F V E f h
t> 0
t>o
=
x
U-ai'F^-,
if
a + b — (i, A) F> t
'1_A+(a+b-t)i'bi
a + b — (i, A) t
ip(a-t). I
1A-(a+b-t)i'^i
These identities follow from 3.1.9. 23.1.4. U n i t a l m o d u l e s . A U-module M is said to be unital if (a) for any ra G M we have 1 Am = 0 for all but finitely many A G X; (b) for any ra E M we have J^Aex
=
m
-
If M is a unital U-module, then we regard it as an object of C as follows. The decomposition M = ®M A is given by Mx = 1AM; the action of u G U is given by um = (ulx)m for any A G X and any ra G M A . Here ulx is regarded as an element of U as in 23.1.3. In this way, we see that to give a unital U-module is the same as to give an object of C.
23.1. Definition and First Properties of U
187
23.1.5. The comultiplication of U induces a similar structure on U. More precisely, for any A'1? A2, X'{, X2 G X, there is a well-defined linear map 1
Ai+A^Uy^// -> (a^UA^) (a^UA^)
such that AA^A'/.A^A-C^Ai+A^Ai'+A^W) = ( ^ . A ' / 0 7Ty2>y^)(A(x)) for all x G U. This collection of maps is called the comultiplication of U, and may be regarded as a single linear map from U to the direct product
AjjAj
II
(aiUA;')®(ASUa»).
23.1.6. The algebra automorphism u : U —> U induces, for each A7, A", a linear isomorphism A'UA" —• _A'U_A". Taking direct sums, we obtain an algebra automorphism u : U —• U such that u;(1a) = 1 - a for all A and u(uxx'u') = u(u)u(x)u(x')uj(u') for all u,u' G U and x, x' G U. It has square 1. The map a : U —• U induces, for each A', A", a linear isomorphism A'UA" —• _A"U_A'. Taking direct sums, we obtain a linear isomorphism a : U —• U such that U induces, for any A, A' G X, a Q-linear map ~ : AUA' —» AUA'- Taking the direct sum of these maps, we obtain a Q-linear map : U —> U with square 1, which respects the multiplication of U, maps each IA into itself and satisfies tst' = tst' for M ' e U and s G U.
23. The Algebra U
188
2 3 . 2 . TRIANGULAR DECOMPOSITION, ,4-FORM FOR U
23.2.1. The U-bimodule structure on U gives us a f f o p p -module structure (x x') : u i—• x+ux'~ on U. Similarly, the U-bimodule structure on on U. From U gives us a f 0 f o p p -module structure (x x') : u h-> X~UX,+ the triangular decomposition for U, we see that U is a free f f opp -module with basis ( l A ) A e x for either of the two f f o p p -module structures. In other words, (a) the elements b+l\b'~ vector space U;
(b,b' G B,A G X) form a basis of the Q(v)-
(b) the elements b~l\b'+ vector space U.
(b,bf G B, A G X) form a basis of the Q(u)-
This is called the triangular decomposition
of U.
L e m m a 23.2.2. (a) The A-submodule of U spanned by the elements x+\\x'~ (with x,x' G a?) coincides with the A-submodule of U spanned by the elements x~l\x,+ (with x,x' G a W e denote it by ^ U . (b) The elements in 23.2.1(a) form an A-basis o / ^ U . The elements in 23.2.1 (b) form an A-basis 0/.4U. (c) .4U is an A-subalgebra of U . This algebra is generated by the elements for various i e I, n> 0 and A G X. (a) follows immediately from the commutation formulas 23.1.3. (b) follows from (a) and the fact that B is an .4-basis of ^ f . (c) follows from the commutation formulas 23.1.3. 23.2.3. From the integrality properties of the maps r, f , it follows easily that the maps AA^A'/.A^A? : l V i + A ^ U l A / ' + A - -> (1A>U1A>') ( l ^ U l ^ ) restrict to ^4-linear maps lAi+AiUU)lAi'+Ai' -
(lAiUU)lAi') X' be the unique morphism of root data. We have an induced algebra homomorphism 0 : U ' —> U between the corresponding Drinfeld-Jimbo algebras (see 3.1.2). Assume that we are given ( e l ; let maps the left ideal U'(K^ of U ' into the left ideal X ^ e v ~ v ^ ' ^ ) of U; hence it induces a linear map on the quotients: Ulc. j>: U ' l c , Prom the triangular decomposition in U ' and U, it follows that the linear map (j) is an isomorphism. Using the definitions, we see that 0 restricts to an isomorphism of .4-modules Aj>:AV'lc* l G f < g > f a s follows: (a) (6 + 6 /_ 1A'-A)(1 1) = 6+( 1 ®b') = b®b' + E ^ & i b'x where c ^ ^ G Q(t>), and the sum is taken over elements b\, b[ in B such that tr |&i| < tr |6|, tr \b[\ < tr |6'| and b[ belongs to the U-submodule of My = f generated by b'. Similarly, the elements b'~b+lx» with 6,6' G B and X" G X form a Q(t>)-basis of U. Such an element (with X" = X' - X) acts o n l < g > l e f ® f as follows: (b) (6 , -6+1 v _A)(1 ® 1) = 6 7 "(6 0 1) = 6 0 b' + £ c'bi b, 6X ® b[ where c'bi b, G Q ( f ) , and the sum is taken over elements 6i,6j in B such that tr |6I| < tr |6|, tr \b[\ < tr |6'| and 6i belongs to the U-submodule of wMA = f generated by 6. Using either (a) or (b), we see that (c) the Q(v)-linear map 7r' : U 1 A ' - A —'- U,MX®MX> given by u is an isomorphism.
U(
1(g) 1)
23.3.2. For any A G X+, we define a ^U-submodule .4 MA of the Verma module Mx as follows. As an ,4-module, it is the .A-submodule Af of f = Mx. This submodule is obviously stable under the action of the operators F ^ G U on MA; using repeatedly the commutation formulas 23.1.3, we see that it is also stable under the operators E\n>> G U on Mx; hence it is
190
23. The Algebra U
a ^U-submodule of M\. The same ,4-submodule is an ^U-submodule of "Mx, denoted The argument used to prove 23.3.1(c) can be repeated word for word for A instead of Q(u) and gives the following result. (a) Given A, A' € X, the map u H> u(l 0 1) defines an ^-linear isomorphism 7r7 of ^U1A'-A onto the ,4-submodule %Mx®a(aMy) oi"Mx®My. 23.3.3. Let "Ax (resp. f = My Ay). By 14.4.11, T (resp. T ) is the subspace of f generated by a subset of the basis B, namely by D = Ui,n;n>ai .4 A a given by x »-> x~rj\ is the left ideal XT of ^ f generated by the elements for various i G / and n > (i, A). We can now repeat word for word the proof of Proposition 23.3.6 for A instead of Q(v) and we obtain the following result, where A, A' G X+. P r o p o s i t i o n 23.3.8. The assignment u u(€-\®r]y) defines a surjective A-linear map tt of aUl^ onto the A-submodule ^A\®a( A Ay) of"A\ M in C' such that /K_A = mSince M is integrable, we can find integers ai, a[ G N such that E\a) m = 0 for all i and all a > ai and ^m = 0 for all i and all a' > a'{. Since the root datum is Y-regular, we can find X € X such that (i, A) > ai and (i, A + C) > Q-i for all i. Let A' = A + Then (i, X') > a'{ for all i. The existence of / follows now from 23.3.6.
CHAPTER
24
Canonical Bases in Certain Tensor Products 24.1.
INTEGRALITY PROPERTIES OF THE
QUASI-7£-MATRIX
24.1.1. Let M , M ' € C be such that either UM £ Chi or M' e Chi (see 3.4.7). We regard M ® Mf naturally as a U 0 U-module and we define M ® M' by 0 ( m m!) = ^ ©i/(m mf) a linear map © : M ® Mf (notation of 4.1.2.) This is well-defined since only finitely many terms of L e m m a 24.1.2. Let M, M ' be as above. We have (b) Assume that we are given Q-linear maps : M —> M and ~ : M ' —> M' such that urn — urn and um! — urn' for all u € U, m E M, m! G Mf. Let - = : M ® M ' -> M®M'. Then A ( u ) B ( m 0 m / ) = ©(A(t2)(m ® ra')) The set of u for which (a) holds is clearly a subalgebra of U containing all K^. Hence it suffices to check (a) in the special case where u is one of the algebra generators Ei, Fi. Applying both sides of the equalities 4.2.5(c),(d)
(Ei ® 1 + Ki ® Ei)Q(m ® m') = ©(£; 1 +
® ^ ) ( m ® ra')
This proves (a). Now (b) is a consequence of (a). The lemma follows. 24.1.3. The following property is just a reformulation of the property in Lemma 24.1.2(b). Let M, M ' be as in that lemma. Then for any ra 6
(a)
uQ(m ® ra') = Q(u(m ® ra'))
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010
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24-1. Integrality Properties of the Quasi-H-Matrix
193
P r o p o s i t i o n 24.1.4. Let A, A' G X. Consider the Verma modules M\, Note that M = " Mx G C and M' = My G Chi; hence
My.
6 : M M ' -> M O M ' is well-defined. Then
6
leaves stable the A-submodule
A
M \
).
Since the ambient space of M and M' is f, we may regard ~ : f —• f as maps : M —• M, ~ : M' —• M'. Using the definition of Verma modules, we may identify M' = U / UEi + - v
194
24- Canonical Bases in Certain Tensor Products
For any v° and any b°, b'° G B^o, we have Pbo,b'° € A. such that b° G B(-tu 0 (A)), b'° G B ^ o t A ' ) ) - By We can find A, A' G "AA ® Aa> maps the .A-submodule ®a UAA') 24.1.5, O : "AA ® AA' into itself. By 21.1.2, we may canonically identify as U-modules ^AA with A_u,0(A) and AA' with u;A_u,0(A') respecting the canonical bases. It follows A—woix)^ maps the .4-submodule that 6 : A_u;o(A)0U;A_u;o(A') ^A_U7o(A) a (J4A_Wo(A')) into itself. In particular, this submodule contains the vector S(rj f ) = J2vYlb,b> Pb,b'b~rj ® b'+£. Here, rj =• r}_WoW and and b' runs over B ^ f l B ^ u ; o ( A ' ) ) . £ = fu>o(A'); b runs over BunB(-w0(X)) Since the elements b~r)<S> b'+£ (for all indices (f, 6, &') as in the sum) are a 6°, b'°) is an index part of an .A-basis of a^-w0(\) ®a (^A_ u;o (A')), and The corollary is proved. in the sum, it follows that Pb°,b'° € 2 4 . 2 . A LEMMA ON SYSTEMS OF SEMI-LINEAR EQUATIONS
L e m m a 24.2.1. Let H be a set with a partial order < such that for any h < h' in H, the set {h"\h < h" < h'} is finite. Assume that for each h < h' in H we are given an element rh,h' € A such that (a) rh,h = 1 for all h; (b) ^2h"-,h 0, we consider the statement Pn which is the assertion of the lemma restricted to elements h < h! such that d(/i, h') < n. Note that property (e) is meaningful for this statement. We prove Pn by induction on n. The case n = 0 is trivial. Assume now that n > 1. Let h < h'. If d(h,hf) < n, then ph,h' is defined by Pn-\. If d(h,h') = n, we note that q = Y^h" hl]b2,b'2 G A, and pbl,b'l]b2,b'2 = 0 unless (bi,b\) > (b2,b2); hence the last sum is finite. Note also that pbl ^ ;bl ^ = 1 and Y Pbi ,b[ ;b2,b'2Pb2,b'2,b3,b'3 = b2€B(\),b'2eB(X)
,63^ »
for any 61, 63 G B(A), 6'1? 63 G B(A'); the last condition follows from \I>2 = 1. Applying Lemma 24.2.1 to the set H = B(A) x B(A'), we see that there is a unique family of elements nbl,b'1-,b2,v2 G Z[v _1 ] defined for 61,62 G B(A),6/1,62 G B(A') such that TTfei ,bi ;6x ,6'x = 1; "b^^b'
e t r ^ t r 1 ] if (biX) =
0
(b2,b'2);
unless (bub[) > (b2,b'2);
Kbub'i-tete =
,63 ^bi ,b[ ;b3 ,b'3 Pb$ ,b'3 ;b2 ,b2 for all (&l,&i) >
(b2,b'2).
We have the following result. T h e o r e m 24.3.3. (a) For any (bi,b[) G B(A) x B(A'), there is a unique element (&iO&i)A,A' € C such that and ( & I O & I ) A , A ' - b+Z-x ® b^rjy G v~lC. A ® b'^rjx' plus a (b) The element {b\§bi)x,x' in (a) is equal to linear combination of elements b2^-x®b2~r}x' with (b2, b2) G B(A) x B(A'), (62,^) < (61, b[) and with coefficients in v~1Z[v~1]. *((&IO&'I)A,A') -
(&IO&;)A,A'
(c) The elements (6i(£) —> C/v~lC
is an isomorphism.
The element (6IO&'I)A,A' = E b ^ ^ . & ' ^ ^ J f - A ® b'2~r)x> (see 24.3.2) satisfies the requirements of (a). This shows existence in (a). It is also clear that the elements (&iO&i)A,A' just defined satisfy the requirements of (b),(c),(d). It remains to show the uniqueness in (a). It is enough to show that an element x G such that x = x, is necessarily 0. But this follows from (d).
24.3. The Canonical Basis ofuA\
24.3.4. The basis W
(g> Ay
197
24.3.3(c) is called the canonical basis of
AAAV.
24.3.5. Let A, A G X+. Let (Yf ,X',...) be the simply connected root datum and let / : Y' —> Y, g : X —• X ' be the unique morphism of root data. Let U ' be the algebra defined like U, in terms of ( y , X 7 , . . . ) . Let A', A' G be defined by A' = g(A), A' = g(A). Then w A a ® A x , defined in terms of U, has the same ambient space as wAA' <S> A^,, defined in terms of U 7 . We have a priori two definitions of the canonical basis of this space, one in terms of U, one in terms of U'. Prom the definitions, we easily see that these two bases coincide.
CHAPTER
25
The Canonical Basis B of U
25.1.1. In this section, the root datum is assumed to be F-regular. P r o p o s i t i o n 25.1.2. Let A, A' be dominant elements of X.
We write r\ =
(a) There is a unique homomorphism of U-modules x • AA+A' —• AAAA'
sum over bi E B(A),6 2 G B(A'), with /(6;6I,6 2 ) G Z j v 1 ] . (c) If b~r)' + 0, then /(fe; 1, b) = 1 and f(b; 1, b2) = 0 for any b2 ± b. If The vector r) rfG A\ Aa' satisfies Ei(rj rj') = 0, K^(rj (g> 7/) = (»,\)+(»,x') ( b y 3.5.8). By the definition of comultiplicav u This implies s u m over tion in U, we can write x(6~r/") = b2 /(&; 6i, b2)b^r} ® b^rfi 6i G B(A),6 2 G B(A'), with f(b;bub2)'eQ{v) satisfying f(b;l,b2) = 1 if By 23.2.3, we have /(&;&i,&2) G A for all bi,6 2 . Hence to prove (b), it suffices to show that f(b; b2) G A for all 6i, b2. We have a commutative
where S is as in 18.1.4, the left vertical map is given by x »—• x~rj" and the right vertical map is given by x ® y —> (x~r]/) 1. We can assume that x = Fix' for some i such that vi > 0 and some a;' £ Then [x,y] = [Fix',y] = -[K^x^Eiy]
~[x\K^Eiy]
=
and (x,y) = (FiX,y) =
Vi(x\K-iEiy).
By the induction hypothesis, we have -\x\k.iEiy) = (-1)tr ^ . ^ ( x ^ K . i E i y ) hence [x,y] = (—1) tr uv_\u\(x,y), which completes the induction. Now let 6, h' be as in (b). We must show that [6—77, b'~rj\ is in v~1Z[v~1] unless b = h' = 1. We may assume that there exists v such that b~r},b'~rj both belong to (A)^. The result then follows from (c) since, by 19.3.3, we have (b~r],b'~r]) £ Z[v - 1 ]. 25.1.5. Let A, A7, A" be dominant elements of X. We define a homomorphism of U-modules t : "Ax+y ® AA'+A"
"AA' AA"
as the composition of W
x' X ' ^AA+A' A V + A "
where x',X
a r e 95
1 0
6y
AA (8) "Ay
® Ax> ® A A " ,
25.1.3, 25.1.2, with 1 : " A A
"Ay
Ay
A A " - >
U
AX
A A " .
25.1. Stability Properties
201
L e m m a 25.1.6. (a) Let b E B(A), b" E B(A"). We have ® b"-rix> + \") = b + Z- X ® b"~rix»
mod
v~lCx,x"
(notation of 24.3.1). (b) Let b E B(A + A'), b" E B(A' + A"). Assume that either b ® b"~rix'+X") = 0 m o d v " 1 ^ ' . (c) t is surjective. In this proof we shall use the symbol = to denote congruences modulo vtimes a Z[i; _1 ]-submodule spanned by the natural basis. Using 25.1.2, 25.1.3, 25.1.4, we see that if b,b" are as in (a), we have t(b+£-x-\> ®b"-r)y+x")
= (1 ®6\> ® 1 )(b+£-x ® =
® V\' ®
+
b £-x®b"-rix».
Using again 25.1.2, 25.1.3, 25.1.4, we see that if b, b" are as in (b), we have = 0. t{b+^x-x>®b"-r)x>+x») Now (c) follows from the fact that £_A r)x' is in the image of t and it generates the U-module "Ax ® Ay (see 23.3.6). Using the definition 24.3.3 of the elements (b^b')x,x', we can reformulate the previous lemma as follows. L e m m a 25.1.7. (a) Let b E B(A), b" E B(A"). We have t(bb")x+x>,\>+\" = (t>0b")x,x»
mod
v~lCx,x».
(b) Let b E B(A + X'), b" E B(A' -I- X"). Assume that either b (£ B(A), or b" £ B(A"). We have t(bb")x+x',x'+x" = 0
mod
v~1Cx,x"-
25.1.8. In the following result we show that the maps u
$ : "Ax+x' ® Ay+x"
Ax+x'
A A ' + A "
and ^ -."AxteAx"
u
Ax 0 AA",
defined as in 24.3.2, are compatible with t.
202
25. The Canonical Basis B o / U
L e m m a 25.1.9. We have tV = Vt. We write £,77 instead of £_A-A', rjy+y. Since any element of wAA+A' 0 AA'+A" IS of the form «(£ 0 rj) for some u G U (see 23.3.6), it is enough to check that tG(u(£ 0 77)) = et(u(£ 0 77)) for all MGU, Using the definition of © and its property 24.1.3(a), we have tG(u(£ 0 77)) = t u ( G ( £ 0 77)) = tu(£ ® 77) = ut(£ 0 77) = u ( f _ A and Gtiui^rj))
= Gu(t({, (8) 77)) = U 9 ( £ _ A 0 77A") = U ( £ - A 0 »?A")-
The lemma is proved. P r o p o s i t i o n 25.1.10. (a) Let b G B(A), b" G B(A"). We have t(b0b") x+x>,x'+x" = ( W ) A,A". (b) Lei 6 G B(A + A'), 6" G B(A' + A"). Assume that either b with /?,/?' G B, is equal to wi(£_a<S>t?a") for some u\ G P( tr \/3\, tr |/?'|); moreover, u\ can be plus an element in P( tr \(3\ — 1, tr \(3'\ — 1). taken to be equal to /3+/3'~ From this we deduce that (e) (b§bn)A,A" £ "Ax AA" is of the form u(£_A ® Vx") f° r some u G plus an P( tr |6|, tr |6"|); moreover, u can be taken to be equal to b+b"~ element in P( tr |6| - 1, tr |6"| - 1). Assume that u is such an element and that u' is another element with the same properties as u. Then (u — U')(£_A rjy) = 0; hence, by 23.3.8, we have u-u'e
J2
AVFfn\+
i,n>(i,X")
^ i,n>(i,
n )
l
C
.
A)
Since u — u' G P( tr |6|, tr |6"|) we deduce that, if {i, A) and (i, X") are large enough (for alH), then we must have u = u'. Thus, for such A, X" the element u above is uniquely determined. We denote it by ux,x"Assume now that A, X" G X+ satisfy b G B(A), b" G B(A") and X" - X = Let X' G X+ be such that (i,X') is large enough for all i, so that v! = itA+A',A'+A" is defined. We show that x ® Vx") = (bW')x,x"Indeed, if t is as in 25.1.5, we have u'(€-x®r)x>>) = u'(t(£-x-x' = t((bOb")
Vx'+x")) = A+A',A'+A") =
x-x
®VX>+X"))
(b0b")x,x»
where the last equality follows from 25.1.10. It follows that UA,A" IS independent of A, X", provided that (i, A) and (i, X") are large enough (for all i); hence we can denote it as u, without specifying A, A". It also follows that u satisfies the requirements of (a). This proves the existence part of (a). The previous proof shows also uniqueness. Thus (a) is proved.
25. The Canonical Basis B o / U
204
Now let A, A" be as in (b). Let A' G X+ be such that (i, X') is large enough for all i, so that WA+A',A'+A" IS defined (hence it is u). We have A ® F ] \ " ) = U ( T ( I - A - A ' 0 *?A'+A")) =
A - A ' 7?A'+A"))
= T ( ( & 0 & " W , A ' + A " ) = 0,
where the last equality follows from 25.1.10. This proves (b). We prove (c). Let u, A, X" be as in (a). We have U ( £ _ A 0 T?A") =
F?A")
= O U ( ^ _ A (regarded as objects of C-) are admissible. For (, )A' this follows from the definition. The same holds for (, )A on AA- One checks easily that applying u to an object of C[ with an admissible form gives a new object of C[ for which the same form is admissible. In particular, (, )A' is admissible for "A\. The lemma is proved. P r o p o s i t i o n 2 6 . 2 . 3 . Let x,y G Ul^. When the pair A, A' tends to oo (in the sense that (i, A) tends to oo for all i, or equivalently, (i, A7) tends to oo for all i, the difference A7 — A being fixed and equal to the inner product _1 (x(£_A *7A')>2/(C-a ® V\'))\,\' E Q(v) converges in Q((v )) to (x,y). Assume first that x = have x{£-X®r)\')
—
where X\,y\ G f. In this case we = f-A ® XiVx'
and y{£-\ <S> 77a') = i-x
®yiV\r,
26. Inner Product on U
212
hence and, by 19.3.7, this converges to (xi,yi) when A —• oo. Since (x\,yi) = (a:]" the proposition holds in this case. Next, we prove the proposition in the case where x = and y is arbitrary. We may assume that y = p(x^)y^ where Xi,yi G f. Using the previous lemma, we have (12/ric(f-A ® vx
and by the earlier part of the proof, this converges to (rr^l^, y f l ^ ) when A oo. Since yf 1^) = (1^,2/), the proposition holds in this case. We now consider the general case. We can write x = ul^ where u G U. Using the previous lemma, we have 0 Vx'),p{u)y{£-\
(U1C(£-A ® 77A'),2/(£-A 77A'))A,A' =
Vx'))x,x>
and by the case previously considered, this converges to (1 £,p(u)y) A —• oo. Since (l^,p(u)y) = (ul^,y), the proposition is proved.
when
2 6 . 3 . A CHARACTERIZATION OF B L) ( - B )
In the following result there is no assumption on the root datum. T h e o r e m 2 6 . 3 . 1 . (a) Let 6,6',bi,b[ G B and let C,Ci € X.
We have
In particular, the canonical basis B o / U is almost orthonormal {or (,). (b) Let (3 G U. We have (3 G B U (—B) if and only if (3 satisfies the following three conditions: (3 G j3 = (3 and ({3,(3) G 1 + v_1A. Note that (a) is trivial when C Ci- Hence we may assume that £ = In this case, using the definitions we are immediately reduced to the case where the root datum is simply connected. Then 26.2.3 is applicable. Hence, using the definition, we see that it is enough to prove that (c) ((&0&')A,A', (6lO&i)A,A')A,A' = S B M ^ M for any A, A' G Since (b+€-x ® b'~r)\,
M O D
V
~'
A
7
such that A - A = £ and such that b G B(A), b' G B(A'). ® b[~r]x)x,x' = (b~r)x,b]~rix)x(b,~rix>,b,1-7)x')x>
G (fa,bi +v~ 1 A)(6 b ,y i + v~*A) =
6b,b16b>,b'l
+
26.3. A Characterization of B U ( - B )
213
we see that (c) follows from Theorem 24.3.3(b). We prove (b). If (3 G ± B then, by (a) and 25.2.1, it satisfies the three conditions listed. Conversely, if (3 G U satisfies the three conditions in (b) then, using (a) and Lemma 14.2.2(b), we see that there exists j3' G B such that f3 — (±/?') is a linear combination of elements in B with coefficients in v - 1 Z [ v - 1 ] . These coefficients are necessarily 0, since (3 — (±/3') is fixed by : U —> U. The theorem is proved. Corollary 26.3.2. If (3 G B, then a((3) G ± B and u((3) G ± B . a and u commute with ~ : U —> U, preserve the lattice ^ U and preserve the inner product (,) (see 26.1.4, 26.1.6). Hence the corollary follows from Theorem 26.3.1(b).
CHAPTER
27
Based Modules
27.1.1. In this chapter we assume that (/, •) is of finite type. Let M e C. We assume that M is finite dimensional over Q ( f ) . For any A £ we denote by M[A] the sum of simple subobjects of M that are isomorphic to A a . Then M = © ^ M [ A ] . We also define for any A £ X+:
27.1.2. A based module is an object M of C, of finite dimension over Q(v)
(b) the *4-submodule j^M generated by B is stable under ^ U ; (c) the Q-linear involution : M —> M defined by fb = fb for all / £ Q(v) and all b £ B is compatible with the U-module structure in the (d) the A-submodule L(M) generated by J5, together with the image of B in L(M)/v~~1L(M), forms a basis at oo for M (see 20.1.1). We say that ~ : M —> M in (c) is the associated involution of (M,B). The direct sum of two based modules (M, B) and (M', B') is again a based 27.1.3. The based modules form the objects of a category C; a morphism from the based module (M, B) to the based module (M', Bf) is by definition
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011
214
27.1. Isotypical Components
215
27.1.4. Let ( M , B ) be a based module and let M' be a U-submodule of M such that M' is spanned as a Q(t>)-subspace of M by a subset B' of B. Then (M', B') is a based module; moreover, M/M' together with the image of B — B' is a based module. For any A G X+, A A together with its canonical basis, is a based module. (See 19.3.4, 23.3.7, 20.1.4.) 27.1.5. Let (M, B) be a based module with associated involution ~ and let ra G M be an element such that ra = ra, ra G ^ M and m G B + v~1L(M) (resp. ra G v~1L(M)). Then we have ra G B (resp. ra = 0). Indeed, we can write m = JZbeB with CF, G A. By our assumption, we have Q> G A for all b. Hence Cb G Z[t>_1] for all b. We have Cb — Cb for all b. Hence c& G Z for all b, except for all b. Moreover, by our assumption, we have CF> G It follows possibly for a single b for which we have Cb = 0 or 1 mod that Cb = 0 for all 6, except possibly for a single b for which we have c& = 0 or 1. Our assertion follows. 27.1.6. Let (M, B) be a based module. Assume that M ± 0. Let Ai G X+ be such that MXl ^ 0 and such that Ai is maximal with this property. Let Bi = B n MXl. It is a non-empty set. Let M' = ©beBiAo^.b G C. Here A\ lt b is a copy of AXl corresponding to b; we denote its canonical generator by rjb. For any b G jE?i, we have Eib = 0 for all i G / by the maximality of Ai. Hence there is a unique homomorphism (j) : M' - > M o f objects in C whose restriction to any summand AXl,b carries 775 to b. Let B' be the basis of M' given by the union of the canonical bases of the various summands AXl,bProposition 27.1.7. In the setup above, B fl M[Ai] is a basis of M[\\] and defines an isomorphism M' = M[X\] carrying B' onto B fl M[Ai]. Thus (b') is fixed by ~ : M —> M. We know from 19.3.5 that there exists a sequence ii, •.. ,ip in I such that b' is equal to Fi1Fi2 • • • Fipr)b plus a v~lA-linear combination of elements of the same kind. Now the action of Fi on M' is compatible with
21. Based Modules
216
the action of Fi on M. Hence (f>(b') is equal to F^F^ ... Fipb plus a linear combination with coefficients in v - 1 A of elements of the same kind. By property 27.1.2(d) of B, we see that either (#) G B + v~lL(M) or 4>(V) G v~~lL(M). On the other hand, by the definition of the canonical basis of M', we have that b' belongs to the ^U-submodule of M' generated by 775; hence (p(b') belongs to the ^U-submodule of M generated by 6; by the property 27.1.2(b), we then have (j>{b') G ^ M . These properties of (br) G B or (f)(b') = 0 (see 27.1.5). The second alternative does not occur: indeed, the restriction of 4> to the summand Aa1;6 is injective since A\ lt b is simple. Thus we have <j)(b') G B. We see that (j> defines a bijection of the canonical basis of A\1,b with a subset B(b) of B. Next we consider an element b € B1 distinct from b. We show that B(b) is disjoint from B(b). Indeed, assume that 61 G B belongs to B(b) C\B(b). Then we have 61 = F^Fiz • ••Fiyb mod v~lL(M) and 61 = FjlFh • • • Fjgb
mod
v~1L(M)
for some sequences ii, «2> • - - , a n d j\,32, • — ijq m I27.1.2(d), we then have b = EjqEjq_l • • • Ej1Fi1Fi2 • • • Fipb
mod
By property
v~lL(M).
Hence b is equal to some element in B(b) plus an element of v~1L(M). It follows that be B{b). In particular, we have b = (6') for some b' G Since b ^ b, we have V / r]b\ hence b' G A ^ b with A7 < Ai. It follows that b G Mx> with A' < Ai. This contradicts the assumption that b G B\. We have proved therefore that B{b) is disjoint from B(b). Since B' is the disjoint union of the canonical bases of the various AAltb and these subsets are carried by <j> injectively onto disjoint subsets of B, it follows that (j) restricts to an injective map B' —> B. Since B' is a basis of Mit follows that 0 : M' —> M is injective. Thus we may identify M' with a U-submodule of M (via cj)) in such a way that B' becomes a subset of B. This submodule is clearly equal to M[Ai]. The proposition follows. Proposition 27.1.8. Let (M, B) be a based module and let A G X+. (a) B fl M[> A] is a basis of the vector space M[> A] and
Then
27.2. The Subsets B[X]
217
(b) B f l M [ > A] is a basis of the vector space M[> A]. First note that (b) follows from (a). Indeed, the vector space M[> A] is a sum of subspaces of form M[> A7] for various A' > A. To prove (a), we argue by induction on dim M. If dim M = 0, there is nothing to prove. Therefore we may assume that dim M > 1. For fixed M, we argue by descending induction on A. To begin the induction we note that if A) is sufficiently large, then M[> X] = 0 and there is nothing to prove. Assume that A is given. If M[X] = 0, then M[> A] is a sum of subspaces M[> A7] with A' > A; hence the desired result holds by the induction hypothesis (on A). Thus we may assume that M[A] ± 0. Then clearly Mx ± 0. We can find Ai G X+ such that Ai > A, M A l / 0 and Ai is maximal with these properties. Let M' = M[Ai] and let B' = B n M'. Then (M',B') G C by 27.1.7. Hence, by 27.1.4, M" = M/M 7 , together with the image B" of B-B', is an object of C. Since M' ^ 0, we have dimM 77 < d i m M ; hence the induction hypothesis (on M) is applicable to M". We see that 5 " n M " [ > A] is a basis of M"[> A]. Since M' = M'[Ax] and Ai > A, we see that M[> A] is just the inverse image of M"[> A] under the canonical map M —> M"\ moreover, a basis for this inverse image is given by the inverse image of B" D M"\> A] under the canonical map B —> B". The proposition is proved. 2 7 . 2 . T H E SUBSETS £ [ A ]
27.2.1. Let (M,B) be a based module. Let b G B. We can find A G X+ such that b G M[> X] and A is maximal with this property. Actually, A is unique. Indeed, assume that we also have b G M[> A;] and X' is maximal with this property. We note that M{> A] fl M[> A7] is a sum of subspaces M[> X"} for various A" such that A < X" and X' < X" and from 27.1.8 it follows that b G M[> X"] for some such X". If A ^ A7, then A77 satisfies A < A77 and A7 < A77, and we find a contradiction with the definition of A. Thus the uniqueness of A is proved. Let Z?[A] be the set of all b G B which give rise to A G X+ as above. These sets clearly form a partition of B. From 27.1.8, we see that, for any A G X+, the set Ux>€x+-,\'>\B[^'] is a basis of M[> X] and the set U A / g X + ; A / > a B[A 7 ] is a basis of M[> A]. P r o p o s i t i o n 27.2.2. Let f be a morphism in C from the based module (M,B) to the based module ( M 7 , B 7 ) (see 27.1.3). For any X G X+, we have /(B[A]) CB'[A]U{0}.
218
21. Based Modules
From the definitions, we see that f(M[> A]) C M'[> A] and f(M[> A]) C M'[> A]. Hence if b G B[A], then either f(b) 6 B'[\'] for some A' > A or f{b) = 0. Assume that f(b) $ B'[A]. Then f(b) G M'[> A]. Using the obvious inclusion f(M) D M'[> A] C / ( M [ > A]), we deduce that 6 G M[> A] -f k e r / . Since both M[> A] and k e r / are generated by their intersection with B, it follows that either b G M[> A] or b G k e r / . The first alternative contradicts b G 5[A]; hence the second alternative holds and we have f(b) = 0. The proposition follows. 27.2.3. Let ( M , £ ) be a based module. Let A G X+. We define B[X]hi to be the set of all b G B such that b G Mx and Eib G v~1L(M) for all i G I. We define B[X]l° to be the set of all b G B such that b G Mw°W and Fib G v _ 1 L ( M ) for all i G / . P r o p o s i t i o n 27.2.4. (a) We have B[X]hi C B[X] and B[X]l° C B[A]. (b) Let p : M[> X] —> M{> X]/M[> X] = M be the canonical map. Note that p defines a bijection of -B[A] with a basis B of M and that (M,B) belongs to C so that B[X]hl and B[X]l° are defined. Then p restricts to bijections B[X]hi B[X]hi and B[X]l° B[X]l°. We prove (a). Let b G B[X]hl. There is a unique A' G X+ such that b G B[X']. We must prove that A = X'. We have b G M[> A']. Replacing M with M[> A7], we may assume that M = M[> X']. Let ir be the canonical map of M onto M" = M/M[> A']. Then B[X'] is mapped by IT bijectively onto a basis B" of M" and we have 7r(b) G B". Moreover, ir(b) belongs to B"[X]ht and we are therefore reduced to the case where M = M". Thus we may assume that M — M[X']. Now 27.1.7 reduces us further to the case where (M, B) is k\> with its canonical basis. In this case, there are two possibilities for b: either b is in the A'-weight space or there exist i and b' G B such that F^b' -be v~1L(M). In the first case we have b G M A '; in the second case we have Eib — b' G v~lL(M)\ hence Eib £ v~lL(M), in contradiction with our assumption on b. Thus we have b G Mx , hence A = A;, as required. We have proved that B[X]hl C -B[A]. The proof of the inclusion B[X]la C B[A] is entirely similar. We prove (b). We assume that M = M[> A]. It is clear that p(B[X]hi) C B[X]hi and p(B[X]l°) C B[X]l°. Assume that b G £[A] satisfies b £ B[X]hi. We show that p(b) ^ B[X]hl. By our assumption, we have that either b e Mx> with Y ± X or that Eib £ v~lL(M) for some i. If be Mx' with X' / A, then p(b) G Mx' with A; ^ A; hence p(b) $ B[X]ht, as required. If Eib £ v~1L(M) for some i G / , then there exists b' e B such that Eib - b' G y-1L(M) and therefore Fib' -be v~lL(M).
27.3. Tensor Product of Based Modules
219
We consider two cases according to whether or not b' G M[> A]. In the first case (ib' G M[> A]), we have Fib' G M[> A] (since M[> A] is a subobject of M) hence b G M[> A]-\-v~ 1 L(M); this implies that b G M[> A] (using that 5 fl M[> A] is a basis of M[> A]). Then we have p(b) = 0 and, in particular, p(b) £ B[X]hl, as required. In the second case (b' £ M[> A]), we have b' e B[A]; hence n(b') G B. Let L(M) be the A-submodule of M generated by B. From Eib — b' G v~1L(M), we deduce Ei(Tr(b)) - 7r(b') G v~lL(M). In particular, we l ht have Ei(7r(b)) £ v~ L(M)\ hence p{b) B[X] , as required. Thus we have proved the equality p(B[X]ht) — B[X]hl. The proof of the equality p(B[X]l°) = B[X]l° is entirely similar. 27.2.5. C o i n v a r i a n t s . Let ( M , £ ) G C. Let M& 0] = Bx^0M[X]. The space of coinvariants of M is by definition the vector space M* = M / M [ / 0]. Clearly, 0] is equal to the sum of the subspaces M[> A'] for various + X' e X - {0}; hence, from 27.2.8, it follows that UAVO^[A7] is a basis of 0]. We deduce that under the canonical map 7r : M —• M* the subset JB[0] of B is mapped bijectively onto a basis B* of M*. Note that n is a morphism in C if we regard M* with the U-module structure such that M* = M* [0]. We see that (a) (M*, B*) is a based module with trivial action of U. P r o p o s i t i o n 27.2.6. We have £[0] = B[0]hi = B[0]l°. This set is mapped bijectively by n : M —> M* onto B*. To prove the first statement, we are reduced by 27.2.4(a),(b) to the case where M = M[0], where it is obvious. The second statement has already been noted. 2 7 . 3 . T E N S O R P R O D U C T OF BASED MODULES
27.3.1. Let (Af, B), (M 7 , B') be two based modules with associated involutions ~ : M —> M, ~ : M' -> M'. We will show that the U-module M M' is in a natural way a based module. The obvious basis B M M' leaves j^C stable; it follows that we have C From 24.1.2 and 4.1.3, it follows that = 1 and V(ux) = uV(x) for all u G U and all x G M M'. We shall regard B x B' as a partially ordered set with (bi,b[) > (62,63) if and only if 61 G M A L , 6 I G M / A ' I , 6 2 G Mx*,b'2 G M'x2 where AI > A 2 ,A'j < A2, Ai -I- Aj = A2 + A2. From the definition we have, for all 61 G B, b[ G B', b[) =
J2
Pbub'iMUh ® 63
b2£B,b'2£B'
where Pbub'i;b2,b'2 e A and also that
= 0 unless (61,6i) > (62,6'2). Note P6i,6'i;6i,6i = 1
and Pbi ,b[ ;b2,b'2Pb2,b'2;b3,bf3
=
M fy ,b'3
b2eB,b'2eB'
for any 61,63 G B and b[, 63 G the last condition follows from ty2 = 1. Applying 24.2.1 to the partially ordered set H — B x B', we see that there is a unique family of elements ^ Z[i>-1] defined for 61,62 G B and 61,63 G B', such that 7r
61,6i;bi,6/1 = Ii G v - 1 Z [ v - 1 ] if (61,6i) ^ (62,6'2);
Ttbub'ite^ = 0 unless (61, &i) > (62,6/2); = ]Cb3, 6^61,6i;63, b'3Pb3, b'3,b2 ,b'2 for all (61,6;) >(62,6^). We have the following result. T h e o r e m 27.3.2. (a) For any (bi,b[) G B x there is a unique element 6j6i G C such that ^ ( 6 i 0 6 ; ) = 6i6i and (6i6i) - 61 6^ G v~lC. (b) The element b\^b\ in (a) is equal to bi®b[ plus a linear combination of elements 62 62 with (62,63) ^ B x B', (62,62) < (6i,6'i) and with coefficients in v~1Z[v~1]. (c) The elements b\with 61, b\ as above, form a Q(v)-basis M 0 M', an A-basis of ^C and a 7*[v~l]-basis of C.
of
27.3. Tensor Product of Based Modules
221
b\O&i j u s t defined satisfy the requirements of (b),(c) and that (d) holds. It remains to show the uniqueness in (a). It is enough to show that an element x G v~1C such that x = x is necessarily 0. But this follows from (d). 27.3.3. The previous result, together with the known behaviour of bases at oo under tensor product, (see 20.2.2) shows that (M M',B) is a based module with associated involution This is by definition the tensor product of the objects (M, 15), (M\ B'). 27.3.4. Let A, A' G X+. Applying the previous construction to M = and M' = Ay regarded as based modules (with respect to the canonical bases), we obtain a basis of uAA Cg> AAS which clearly is the same as that constructed in 24.3.3. Thus, U A\ Ay, together with its canonical basis in 24.3.3, is a based module. P r o p o s i t i o n 27.3.5. Let A, A7, A" G X+. (a) The U-modules M = ^AA+A' ® AA'+A" and M' = U A \ ® Ay with their canonical bases B,B' constructed in 24-3.3, are in C; moreover, t : M —> M' (see 25.1.5) is a morphism in C. (b) For any Ax G
we have t(B[Ai]) C B'[Ax] U {0}.
The fact that (M, B), (M', B') are objects of C has been pointed out in 27.3.4. The second assertion of (a) follows from Proposition 25.1.10. Now (b) follows from (a) and 27.2.2. 27.3.6. Associativity of t e n s o r p r o d u c t . Let (M, J3), (M', B'), and ( M " , B " ) be three based modules. On the U-module M M' M", we can introduce two structures of based module: one by applying the construction in 27.3.2 first to M M' and then to (M 0 M') ® M"\ the second one by applying the construction in 27.3.2 first to M' M" and then to M (M'®M"). Let Bi, B2 be the bases of MM'®M" obtained by these two constructions. We show that B\ = B2. By definition, the associated involutions to these two structures are given by ^
(A
^ L X E ^ E ^ R ® - & > - )
and J2 (1 ® A ) ( B ^ ) 0 ^ ( ~ ® _ ® " )
222
21. Based Modules
respectively. These coincide by 4.2.4. Next from the definitions, we see that the Z[v _1 ]-submodules of M M' ® M" generated by B\ or B2 coincide; they both coincide with the moreover, Z[v~1 ]-submodule C of M M' ® M" generated by B®B'®B"; if 7r : C —* C/v~lC is the canonical projection, then n(B\) = 7r(B2) = tt(B ® B' B"). To show that Bx = £ 2 , it suffices to show that (bb')b" = K>(&'0&") for any b G B,b' G B',b" G B". Let bx = (bb')Ob" G Bx and b2 = b0(b'0>b") G B2. From the definitions, we have that n(bi) = 7r(6 6' 6") and 7t(62) = n(b b' 6"). Hence 7r(&i) = 7r(fe2). Then b\ — b2 G and 6i — b2 is fixed by the associated involution. This forces &i = 6 2 , as required. Thus we may omit brackets and write b()b'()b" instead of ( b § b ' ) W or b§(b'§b"). This implies automatically that the analogous associativity result is also true for more than three factors. 27.3.7. C o i n v a r i a n t s in a t e n s o r p r o d u c t . Let (M, B),(M', B') be The two based modules. We form their tensor product (M M',B$). following result describes the subset [0] of B^. P r o p o s i t i o n 27.3.8. Let be B,b' G B'. We have Bol0] = Uy e X +{bOb'\b e B{-w0(X')}lo,b'
G
B'[\']hi}.
Let b e B,b' e B' be two elements such that b G M A , b' G M ' A \ According to 27.2.6, the condition that b$b' belongs to £[0] is that A + A' = 0 and Fi(b^}bf) G v~1L(M 1. Let ( n , . . . ,ijv) and
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011
224
28.1. Monomials
225
si2si3 * • • siN so that w = Siiiv'. Let bf (resp. b) be the unique element in the canonical basis of A A which lies in the tz/(A)-weight space (resp. the w( A)weight space). Using the induction hypothesis, we see that it is enough to prove that b = F^b'. We have w{A) - s^w^X) = w'(X) - {ii,w'(X))i\ = w'(A) — a\i[ so that f / " 1 V is a non-zero vector in the same weight space as 6; thus, F^l]b' = fb for some / G A - {0}. Next we note that Ei x b' — 0, since the (w'(A) + i'J-weight space is zero. Otherwise, the w/~1(w'(X) + ii )-weight space would be non-zero, hence the (A -f u; /-1 (z / 1 ))-weight space would be non-zero, contradicting the fact that A is the highest weight, since ty / - 1 (i / 1 ) > 0. From the definition of F{ lt it then follows that = F ^ b ' . By the properties of the basis at oo of AA, the previous equality implies that f = c mod v~*A where c is 0 or 1. Hence we have / = c mod v~l7*[v~l] where c is as above and / ^ 0. The involution
> AA keeps b, b' fixed and we have F£x))V
: AA
=
pMy fb = Jb = fb. It follows that / = / ; hence f = c. = FMh>. hence Since / ^ 0 and c is 0 or 1, it follows that / = 1. The proposition is proved. P r o p o s i t i o n 28.1.5. We have
A)) =
A(9(W0,
9(WQ,
—
WQ(X)).
Let i = (ii,«2» • • • i ^ n ) be a sequence in W such that SilSi2 • • • S{N is a reduced expression of Define a 1,02,...,ajv as in 28.1.1. Then i' = • • • s^ is a reduced expression of WQ. (^tsMat-i, .. - , i i ) is such that S{NSIN_1 Let 61,62,..., 6AT be defined by bi = {sh • • •
(iN), -iu 0 (A)), • • • , bN-1 = (sh (i 2 ),
-w0(X)},
bN - (ii, -^o(A)); then 61 = ajv,&2 = g n - i , . . .
= a\. Using 28.1.2, we have
a(e(w0, A)) = A]. Replacing M by M[> A], we are reduced to the case where M = M[> A] (see 27.2.4(a)). Then replacing M by M/M[> A], we are reduced to the case where M = M[A] (see 27.2.4(b)). Using 27.1.7, we are reduced to the case where (M, B) is A with its canonical basis. In this case, we have B[X]hi = {rj} and B[X]l° = {£} and the result follows from 28.1.4.
28.2. The Isomorphism P
229
28.2.8. P r o o f of P r o p o s i t i o n 28.2.4. Let B'^ (resp. ) be the basis of A ® Af (resp. M A)) defined as in 27.3.3, in terms of the based modules (A, P i ) and (M, P ) . Here P i is the canonical basis of A. Let 7r : A ® M — (A ® M)* and 7r' : M A —• (M ® A)* be the canonical maps. We know that TT defines a bijection of P^[0] onto a basis (P^)* of (A M)* and TT' defines a bijection of Pj>[0] onto a basis (P
-w0(X)].
Then we have P(b) = b'2r} modulo the kernel of 7r'. Note that M[> -iw0(A)l (8> A is contained in the kernel of TT' . By Lemma 28.2.6, we have b'2 rj = b'2 A'; using the definition of P , it would follow that A G P , a contradiction. We have proved that U[P] acts as zero on AA, hence AA may be regarded as a U/U[P]-module. This module is simple. Indeed, even as a U-module it has no proper submodules. It is clear that for A ^ A' in X+ — P , the U/U[P]-modules AA,AA' are not isomorphic (they are not isomorphic as U-modules). By the standard theory of finite dimensional algebras, it follows that dim(U/U[P]) > ^ ( d i m A A ) 2 A
(sum over all A G X+ — P). On the other hand, by 29.1.6(a), we have dim(U/U[P]) = ] T u [ > A]/U[> A] < ^ ( d i m A A ) 2 A
A
(both sums over all A G — P). Comparing with the previous inequality, we see that dim(U/U[P]) = ^ ( d i m A A ) 2 A
and dimU[> A]/U[> A] = (dimA A ) 2 for any A G X+ — P. This implies the following result. P r o p o s i t i o n 29.2.2. (a) The algebra (with 1) U / U [ P ] is semisimple and a complete set of simple modules for it is given by AA with A G X+ — P. (b) For any A G X+, the homomorphism U[> A]/U[> A] —• End(A\) (see 29.1.6(a)) is an isomorphism. Actually, we get (b) for A G X+ — P; but for any A G X+ we can find P as above, not containing A.
29.3. The Refined Peter-Weyl
Theorem
233
29.2.3. From the definition, we see that the finite dimensional semisimple algebra U / U [ P ] inherits from U a canonical basis, formed by the non-zero elements in the image of B. 2 9 . 3 . T H E REFINED P E T E R - W E Y L T H E O R E M
L e m m a 29.3.1. Let A G X+. (a) The anti-automorphisms U [ > -wo( )}.
S, 5',cr : U —• U carry U[> A] onto
(b) The automorphism u : U —> U carries U[> A] onto U[> — wo(A)]. Ay are such that Let u U[> A]. Assume that A' X+ and m S(u)m ^ 0. The map 6y : "Ay ® Ay -> Q(v) (see 25.1.4) may be considered as a non-degenerate pairing; hence there exists m! "Ay such that 8y(m' ® S(u)m) ± 0. Using 28.2.1, we see that Sy(m' ® S(u)m) = 6y(um' m); hence um' / 0. Since "Ay = A_ Wo (a'), we see using 29.1.3, that — wo(X') > A, or equivalently, that A' > — ^o(A). Using again 29.1.3, we deduce that S(u) U[> — wo(A)]. An entirely similar proof shows that S'(u) U[> —i«o(A)]. Thus ( U [ > A]) U[> -w0(\)] for all A and S'(;U[> -^o(A)]) C U[> A] for all A. Since SS' = S'S = 1, the assertions about S and S' in (a) are proved. The assertion about a follows from the assertion for 5, using 23.1.7 and the fact that U[> A] is generated by elements in B which are contained in the summands of the decomposition 23.1.2 of U. We prove (b). Let u G U[> A]. Assume that G X+ and ra G Aa' are such that a ) ( u ) m ^ 0. Then um ^ 0 in "'Aa' which is isomorphic to A_u,0(a')5 hence, by 29.1.3, we have WQ > A or equivalently, Using again 29.1.3, we deduce that u(u) G U[> — woA]. Thus, U( [> A]) C U [ > -w Similarly, w(U[> - ^ o ( A ) ] ) C U [ > A]. The lemma follows. 29.3.2. We shall use the following terminology: an element (3 G B is said to be involutive if au;(/3) = ±{3. (Recall that era; = ua maps B to ± B . ) The following theorem is, in part, a summary of the results above. T h e o r e m 29.3.3. Given X G X+, we define U[> A] (resp. U[> A]j as the set of all u G U with the following property: if X' G X+ and u acts on Ay by a non-zero linear map, then X' > X (resp. X' > X). (a) U[> A] and U[> A] are two-sided ideals of U, which are generated as vector spaces by their intersections with B . The quotient algebra
234
29. A Refinement of the Peter- Weyl Theorem
U[> A]/U[> A] is isomorphic (via the action of U on A\) to the algebra End(A\); in particular, it is finite dimensional and has a unit element, denoted by 1A- Let n : U[> A] —• U[> A]/U[> A] be the natural projection. (b) There is a unique direct sum decomposition of U[> A]/U[> A] into a direct sum of simple left U-modules such that each summand is generated by its intersection with the basis 7r(B[A]) of U[> A]/U[> A]. (c) There is a unique direct sum decomposition of\J[> A]/U[> A] into a direct sum of simple right U-modules such that each summand is generated by its intersection with the basis 7r(B[A]) of U[> A]/U[> A]. (d) Any summand in the decomposition (b) and any summand in the decomposition (c) have an intersection equal to a line consisting of all multiples of some element in the basis 7r(B[A]). This gives a map from the set of all pairs consisting of a summand in the decomposition (b) and one in the decomposition (c), to the set 7r(B[A]). This map is a bijection. (e) Each summand in the decomposition (b) and each summand in the decomposition (c) contains a unique element of the form 7t(/3) where (3 G B[A] is involutive. (f) Let b,bf G B[A]. There exists b" G B[A] and Cb,b',b" £ A such that bb' = c6f6',b"&" mod U[> A]. (a) has already been proved, (b) follows from the definitions, using 27.1.7, 27.1.8 with M = (UAX, ® A A ")[> A]/(WAa' ® A a «)[> A] and with Ai = A (for various V, A" G X+). (c) follows from (b) using the anti-automorphism a u = uj<j of U which maps B into itself, up to signs, (see 26.3.2) and maps U[> A] and U[> A] into themselves (see 29.3.1). We prove (d). The two subspaces considered in the first sentence of (d) are a minimal left ideal and a minimal right ideal in the algebra U[> A]/U[> A] which has 1, and is finite dimensional and simple (by (a)). Their intersection is therefore a line. Since both these subspaces are spanned by a subset of the basis 7r(B[A]), the same is true about their intersection, and the first assertion of (d) follows. The map in the second sentence of (d) is obviously surjective. It is a map between two finite sets of the same cardinality (dim AA)2 (see 29.2.2); hence it is a bijection. We prove (e). Let G b e a summand in the decomposition (b). The map aw : U —• U induces an involution t of the vector space U[> A]/U[> A]. The image of G under i is a summand in the decomposition (c), which by (d) intersects G in a line spanned by a vector in 7r(B[A]). This line is necessarily stable under L (since t is an involution); hence our vector in this
29.4-
Cells
235
line is preserved up to a sign by i. This proves (e) as far as G is concerned. The same proof applies to summands in the decomposition (c). We prove (f). The product 7r(6)7r(6/) is in the intersection of the left ideal of U[> A]/U[> A] generated by ir{b') with the right ideal generated by 7r(b), hence, by (d), is of the form ,b"7r(b") for some b" G B[A] and some Cb,v,b" £ Q(v), which is necessarily in A since the structure constants of the algebra U with respect to B are in A. The theorem is proved. 29.4.
CELLS
2 9 . 4 . 1 . The subsets B[A] (for various A G are called two-sided cells; they form a partition of B. For each A, the two-sided cell B[A] is further partitioned into subsets corresponding to the bases of the various summands in the decomposition 29.3.3(b) (these are called left cells) and it is also partitioned into subsets corresponding to the bases of the various summands in the decomposition 29.3.3(c) (these are called right cells). Then 29.3.3(d) asserts that any left cell in B[A] and any right cell in B[A] have exactly one element in common; 29.3.3(e) asserts that any left cell and any right cell contain exactly one involutive element. Since the number of left cells (or right cells) in B[A] is dimA A , it follows that the number of involutive elements in B[A] is also dimA A .
29.4.2. Let A be an associative algebra over a field K with a given basis B as a K-vector space. We do not assume that A has 1. The structure constants Cb,b',b" £ K oi A (where b, b', b" G B) are defined by bb' = c b,b',b"b". Generalizing the definition of cells in Weyl groups (which goes back to A. Joseph), we will define certain preorders on B as follows. If 6, b' G B, we say that b' {,a,6a+1 ^ 0 (resp. Cbaipatba+1 0) for s = 1 , 2 , . . . , n— 1. We say that b' 0, we consider the subset
6 ( n ) = { ^ a ) l _ n J F ; ( 6 ) ; n > a + b} U {F^ lnE$a) ;n > a + 6} of the canonical basis B (with the identification 25.3.1(c)). Note that 6(N) consists of (n + 1 ) 2 elements. The product of two elements of 6 ( n ) is given by the following equalities (modulo a linear combination of elements in 6(n + l ) U 6 ( n + 2 ) U - ) : E\a)\.nFld) E\ah^nFiu>Eri-nFr
M) =
a
Fln~a} 1nE\n~d)
+d
\ib = c,n 0,2a < n and with a > 0,2a < n, with (a) _ r W a) a ) if % _ , the identification = F l \ n E \ if 2a = n.
29.5. The Quantum Coordinate 29.5.
Algebra
237
T H E QUANTUM COORDINATE ALGEBRA
2 9 . 5 . 1 . Let O be the vector space of all Q(v)-linear forms / : U —> Q(V)
with the following property: / vanishes on U[> A] for some A G X+. If all a G B, then the linear form a : U —• Q(t>) given by a(a') = 8a,afor a G B, belongs to O and {a|a G B} is a basis of O. This follows from 29.3.3. We define an algebra structure on O by the rule ab = Xlc7™?6^, where c runs over B and rh"b are as in 25.4.1. The previous sum is well-defined: all but finitely many terms are zero. This product is associative by 25.4.1(b). c The linear map A : O —» O ® O given by A(c) = b m aba ® b (with c m ab as in 25.4.1) is well-defined. All but finitely many terms in the sum are zero. This map is called comultiplication. It is coassociative by 25.4.1(a) and it is an algebra homomorphism by 25.4.1(c). The element lo is a unit element for this algebra. Consider the linear function O —• Q ( f ) which takes a to 1 if a = 1a for some A G X, and otherwise, to zero. This is an algebra homomorphism. Thus O becomes a Hopf algebra called the quantum coordinate algebra. It is easy to see that this definition is the same as the usual one. We call {a|a G B} the canonical basis of O. 29.5.2. Let ^ O be the ,4-submodule of O generated by the basis (a). Since the structure constants are in A, it follows that ^ O inherits from O a structure of Hopf algebra over A. If now R is any commutative ^-algebra with 1, we can define a Hopf algebra over R by ^ O = R®a (AO).
The Canonical Topological Basis
30.1.
T H E DEFINITION OF THE CANONICAL TOPOLOGICAL
BASIS
In this chapter we assume that (/, •) is of finite type. We denote by (U~ ® U+)* the closure of U~ (8) U+ in (U U)f (see 4.1.1). The elements of (U~ ® U + )f are possibly infinite sums of the form Ylb b'eB cb,b'b~ ® with c^b' £ I n this chapter we shall construct + a canonical topological basis of (U~ (g>U )T which gives rise simultaneously to the canonical bases of all tensor products of type A a w Av. Let ~ : (U~ & > (U~~ ® U + )f be the ring involution defined as the unique continuous extension of ~ ® ~~ : U~ U + —> U~ (g> U + . Note that we have G E (U~ U + J (see 4.1.2). By 24.1.6, we can write uniquely 30.1.1.
30.1.2.
From 4.1.2, 4.1.3, it follows that the Q-linear map
given by >P(x) = Ox (product in (U~ U + J) satisfies \I>2 = 1. We clearly have ^ ( / x ) = f*f?(x) for all / E A and all x. Hence if we set
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © © Springer Springer Science+Business Science+Business Media, LLC LLC 2010 2011
238
30.1. Definition
of the Canonical
Topological
Basis
239
The last sum is finite by the previous statements. Applying 24.2.1 to the set H = B x B we see that there is a unique family of elements PbiMfaM £ Z[F -1 ] defined for bi,b'1,b2,b'2 G B such that Pbifi'^biM — 1; PbxKM
€ v~1Z[v~1] if (61,14) +
(hMY,
Pbub'i;b2,b'2 = 0 unless ( 6 1 , b [ ) < (b2,b'2); PbuKfaM = 52b3,b'3Pbi,b>1\b3,b'3rb3,b'3]b2,b!2 for all (bijb'i) < (b2,b2). Thus we have the following result. P r o p o s i t i o n 30.1.3. For any G B x B, there is a unique element + G (U~U y such that Qf3bl,b\ = Pbxft and such that fo^b^-hi ®b[+ is an (infinite) linear combination of elements b2 b2+ with (b2,b2) > (bi,b\) and with coefficients in v~1Z[v~1]. We have /?6l)6> = E b ^ P & i . & i ®
b +
2 -
30.1.4. The elements /?6l>6/ G (U~ 0 U+)T, for various (61, b[) G B x B, are said to form the canonical topological basis of (U~ U + )f. This is not a basis in the strict sense. = where Taking 61 = b'x = 1, we obtain an element T = Tu G U~ U+ for all v and (a)
r 0 = 1 ® 1.
Hence T is an invertible element of (U~ U + )f. By definition, we have GT = T; hence (b)
e = rf~1.
Note also, that if v ^ 0, then Tu is a linear combination of elements b~ bf+ (b,b' G B„) with coefficients in v - 1 Z[u - 1 ]. This property, together with (a),(b), characterizes T. By the general construction in 27.3.2, the 30.1.5. Let A, A' G X+. U-module A\ R (#f opp )-module structure on denoted in the same way. Similarly, the ^ f f opp )-module structure (xxf) : u 1—• x~ux,Jt on ^ U , by change of scalars, induces a R{^R(R{° module structure on R\J denoted in the same way.
PP
(a) the elements b+l\b'~
(b,b' G B,A G X) form a basis of the R-
(b) the elements b~l\b'+
(6,6' G B,A G X) form a basis of the R-
(c) the i?-algebra R\J is generated by the elements E^LX,
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © Springer Science+Business Science+Business Media, LLC 2010 2011
for
245
)-
246
31. The Algebra R\J
31.1.3. We will give an alternative construction of in terms of Let RA be the algebra generated by the symbols ,x~L^x'+ with x £ i?f/y, x' G flfi/', for various i/, i/, and £ G X; these symbols are subject to the following relations: (fl{'>)+lc(tfW)- = (0f>)-lc+oiW(0,)-= £ > ( t> o
0" i
a + b- (z,C) t
t>0
x+l^ =
~a + b-(i, t
= l£_„x~ for x G f„;
(x+l c )(l c >x'~) = Using the relations of RA, we easily see that the symbols x + l ^ x / - generate RA as an .R-module. In other words, the elements b+l^bf~, with b,b' G B and ( E l , generate RA as an -R-module. Since they form an R-basis of RXJ, they must also form an R-basis of RA and we deduce that: (a) the natural algebra homomorphism RA —> RU is an isomorphism. 31.1.4. There is a natural i?-linear involution cr : R{ —> it is given by a change of rings from the analogous involution for R = A, which is the restriction of a : f —> f. The automorphism u : U —> U restricts to an automorphism u : ^ U —> ^ U ; tensoring with R, we obtain an i?-algebra automorphism u : RU —> KU.
31.1.5. As in 23.1.4, we say that a ^U-module M is unital if (a) for any m G M we have \\m = 0 for all but finitely many A G X; (b) for any m G M we have XIagx
=
m
-
31.1. Definition of R\J
247
We then have a direct sum decomposition (as an abelian group) M = where Mx — 1\M; we can regard M as an i?-module by rm = ®\exMx ^ A ( r l A ) ( m ) for r G R,m G M. Then the decomposition above is as an /2-module. The unital # U-modules are the objects of an abelian category RC with the morphisms being homomorphisms of ^ U-modules. 31.1.6. Let M G RC, let i G I and let n € Z. We define iMinear : M ^ M by E\n)m = E A C ^ 1 * ) ™ and maps E\n) : M 1 F^m = E A C ^ * ) ™ for all m G M. (Recall that E\n) 1 a and i^ ( n ) l A are elements of B C U, hence are well-defined in /?U.) It follows immediately : M —> M) and 0?n) ^ (*\ : M from the definitions that D\N) H-> i > x+m and M) define two Rf-module structures on M, denoted by x,m — x, m H-> x~m respectively. We have (a) E\n)Mx
C Mx+ni\
F^n)Mx
C Mx~ni'
for any i G I,n G Z and
x e x. Moreover, for any £ G X and any m G (b) E^F^m
= F^E^m
(c) E ^ F ^ m =
E
t
we have
if i ± j; >
0
^ r ^ ^ F ^ E ^ m ,
(d) F ^ E ^ m = E t > o H [ ~ a + b 7 { i , ° ] i ) E i a ~ t ) F ? ~ t ) m ' 31.1.7. Conversely, let M be an i2-module with a given direct sum decomand given /^-linear maps E\N),FLN) (for position M = (B^ZXM^ i G J , n G Z) satisfying 31.1.6(a)-(d) and such that E\N) = F^N) = 0 for {E\U) : M - M) and 0{'^lcb)ax ® bx'. (All but finitely many terms in the last sum are zero.) The fact that the rule above defines an # U-module structure follows from the identity 25.4.1(c). This U-module is unital, by the identity 25.4.1(d). Thus M ®R M' is naturally an object of RC. Now let M, M', M" be three objects of RC. By the previous construction, the H-module M <S>R M' ®R M" can be regarded as an object of RC in two
248
31. The Algebra R\J
ways, (M ®R M') ®R M" and M (Mf ®R M"). In fact these two ways coincide; this follows from the identity 25.4.1(b). 31.1.9. From the definitions it is clear that, in the case where R = Q(v),v = v, we have RC = C and the tensor product just defined coincides with the one introduced earlier for C. 31.1.10. To any object M of RC, we associate (as in 3.4.4) a new object of RC as follows. u M has the same underlying jR-module as M. By definition, for any u G /{U, the operator m on W M coincides with the operator aJ(U) on M. 31.1.11. If R' —R is a homomorphism of commutative *4-algebras with 1, we have ^ U = R R> ( k ' U ) and for any object M G R>C, we may regard R ®R> M naturally as an object in RC with the induced # U-module structure. This gives a functor R>C —> RC called change of rings, or change of scalars. It commutes with tensor products (as in 31.1.8) and with the operation u in 31.1.10. 31.1.12. Let (Y',X',...) be another root datum of type (/, •) and let / : Y' —> y, g : X —> X' be a morphism of root data. This induces a homomorphism : U 7 —• U between the corresponding Drinfeld-Jimbo algebras (see 3.1.2). For each oUUlA+ni') ->
flUlA
MX
0,
where the first map has components given by right multiplication by 1 x + n i ' E ^ and the second map is given by u — i > ul (1 is the canonical generator of M\). This is deduced by tensoring with R from the analogous exact sequence over
31.2. Integrable RU -modules
249
Let M e RC and let M e Mx be such that E\n)m = 0 for a l M e l and all n > 0. From the previous exact sequence, we see that there is a unique morphism t : M\ —• M such that t(l) = ra. 31.2.
INTEGRABLE
/JU-MODULES
31.2.1. In this section we assume that the root datum is y-regular (except of A v is a unital ^ U in 31.2.4). Let A, A' e X+. The ,4-submodule submodule (see 23.3.7); by change of scalars, it gives rise to an object RAX>
— R A U A
v
)
of
RC.
Similarly, the .4-submodule ^AA r a " ® ra m' for all ra G M A ,ra' G M A ',ra" G M A ". Proposition 32.2.2. (a) The map
256
32. Commutativity
Isomorphism
coincides with the composition 1
1
—> M (8> M M
M®M ®M
• M M ® M .
(b) The map
coincides with the composition M " ® M ® M '
M ® M " ® M '
' " P ' * " ' - " " ,
M ® M ' ® M " .
Let ( f ' , d f ) be another pair like (/, d), but with d! = 1; thus / ' has values in Z. Assume that the proposition holds for / replaced by / ' ; as in the proof of Theorem 32.1.5, we see that it also holds for / ' . Thus, in the rest of the proof, we shall assume that d = 1 so that / takes values in Z; we then have v = v. Using the characterization of integrable objects given in 31.2.7, we are reduced to the case where each of M, M ' , M" is of the form ^A A ®R (RA\>)-, since such objects are obtained by change of rings from the analogous objects over A, we may assume that R = A; this case can be obviously reduced to the case where R = Q(f). We may assume therefore that R = Q(v). Let me Mx,m' e M'x\m" e M"x". Using the definitions and 4.2.2(b), we have fI^M",M®M'{m 0 m ' ® m") = t/( a "> a + a ') ^
©„(ra" (ra ra7)) V
and (/7£m",M ® 1 M')(!M ® fI^M",M')(m = vfix">x,) ^2(fnM",M
® m ' ® m")
® 1 M>)&l3(rn m" ra')
On the other hand, using the definitions and 4.2.2(a), we have fft<M®M>,M"(m" ®m®rn') v'y
= vf(x+x^x")
y^ Q„((m ® ra') ® ra")
32.2.
The Hexagon
Property
257
and ( 1 M ® j1Z>M',M")(f'R>MYM"
f{x x,,)
= v ' (iM
®
1
M'){M"
m
ra')
The proposition follows since /(A" - i/", A) - /(A", A) = /(0, A) - /(*/', A) and / ( V , A" + v") - /(A', A") = /(A', i/") - /(A', 0) for i/" G Z [ / ] . L e m m a 32.2.3. Let d > 1 be the order of the torsion subgroup of X/Z[I]. There exists a symmetric 7i-bilinear pairing f : X x X —> ^Z which satisfies 32.1.3(a). We can find a direct sum decomposition X = X\ © X2 such that Z [I] is contained in X\ as a subgroup of index d. There is a unique symmetric bilinear pairing fi : X\ xXi —> ^Z such that fi(i', j') = -i-j for all i,j G I. fi(xi,x[). For xi,x[ G Xi and x2,x'2 G X2, we set f(xi 4- x2,x[ -f x2) — This has the required properties. P r o p o s i t i o n 32.2.4 ( H e x a g o n p r o p e r t y ) . Let M , M ' , M " G RC. Let ( f , d ) be as in the previous lemma. Assume that we are given an element v G R such that vd = v. Then the map /7£M",MM' : M <S> M' ® M" —• M " M ' coincides with the composition M ® M ' ® M "
I
M
%
,
1
I
M
"
M
\ M ® M " ® M '
M " ® M ® M '
and the map /^MM',M"
: M"
® M ® M '
M ® M '
®M"
coincides with the composition
This follows from Proposition 32.2.2, since in our case, /II 7 , / I I " are the identity maps.
CHAPTER
33
Relation with Kac-Moody Lie Algebras 33.1.
T H E SPECIALIZATION v =
1
Let j / f be the free associative algebra over R with generators OI (i E I). As for 'f, which corresponds to the case R = Q(^), we have a natural direct sum decomposition = S i / ^ ' f u ) where v runs over N[/]; each % is a free /2-module of finite rank. 33.1.1.
Let be the quotient of the algebra R{ by the two-sided ideal of R{ generated by the elements .
E
(-1)p>(
P + P .
P
for various i ^ j in I. Recall that : A —> R is given. Let nf u be the image of R i u under the natural map —> It is clear that we have a direct sum decomposition = @ u (Rf l/ ). From the definition, we have, for any z/, an exact sequence of i?-modules (K%") —•
v
fifv
—+ 0
where the indices satisfy i / , i / ' G N[J] and z/ + z/' + (l — (z, j'))i+j = v\ the first map has components x,x' x$ijx'. If we take this exact sequence for R = Q[v, v" 1 ] and we tensor it over Q[i>,i;-1] with Q(v) or with RQ (a field of characteristic zero, regarded as an *4-algebra or Q[i7, -algebra via v i—• 1), we obtain again exact sequences, by the right exactness of tensor product. We deduce that RQf„ — RQ (Q[v,t;-i]f|/) and Q(v)lv
= Q ( V ) ® (Q[V,V-I]FI/).
Since Q[V)V-i]f„ is a finitely generated Q[t;,t>_1]-module and Q(v) is the quotient field of Q[v, v - 1 ], we deduce that (a) d i m Q W ( Q W f t , ) < dim
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010 DOI 10.1007/978-0-8176-4717-9_30, © Springer
258
33.1. The Specialization v =1
259
By 1.4.3, there is a unique (surjective) algebra homomorphism Q(v)f —• f which takes Qi to Qi for all i (and preserves 1). It is clear that this homomorphism maps Q( v )f u onto f„, hence (b) dimQ(v) f„ < dimQ( v )( Q ( v )f I/ ) for any v G N[/]. Note that ^ f is the ifo-algebra defined by the generators Qi (i G I) and the Serre relations p+p' = l-(i,j')
for various i ^ j in I. Thus it is the enveloping algebra of (the upper triangular part of) the corresponding Kac-Moody Lie algebra over Ro. 33.1.2. Assume now that the root datum is y-regular and X-regular. Let A G X+ and let M — R^A* G R^C'. The linear maps EiyFi : M —• M satisfy in our case: (a) EiM< C FiM< C for any i G I and ( G l ; (b) (EiFj - FjEi)m = Sij(i, £)m for any i, j G I and m G M c ; (c) T,p+p>=i-(i,j>)(-1)p'(Ei/Pl)Ej(Ef in / ;
/P'1) = 0 : M
M for any i ^ j
(d) = 0 : M - M for any i ? j in I. This shows that M is an integrable highest weight module of the KacMoody Lie algebra attached to the root datum. By results in [3], namely, the complete reducibility theorem of Weyl-Kac and the Gabber-Kac theorem, M is simple as a module of that Lie algebra and the Ro-\meoi map i
given by KK--K
^FilFi2'--Fivr)x
is an isomorphism. It follows that (e) RqA\ is a simple object of R0C and (f) for given v G N [/], we have dim/?0(/i0fly) = dim R o ( f l o A A -") provided that (i, A) are large enough for all i.
260
33. Relation with Kac-Moody
Lie Algebras
From the definition of A\ and R0A\, it is clear that (g) dim f l 0 (fl 0 Aj" , / ) = dim Q ( v ) A ^ - " for any A, v and (h) for given v £ N[/], we have dimq( v )(f I/ ) = dimq( v ) A* - " provided that (i, A) are large enough for all i. Prom (f),(g),(h) we deduce that dimQ( v )(fj,) = dim/^tfofj,) for all v. Combining this with the inequalities 33.1.1(a),(b), we see that those inequalities are in fact equalities. In particular, the natural surjective f must be an isomorphism. Similarly, the natural homomorphism Q(v)f surjective homomorphism n 0 f —• # 0 f is an isomorphism since dimHoUo^) > dim flo (f I/ ) = dim Q ( t; )(f I/ ) = d i m ^ ^ f , , ) for all v. Thus we have the following result. T h e o r e m 33.1.3. (a) The natural algebra homomorphism Q(„)f —> f is an isomorphism. (b) We have diniQ^) iu = dim^ 0 (fl 0 f I/ ) for any v. (c) The natural algebra homomorphism
Roi
—•
is an isomorphism.
(d) If A £ X+, then the dimension of the weight spaces of A\ are the same as those of the simple integrable highest weight representation of the corresponding Kac-Moody Lie algebra. 33.1.4. R e m a r k . Parts (a), (b) and (c) of the theorem hold for arbitrary root data, since only the Cartan datum is used in their statement. Corollary 33.1.5. The algebra U can be defined by the generators Ei (i £ I), Fi (i £ / ) , KM (p £ Y) and the relations 3.1.1(a)-(d), together with the quantum Serre relations for the Ei's and for the Fi's. 3 3 . 2 . T H E QUASI-CLASSICAL CASE
33.2.1. In this section we assume that the ^4-algebra R is a field of characteristic zero and {yi) — ± 1 in R for all i £ I. We then say that we are in the quasi-classical case; this is justified by the results in this section. We also assume that (/, •) is without odd cycles (see 2.1.3). Then, by 2.1.3, we can find a function
33.2. The Quasi-Classical
Case
261
(a) i t—• ai from I to {0,1} such that ai + a j = 1 whenever (i, j') < 0. Let Ro be the ring R with a new ,4-algebra structure in which v £ A is mapped to 1 € R. We want to relate the algebras and R0f. We cannot do this directly, but must enlarge them first as follows. Let A be the i?-algebra defined by the generators Qi, Ki (i £ I) subject to the following relations: the Qi satisfy the relations of the Ki commute among themselves and KiQj = ^ [ v i ^ ^ O j k u for all i,j £ I. Let Ao be the i?-algebra defined by the generators Qi, ki (i £ I) subject to the following relations: the Qi satisfy the relations of j ^ f , the k i commute among themselves and kiQj = (vi)(l,:> )Qjki, for all i,j £ I. It is clear that, as an /^-vector space, A (resp. ylo) is the tensor product of ftf (resp. # 0 f) with the group algebra of Z1 over R, with basis given by the monomials in k i . P r o p o s i t i o n 33.2.2. (a) The assignment Ei E[ = Eikand for all i, defines an isomorphism of R-algebras AQ —• A.
ki
> ki
(b) dim f l ( f l f I / ) = dimfl(/ Jo f |/ ) for all v. (c) The natural algebra homomorphism
—> r{ is an isomorphism.
Let i, j £ / be distinct. A simple computation shows that we have in A: =
E'fE'jE'f
The factor ^(VJ^CP+P'XP+P'-1) is 1 since (vi) = ± 1 and the exponent is even. By the definition of ai, we have
and this equals < j > ( i i f p + p' = 1 — {i,j'}- Note also that 4>([n]\) = since (vi) = ±1. Hence, if p + p' = 1 - (i,j;), then
=
<j)(vi)(p+p'Kp+p'-1)/2(j>(vi rpp'ci>(vi)p^-p)p\p,}-
It follows that E
(-ir'(E^/p\)E'J(El"'/p'\)
p+p' =
l~{i,j')
=
ttepSM-VJ'))/'
262
33. Relation with Kac-Moody Lie Algebras
This shows that the assignment in (a) preserves relations, hence defines an algebra homomorphism A0 —• A. The same proof shows that the asKi defines an algebra homomorphism signment Ei »-> EiK~ a i and Ki A — i t is clear that this is the inverse of the previous homomorphism. This proves (a). For any v, the isomorphism in (a) maps the subspace f u isomorphically onto the subspace (NFU)K where A' is a monomial in the KI depending only on v. This proves (b). The homomorphism in (c) maps onto Riu for any V. Using (b), it is an isomorphism. This implies (c). follows that this restriction RIv —> The proposition is proved. The next result compares the algebras # 0 U and Proposition 33.2.3. There is a unique isomorphism of R-algebras f : H0U —> such that f(Eil c) = cf>(vi)a^0Eilof(Filc)
=
= lc
for alii € I, C G X. Here ai is as in 33.2.1(a). To construct / , we take advantage of the fact that for the algebra # 0 U, we know a simple presentation by generators and relations, while for we do not. It will be easier to first construct a functor from RC to RqC and then to show that it comes from an algebra homomorphism. Let M be an object of RC. The linear maps EI, FI : M —*• M (i G I) satisfy (a) EiM x C Mx+i',
FiM x C M x - { ' for any iel
(b) (EiFj - FjEi)m = M;
and A G X;
A)m for any i j G I and m €
x
= 0 :M
(c) any i ± j in I;
M for
(d) =0:M M for any i ^ j in I. For any a G Z, we define linear maps Pi,a ' M —• M by Pi>a(m(vi)a(l'x)m for m G Mx. We define linear maps E[, F[ : M —> M in terms of the function 33.2.1(a) by E\ = EiP^ai,F( = <J>(vi)FiPiti_a.. As in the proof of the previous proposition, we can check that: (al) E[MX C M A + i \
F(MX C MA-*' for any iel
and A G X;
(bl) (E{Fj - F<E%)m = 6ij(i, A )m for any i j el and m G M A ;
33.2. The Quasi-Classical
Case
263
(cl) E p + ^ i - ^ ^ t - i F ' f ^ / p O ^ t ^ ' / p ' ! ) = 0 : M i ^ j in / ;
M for any
(dl) = 0 : M - M for any i j in / . Since /^f = # 0 f (see 33.1.3) has the presentation given by the Serre relations, it follows that M with its weight space decomposition and with the linear maps E'I} F( is an object of R0C. Thus, we have defined a functor T : RC —• R0C. This functor has the following obvious property: it associates to M an object T(M) with the same underlying R-module as M and any endomorphism of M in RC is at the same time an endomorphism of F(M) in RqC. Applying the functor T to flU, regarded as a left module over itself, we obtain a structure of a unital left U-module on Thus, we have an i?-bilinear pairing ^ U x —• ftU, denoted by a, b a * 6; this has the properties (aa') *b = a* (a' *b) and (a * b)b' = a * (bb'). The last property holds since right multiplication by b is an endomorphism of in RC, hence also in R0C. We define a map / • R 0 U —• flU by f(a) = E c g x a * ^C' o n ^y finitely many terms in the sum are non-zero. We have /(a)/(a') = $ > *
lC)(a'*lC')
= Ea*E1c(«,*1c) c c * (a' *
= c
=
E
a a
' *
=
f(aa')-
c
It is clear that / has the specified values on the algebra generators ^ l c , F a c , l c °f u. We show that / is an isomorphism. The elements are algebra generators of ^ U , since 4>([N]\) = ± n ! is invertible in R for any i £ I and any n > 0. It follows that / is surjective. As in 31.1.2, (resp. RQIJ) is a free RF ^f o p p -module (resp. /j 0 f ® R0fopp-module) with generators (under (x x') : u »-*• x+ux'~ ), and / carries the subspace of ^ U spanned by x+l^x'~ with X £ © t r V 1 . In this chapter we assume that the AR and with v = <j)(v), is such that R is an integral algebra R, with (j) : A domain and the following hold vzl = 1 and wlt ^ 1 for all 0 < t < Z. All the identities proved in 1.3 for Gaussian binomial coefficients imply, after applying , corresponding identities in R. However, certain identities will be satisfied only in R. We shall now give some examples of such identities. L e m m a 34.1.2. (a) If t > 1 is not divisible by I, and a G Z is divisible by I, then 0([®]) = 0. (b) If a\ G Z and t\ G N, then we have
(c) Let a G Z and t G N. Write a = ao + la\ with ao> a i G Z such that 0 < ao < I ~ 1 and t = + Hi with to^ti £ N such that 0 < to < I — 1. We have 2 a a ^ ^ = v (aoti-aito)«+(ai + l)tii 0 ^ ° ^ Here ("*) G Z is an ordinary binomial coefficient. We prove (a) for a > 0 by induction on a. If a — 0, we have trivially ["] = 0; if a = Z, the equality (f>{ [*]) = 0 follows directly from the definitions. Assume now that a > 21 and that (a) holds for a — I instead of a. By 1.3.1(e), we have
\L J /
\L
J/
\ L J /
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © © Springer Springer Science+Business Science+Business Media, LLC 2010 2011
265
266
34-1- Gaussian Binomial Coefficients at Roots of 1 266
For each term in the sum, we have that either t' or t" is not divisible by /; hence the sum is zero by the induction hypothesis. This proves (a) for a > 0. We now prove (a), assuming that a < 0. Write t = to + lt\ with 0 0. We have MP\-/(MP)=P]-vl2p{p-1)/2>
(a)
We prove (a) by induction on p. If p = 0 or 1, then (a) is trivial. Assume that p > 2. We have 4>m'/(mp) = m i p - ^ / ( w r v
Q'f]) •
Using 34.1.2 and the induction hypothesis, we see that
This proves (a).
268
34-1- Gaussian Binomial Coefficients at Roots of 1 268
L e m m a 34.1.4. Assume that 0 < r < a < 1. We have (a) E U " ' 1 ( - 1 ) ' " ' + 1 + < ' v " ( ' " r ) ( a " ' + 1 + < ' > + ^ ( [ i 7 ] ) =
v (
' °~rV(["])-
In the left hand side of (a) we may replace v i2 ~ z by (—l) z+1 . Note also that I — r > 1; hence l-r i)v q=0
l-r .
Q .
= 0
(see 1.3.4). Hence the left hand side of (a) equals (-ly+lv^a+l-O-'a
l-r
l-r
( - 1 )9V9(1~Z+rV
q—l — a
.
Q .
q' = 0
\l
Q
J/
For any q' in the last sum, we have by 34.1.2(c), {[l~,r]) = v' 9 V([~/ r ]) • Thus the left hand side of (a) is a-r ^ ^^a-r-q'^Iq'+
(_iy+l-r+lvr(a+l-l)-la
(l-r-q')(l-l+r)^
q'= 0 =
=
(_l}a+l-r+lwl-(l-r)l-la
^
1
,r
r
^L9 ^-q'+r(a-r- =0 - r - 1 (_1 \a+l-r+l l-(l-r)l-la a—r a (^y+lvl-d-rV-la^ a—r
We have used vl = v
/
. The lemma is proved.
CHAPTER
35
The Quantum Frobenius Homomorphism
35.1.1. In this chapter we fix an integer I > 1. Then the integers h > 1, the new Cartan datum (/, o) and the new root datum (y*, X * , . . . ) of type (/, o) are defined in terms of Z, (/, •), (y, X , . . . ) as in 2.2.4, 2.2.5. 35.1.2. The assumptions (a),(b) below will be in force in this chapter, (a) for any i ^ j in I such that lj > 2 , we have li > — (z, j'} + 1; Note that (a) is automatically satisfied in the simply laced case: in that case, we have li = Z for all in the general case, the assumption (a) can be violated only by finitely many Z. Note also that (b) is automatically
35.1.3. Let Z' be one of the integers Z, 2Z, if Z is odd, and let I' be equal to 2Z, if Z is even. Let A' be the quotient of A by the two-sided ideal generated
In this chapter, we assume that the given ring homomorphism <j>: A —» R factors through a ring homomorphism A! —* or that R is an *4'-algebra 35.1.4. When (/, •) is replaced by (/, o), the element Vi G A, whose definition depends on the Cartan datum, becomes v* = v l ° 1 / 2 = v1^. For any P G A, we denote by P* the element obtained from P by substituting v by v*. For each i G / , we set v* — (vi) and v* — (f>(v*) =
L e m m a 35.1.5. (a) Letae
ZfZ andtG Z*N. We have ([?];) =
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2010 2011
[?/£]*)
269
270
35. The Quantum Frobenius
Homomorphism
(b) Let a G UZ and lett G N be non-divisible by U. We have (f>( [®]i) = 0 (equality in R). It suffices to prove this for R = A'. This is an integral domain in which v f < = 1 and v f ^ 1 for all 0 < t < U. We prove (a). Applying 34.1.2(b) to Vi and li instead of v and I, we see that ^(["j^) = vj0"*"'*^
Applying
the same result to v* and 1 instead of v and /, we see that tf>( [^J*] ) = v *(o+ii)t/i?
(ajhy
It remains to use the
equality v* = v f . The proof of (b)
is entirely similar; it uses 34.1.2(a). 35.1.6. Let f* (resp. U*) be the Q(v)-algebra defined like f (resp. like U), in terms of the Cartan matrix (/, o) (resp. in terms of (F*, X*, ...))• Then the .R-algebras r{, Rf*, r U , are well-defined. We state the main results of this chapter. T h e o r e m 35.1.7. Recall that R is an A'-algebra. There is a unique Ralgebra homomorphism Fr : ^f —> rV such that for all i e I and n G Z, Fr(e\n)) equals 6^ n//i) if n G UTi, and equals 0, otherwise. T h e o r e m 35.1.8. There is a unique R-algebra homomorphism n) = e\nli) for all i € I and all n G Z. Rf* -> Ri such that Fr'{0l )
Fr' :
T h e o r e m 35.1.9. There is a unique R-algebra homomorphism Fr : RU —> ftU* such that for all i G / , n G Z and £ G X, we have: F r ( ^ n ) l c ) equals E\n/li)lc wise;
if n G UZ and < G X*, and equals 0, other-
Fr(F^l^) wise.
if n G kZ and £ G X*, and equals 0, other-
equals
We give a proof of the last theorem, assuming that theorem 35.1.7 is known. Using the presentation of the algebra RU in terms of RF given in 31.1.3, and the analogous presentation of the algebra in terms of Rf*, we see that it is enough to prove that the assignment rc+l^x- ' • Fr(x+)l(v2) in the multiplicative group of R is /. Thus, we have v 2 i = 1 and v 2 t ^ 1 for all 0 < t < I. By the definition of k, we have v 2Zi = 1 and v?* ± 1 for all 0 < t < U. In particular, ([n]\) is invertible 212 in R, if 0 < n < For any i we have v* = ± 1 since v i 1 = 1 ; hence, when dealing with the algebras RU*, we are in the quasi-classical case (see 33.2). L e m m a 35.2.2. The R-algebra Rf is generated by the elements of^ I) and by the elements 9{ for i G I such that U>2. Recall that the i?-algebra R{ is generated by the elements i and n > 0. Writing n = a + lib with 0 < a < k and be ^ s i n g 34 1 2 ) . On the other hand, e(n) = vabiie(a)e(hb) (a) e\ a ) = ^ ( H i ) - 1 * ? (see 35.2.1) and (b)
= (6!)"ivf
The equality (b) follows from the equalities (0/2
f
and M M / A U M
(see 34.1.3). The lemma is proved.
-
)
(i G
for various N, we have
272
35. The Quantum Frobenius
Homomorphism
35.2.3. We now give the proof of Theorem 35.1.8. As observed in 35.1.11, it is enough to prove the existence statement in 35.1.8, assuming that R is as in 35.2.1. We will show that there exists an algebra homomorphism Rf* —> such that Oi H-> for all i. Since the assumption 35.1.2(b) is in force, the algebra r{* has a presentation given by the generators 6i and the Serretype relations; this follows from 33.2.2(c) . Since ) = 0It remains to prove (b). Let a = —(i,j'). For any q G [0, U — 1] we set gq =
£
(-i)S(''-1-,)«{r)ej,,)«<w e
^
r+s=alj+li — q
This is fij-ijfaij-Mi-g;-i m the notation of 7.1.1. Let g = Y!qZo{-l)qvfiq+liq~qgq0\q). By the higher order quantum Serre relations (see 7.1.5(b)), we have gq = 0 for all q € [0, li — 1]; hence g = 0. On the other hand, setting s' = s + q, we have g = E r ^ r + y ^ - N , CrA^'W* ^ li-1 Y^(-iy+*vri(li-1-qHotljq+liq-q s Crs, = L9J
9=0
Taking the image of g = 0 under the obvious map ^f —> Rf, we obtain r,s' \r+s'=alj +li
For fixed s', we write s' = a + lib where 0 < a < k — 1. We have = v r ^ ' ^ C l i ) ( s e e 34.1.2); hence
ftcrs) = (-l)rv-(,i_1> E ( " l ) < v f < p q=0
q=0
^ L9-1 »'
35.3. Highest
Weight Modules of RU
273
We have used the identity v | i ( / < - 1 ) = see 34.1.2(e). (b) follows. Thus, we have an algebra homomorphism Fr' : Rf* —• Rf such that Fr'(6i) = o\li) for all i. To complete the proof we must compute Fr'(9^) for n > 0. We have Fr'(9^n)) = mrfYir'Fr'iOir
= vtrI?n(n-1)/2(n!)-1(^, ~{i,j') + 1. The following identity holds in Ri:
r=0
^L
r
-!«'
We set a = ~(i,j'). Using Corollary 7.1.7 with m = li,n = 1, we see that we are reduced to checking the identity h-a-l
E
for any r G [0, a]. This follows from Lemma 34.1.4. P r o p o s i t i o n 35.3.2. Assume that the root datum is X-regular. Let A G X be such that (i, A) G UZ for all i G I. Let M be a simple highest weight module with highest weight A in RC and let rj be a generator of the R-vector space Mx. (a) / / ( G X satisfies M^ ^ 0, then £ = A — particular, (i,Q G UZ for all i G I.
knii', where ni G N . In
(b) If i G / is such that li > 2, then Ei, Fi act as zero on M. (c) For any r > 0, let M'r be the subspace of M spanned by the vectors F^^F^1^ • • • F^^rj for various sequences 2i,«2, ... ,ir in I. Let M' — M
'v
Then M' = M.
Clearly, M'r is spanned by vectors in M^ where C is of the form £ = A — ^iUnii', with ni G N. Such £ satisfies (i,() G UZ for all i G I. We use the fact that, for j G I, the integer lj(i,j') is divisible by UWe show by induction on r > 0, that
274
35. The Quantum Frobenius
Homomorphism
(d) EiM'r = 0, FiM'r = 0 for any i e I such that U > 2. Assume first that r = 0. Then E{MQ = 0 is obvious. Assume that for some iel such that li > 2, we have x = Fit] ^ 0. For any j e / , we have EjX = EjFirj = FiEjf] + 6id ([ 2, we have ^ ( [ ^ L ) = > Ejx = n > 2, then = E^Firj is an iMinear combination of FiEj^rj and of
Ejn~^rj, hence is again zero. Thus, E^x = 0 for all j e I and all n > 0; since x e Mx~l , there exists a unique morphism in RC from the Verma module M\-i> into M which takes the canonical generator to x; its image is a subobject of M containing x but not rj. Since M is simple, we must have x = 0. Thus (d) holds for r = 0. Assume now that r > 1 and that (d) holds for r — 1. To show that it holds for r, it suffices to show that E i F ^ m = 0, F i F ^ m = 0 for any i,j in I such that k > 2 and any m e If lj > 2, then E i F ^ m is an R-linear combination of F^Eim and of Fj3~1m, hence is zero since Eim = 0, Fjm = 0, by the induction hypothesis. If lj = 1, then EiFjm = FjEim + Sijfi)^'®].)™, where (i,f) e hZ, hence ^ ( [ ^ L ) = 0, as above; since ^ m = 0, by the induction hypothesis, we have again EiFjm = 0. If i ^ j , then from the identity in 35.3.1, we deduce by interchanging i,j and applying cr, that FiF^m is an .R-linear combination of FiF^m for various r with 0 < r < —(j,if) < lj. For such r we have FiF$ r ) m = 0. (Indeed, if lj > 2, then F i F ^ m = 0, by the induction hypothesis; if lj = 1, then r = 0 and F i F ^ m = Fim = 0, again by the induction hypothesis.) Thus, we have FiF^m = 0. If i = j, then F i F f j ) m = F f j ) F i m = 0, by the induction hypothesis. This completes the inductive proof of (d). Next we show by induction on r > 0 that (e) E\li)M'r C M;_ for any i e / , where, by convention, M'_x = 0. This is clear for r = 0. Assume now that r > 1. We must show that
Fj1^m' e M'r_l for any j and any m e
is an iMinear combination of F^E^m! (which M'r_x. Now E^F^m! is in M'r_x by the induction hypothesis) and of elements E^^m' with t > 0 such that t . Thus, we have a unital #U*-module structure on M . By 35.3.2(c), this is a highest weight module of R \J* with highest weight A. This ftU*-module is simple. Indeed, assume that M" is a non-zero ^U*-submodule of M. Then M" is the sum of its intersections with the various (with ( 6 X ) and is stable under all : M —• M (in the ^U-module structure). Now M" is automatically stable under E f , F? (in the /^U-module structure) for any i and a such that 0 < a M' ® M\ to the equality (c). We obtain =0 (d) Zbvn(b)y;(n')®Fr'(b)-(r,) where the sum is taken over b subject to tr \b\ = N. By 35.3.3, we may regard M as a #U*-module; this is a simple highest weight module of RU* which is just RA\ (see 35.3.4). Note also that Fr'(b)~r) in the RU-module structure is the same as b~r) in the ^U*-module structure. We shall assume, as we may, that (i, A) are not only divisible by li, but are also large for all i, so that the vectors b~rj G M are linearly independent when b is subject to tr \b\ = N (a finite set of b's). Here we use that M = RA\ as a /?U*-module. Then from (d) we deduce that Y^(RJ') = 0, hence 2/6 = 0 for all b such that tr |6| = N. (We use the fact that M' is a Verma module.) This is a contradiction. The theorem is proved. 35.4.3. We assume that the root datum is simply connected. Then there is a unique A G X+ such that (i, A) = li — 1 for all i. Let 77 be the canonical generator of RA\. P r o p o s i t i o n 35.4.4. The map x
x~rj is an R-linear isomorphism f —>
Let J = Ei ! n>/ i (H f 6 , i U ) )- lt suffices to show that J © f = R f . An equivalent statement is that a( J) © f = since f is cr-stable. We have a are suc n ( J ) = Y2in>i ^i^flf- If h — then we can write n = a + kb with 0 < a < li and b > 1 and we use the formulas in the proof of Lemma 35.2.2. We see that 0\n) C 0\li)Rf. It follows that cr (J) = S i ^i'^fif- The fact that and f are complementary subspaces of R{ follows easily from Theorem 35.4.2(b). The proposition follows.
278
35. The Quantum Frobenius
Homomorphism
3 5 . 5 . PROOF OF THEOREM 3 5 . 1 . 7
35.5.1. As we have seen in 35.1.11, we only have to prove existence in 35.1.7, assuming that R is as in 35.2.1. By Theorem 35.4.2, there exists a unique i^-linear map P : Rf —• such that P(0^l) • • • 9?*n)9jl • • • 9jr) is equal to 9h • • • 0in if r = 0 and to 0 if r > 0. (Here z'i,..., zn is any sequence in I and j i , . . . , j r is any sequence 2.) in I such that lh >2,...,ljr> We show that P is an algebra homomorphism. It suffices to show that (a) P(x9i) — P(x)P(9i) for any x E R{ and any i such that h>2
and
(b) P(x9^li)) = P(x)P(9\li)) for any x e Rf and any i. (a) is obvious. We prove (b) for x = of*1^ • • • • • • 9jr by induction on r > 0. The case where r = 0 is trivial. Assume that r > 1. We have x = x'Oj where j = jr and x' = e[[11} ••• 9?*n)9jl • • • 9ir_x. If i ^ j then, using the identity in 35.3.1 (after applying J6'f-t)i'*i
X
l
X
For each £ in the sum we have 0 ' = t t (see 34.1.2); here we use the hypothesis that Z is odd. Hence we have j f >« =
E * t>0;t (Z/IZ)1. Prom the definition, we see that is the free 0 implies b' > 0 since a" — b" + c + g -f an > 0. The sum becomes ^_1y"ve((a"c-b")+(a"+c)(-a"+b"-c-g)) a",b",c€H;0—• u" is an algebra homomorphism. In an entirely similar way we see that (c) holds and that u' is an algebra homomorphism. It remains to show that (u")' = u and (u')" = u for all u. For any M, 2 as above, we have, using 5.2.3(a): (u'Y'z = T[ieTi:_e(u')"z
=
T ^ T ^ Z )
=
=
UZ.
Thus (u')" — u annihilates any integrable U-module, hence (u')" = u. The identity (u")' = u is proved in the same way. The proposition is proved.
292
37. The Symmetries
T'i e
,T{e
37.2.4. We have UT^LJ
= T[[e : U
U and aT[ea = T['_e : U
U.
The symmetries T/ e , T"e are related to the involution ~ : U —> U as follows: KM Let
= ir,-.(«).
for a11
=
G U be such that KiuK-i
«
€
u
-
= v"u We have
= (-i)B«rii», or equivalently, These formulas are proved by checking on generators. Proposition 37.2.5. Let i / j in I. For any m G Z, we have (a) -^i.eC^iJjl.mje) = xi,j;l,— (i,j')-m;e> (b) 37'_e(aJ*,j;l,m;c) = ^i,j; 1, — < i ) — m.;e * The two formulas are equivalent, by 37.1.2(d), so it suffices to prove (a). For m < 0 or m > —(i,j'}, both sides of (a) are zero. So we may assume that 0 < m < —(i,f). We argue by descending induction on ra. For ra = — (i,j'), (a) follows from 37.2.2(b). Assume now that (a) is known for some ra such that 1 < m < ~{i,j'). We show that it is then also true for ra — 1. Recall from 7.1.2(b) the identity x
ITI 1]iK—eixi,j',l,m—l;e'
i,j;l,m\eFi — [ (i, j )
Applying to it the algebra isomorphism a : U —> U o p p , which interchanges x i,j;i,m;e and x'jj.i )Tn;e , we deduce that ~Xi,j-,l,m-,eFi + FiXi,j;l,m;e = H » , / } ~
m
+ 1]
We apply T[ e to this; using the induction hypothesis, we have H i , j') - m + 1] =
Tle{xfi j ; l
=
x
r n . e )EiK_ e i
- E i K ^ l M ^ e )
i,j\\, — (ij')— m,eEiK— ei — EiK— eixi,j;l,
/
-rp
e((ij')+2m) jp
— {ij')—m;e
\f>
37.3. Relation
of the Symmetries
with
Comultiplication
293
The last equality follows from 7.1.2(a). Since [— ( i , f ) — m +1]* 0, we deduce that T / e ^ m _ 1 ; e ) = x i ) j ; i ) _( i ) j /)_ m + i ; e . This proves the induction step. The proposition is proved. 3 7 . 3 . RELATION OF THE SYMMETRIES WITH COMULTIPLICATION 37.3.1.
We define two elements 14' = £ ( - i ) X B ( B " 1 > / 9 { » } i * i B ) ® L/ =
^vn(n-l)/2{n].F(n) ^
E(n)
of (U V J (see 4.1.1 and 4.1.2). P r o p o s i t i o n 37.3.2. The following equality holds in (U ® UJT, for any uG U : (a)
( T ^ t t T ^ A i T ^ u ) = L5A(ti)L?.
It suffices to show that for any integrable U-modules M, N, the two sides of (a) act in the same way on M7V. Let x G M, y G N. Using 5.3.4 twice, we have
L$A(u)L"(x ® y) = L'!-\uL'l(x
The proposition is proved.
® y))
Symmetries and Inner product on f
38.1.1. In this section we fix i £ I. For any j € I distinct from i and for
(fihj'i7?1)
is the same as the element /i,j ; i,m;-i in 7.1.1.)
Let f [i] (resp. af[i]) be the subalgebra of f generated by the elements f(i,j;m) (resp. f'(i,j-,m)) for various j £ / , distinct from z, and various ra £ Z. Since a : f —> f interchanges f(i,j;m) and f'(i,j\m), we have
Note that (b) follows from (a) by applying cr. We prove (a). We consider a product y\V2" 'Vn of elements in f in which each factor is either Oi or is of the form / ( i , j\ ra) for some j different from i and some ra £ Z. Assume that there are two consecutive factors ya = / ( i , j\m) and ya+1 = Oi. Using
which follows from 7.1.2(a), we see that we may replace yaya+1 by a scalar multiple of ya+i2/a plus a scalar multiple of f(i,j;m + 1). Using this procedure repeatedly, we see that y\y2 • • • yn is equal to a linear combination of products y[y2 • • • y'n, which do not have consecutive factors of the form described above, hence have the property that for some s > 0, we have various j,ra. Thus, y\y2 •••y n belongs to Y 2 t > a n y wor< ^ the generators 0j ( j £ I) is of the form yxyi- — yn with ya as above, since
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © Springer Science+Business Media, LLC 2010 2011
294
38.1. The Algebras f[i),
a
f\i\
295
L e m m a 38.1.3. There is a unique algebra isomorphism x > g(x) of f[t] onto ai[i] such that T/ / 1 (x + ) = g(x)+ for any x e f[i]. We have Vi_1(x,+) = g~l{x')+ for any x' E ai[i).
Since T"\,T'i _l are inverse algebra isomorphisms, it suffices to note that
and Tl,-i(f(iJ;m)+)
-(i,j')-m)+,
= f(i,j;
for any j ^ i and any ra E Z. (See 37.2.5.) L e m m a 38.1.4. Assume that x e f satisfies T"1(x+) G U + . Then ir(x) = 0. We may assume that x is homogeneous. By 3.1.6, we have (a)
=
Vi-Vi
By 38.1.2, we can write ri(x)/(vi~v71)
=
J2e\% t> 0
and t>o
where yt,zt belong to f[i] and are homogeneous. By 38.1.3, we have T"\{yt) e u + and Tl'^zi) e U+ for all t > 0. Applying T^ to both sides of (a), we obtain - T^ix^K-iEi (b)
= £(-1) t> o
+
k.iEiTl'il(x+) ~
KiTl'M))-
By our assumption, the left hand side of (b) is in K - i U + , hence so is the right hand side. Using the triangular decomposition in U, we deduce from (b) that Tl'x{y+) = 0 for all t > 0 and T ^ z f ) = 0 for all t > 0. Since T'A is bijective, this implies zf = 0 for all t, hence zt = 0 for all t and »r(:r) = 0.
296
38. Symmetries
and Inner Product on f
L e m m a 38.1.5. Let xt (t > 0) be elements in the kernel of ir, which are zero for all but finitely many t. Assume that 0^xt = 0. Then xt = 0 for all t. This follows from 16.1.2(c) in the setup of 17.3.1. P r o p o s i t i o n 38.1.6. (a) The following three subspaces o f f coincide: f[t]; {x G f\Tl'yl(x+) G U + } ; and {x G f| w h e r e xt G f[i] for all t. By 38.1.3 and 38.1.4, we have ir(xt) = 0~for all t. We then have 0 = (x0 -x) + J2t>i ^ x t where x0 - x and xt for t > 0 are in the kernel of ^r. Using 38.1.5, we deduce that XQ — x = 0 and xt = 0 for t > 0. In particular, we have x = x0 G f[i]. The proposition is proved. L e m m a 38.1.7. r(f(ij;m))
= 1® m m—t—1
f(i,j;m)
+ E II (i-vrafc+am-2a-2K(m"t)/(i,j;i)®^ra-0;h,h'EH
n;h,h'Zh Oin)Zh>, dnih,h'9{zh)9\n)®g{zh,)t
r( x')= n>0;h,h'€H
where c(n; h, h'), d(n; h, h') G Q(v) are zero for all but finitely many indices. By 3.1.5, we have (c) A(x+) = (d) A(x'+) = Zn>0;h,h>eH dn,h,h. g{zh)+E^ Knht) ®g{zh>)+. By definition, we have A(V+) = A ( T ^ ( x + ) ) . Applying (d) we obtain
to
n>0;h,h'eH
By 37.3.2, we can replace ( T j . j 07; , ) _ 1 )A(r/ , 1 (rE + )) by L / i A(x + )Lf, which by (c) equals n>0,h,h'eH
Thus, we have n>0;h,h' = L5(
J2 n>0,h,hfeH
cn.hih,z+km)+ni&E^z+)b':
38.1.
The Algebras f[«], ai[i]
299
or equivalently, n>0;h,h.'(zH n>0;h,h'EH
(equality in (U ® UJ). We consider the U ® U-module u MQ ® w Mo, where Mo is a Verma module. Both sides of the previous equality act naturally on this U ® Umodule. Applying them to the vector , where f G w Mo is the canonical generator, we obtain the equality
E
n,t>0;h,h'eH
vf-^m-irvr^dn-,^
(e) n,t>0;h,h'eH
(equality in u M Q ) . Since F = 0, only the summands with n = t — 0 contribute to the left hand side of (e). Let 7r : " M Q / E ^ M Q be the canonical map. After applying 1 7r to (e), the summands with (n,t) ^ (0,0) (in the right hand side) go to zero; hence (e) implies the equality
h,h'EH
h,h'eH
U
in M 0 ( " M 0 / E i " M 0 ) . The vectors are linearly independent in u MQ; hence we deduce that (d0;h,h' ~ co;hthf)Azh'0
=
0
h'eH
in "MQ/E^MQ), for all h. Hence y:
(do;h,hf ~ co-h,h')zh' G Oii,
h'EH
for all h. Using the fact that f [i] n ( ^ f ) do,h,h' ~ co;h,hf = 0 for all h, h!. By definition, we have 'r(x) =
=0 c
(see
38.1.5), we
deduce that
o-,h,h'Zh ® zh>, and "r{x')
=
Ylh,h'eH do-,h,h'9(zh) ® 9{zh = CQ.h,h> imply that (9 ® 9)('r(x)) — "r(x'). The lemma is proved.
300
38. Symmetries
38.2.
and Inner Product on f
A COMPUTATION OF INNER PRODUCTS
P r o p o s i t i o n 38.2.1. For all x,y G i[i], we have (a) (g(x),g(y))
= (x,y).
Assume that we are given two elements z', z" G f[i) such that (g(z'),g(y')) and
= (z',y')
W),g{y")) = (z",y")
for all y', y" G f [?]. We show that we then have (g(z'z"),g(y))
=
(z'z",y),
for any y G f[i]. By definition, we have (z'z",y)
=
{z'®z",r(y))
and (g(z'z"),g(y))
=
(g(z')®g(z"),r(g(y))).
We have (z",0,f) = 0 since ir(z") = 0 (see 1.2.13(a)); hence (z'®z",r{y)) = (z'®z"!r{y)) (with the notations of lemma 38.1.8). Similarly, ( g ( z ' ) , f 0 i ) = 0, hence W ) ®9{z"\r{g{y)))
= (g(z') ® g(z"),"r(g(y))) = ({9®9)(z'®z"),(g®g)'r(y)).
The last equality comes from lemma 38.1.8. By our hypothesis, we have ((
