Communications in Mathematical Physics - Volume 204

Commun. Math. Phys. 204, 1 – 16 (1999) Communications in Mathematical Physics © Springer-Verlag 1999 Regular Nilpote...

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Commun. Math. Phys. 204, 1 – 16 (1999)

Communications in

Mathematical Physics

© Springer-Verlag 1999

Regular Nilpotent Elements and Quantum Groups Alexey Sevostyanov? Institute of Theoretical Physics, Uppsala University, Box 803, S-75108 Uppsala, Sweden. E-mail: [email protected] Received: 29 May 1998 / Accepted: 12 January 1999

Abstract: We suggest new realizations of quantum groups Uq (g) corresponding to complex simple Lie algebras, and of affine quantum groups. These new realizations are labeled by Coxeter elements of the corresponding Weyl group and have the following key feature: The natural counterparts of the subalgebras U (n), where n ⊂ g is a maximal nilpotent subalgebra, possess non-singular characters.

Introduction Let g be a complex simple Lie algebra, b a Borel subalgebra, and n = [b, b] its nilradical. We denote by ( , ) the Killing form on g. An element f ∈ n is called regular nilpotent if its centralizer in g is of minimal possible dimension. Any regular nilpotent element f ∈ n defines a character χ of the opposite nilpotent subalgebra n = [b, b], where b is the opposite Borel subalgebra. Naturally, the character χ extends to a character of the universal enveloping algebra U (n). Recall that U (n) is generated by the positive simple root generators Xi+ , i = l, . . . , rank g of the Chevalley basis associated with the pair (g, b). On these generators the character χ takes values ci 6= 0, χ(Xi+ ) = ci . Conversely, any character of this form determines a regular nilpotent element in n. Regular nilpotent elements are of great importance in the structure theory of Lie algebras and in its applications. In particular, there is a relation between regular nilpotent elements of a complex semisimple Lie algebra and Coxeter elements of the corresponding Weyl group [9]. Other applications of regular nilpotent elements include the theory of Whittaker modules in representation theory of semisimple Lie algebras [10], the integrability of the Toda lattice [11], and the remarkable realization of the center of the universal enveloping algebra of a complex simple Lie algebra as a Hecke algebra [10]. ? Address after September 1, 1999: Max-Planck Institut für Mathematik, Box 7280, D-53072 Bonn, Germany

2

A. Sevostyanov

This provides a motivation to look for counterparts of regular nilpotent elements in the theory of quantum groups. Let Uq (g) be the quantum group associated with a complex simple Lie algebra g, and let Uq (n) be the subalgebra of Uq (g) corresponding to the nilpotent Lie subalgebra n ⊂ g. Uq (n) is generated by simple positive root generators of Uq (g) subject to the q-Serre relations. It is easy to show that Uq (n) has no nondegenerate characters (taking nonvanishing values on all simple root generators)! Our first main result is the family of new realizations of the quantum group Uq (g), one for each Coxeter element in the corresponding Weyl group. The counterparts of U (n), which naturally arise in these new realizations of Uq (g), do have non-singular characters. Thus, we get proper quantum counterparts of U (n) and of its non-singular characters. As a byproduct, we derive an interesting formula for the Cayley transform of a Coxeter element. Next, we generalize our consideration to the case of affine Lie algebras. Similar to the finite-dimensional situation, the subalgebra Uq (n((z))) in the affine quantum group g) naturally corresponding to n((z)) has no characters taking nonvanishing values Uq (b on the quantum counterparts of the loop generators of n((z)). Again, we introduce new g), labeled by Coxeter elements, such that the natural counterparts realizations of Uq (b of U (n((z))) acquire such characters. Our realizations are variations of the Drinfeld’s “new realization” of affine quantum groups [3]. The perspective application of our construction is the Drinfeld–Sokolov reduction for affine quantum groups.

1. Non-Singular Characters and Finite-Dimensional Quantum Groups In this section we construct quantum counterparts of the principal nilpotent Lie subalgebras of complex simple Lie algebras and of their non-singular characters. We follow the notation of [7]. Let h∗ be an l-dimensional complex vector space, aij , i, j = 1, . . . , l a Cartan matrix of finite type , 1 ∈ h∗ the corresponding root system, and {α1 , ..., αl } the set of simple roots. Denote by W the Weyl group of the root system 1, and by s1 , ..., sl ∈ W reflections corresponding to simple roots. Let d1 , . . . , dl be coprime positive integers such that the matrix bij = di aij is symmetric. There exists a unique non-degenerate W -invariant scalar product (, ) on h∗ such that (αi , αj ) = bij . Let g be the complex simple Lie algebra associated to the Cartan matrix aij . Denote by n ⊂ g the principal nilpotent subalgebra generated by the simple positive root generators of the Chevalley basis. Definition 1. A character χ : n → C is called non-singular if and only if it takes non-vanishing values on all simple root generators of n. Note that any non-singular character is equivalent (up to a Lie algebra automorphism of n) to χ0 which takes value 1 on each simple root generator. Any character of n naturally extends to a character of the associative algebra U (n). It is our goal to construct quantum counterparts of the algebra U (n) and of the non-singular character χ0 . Let q be a complex number, 0 < |q| < 1. Put qi = q di . We consider the simplyconnected rational form UqR (g) of the quantum group Uq (g) [2, Sect. 9.1], This is an associative algebra over C with generators Xi± , Li , L−1 i , i = 1, . . . , l subject to the

Regular Nilpotent Elements and Quantum Groups

3

relations: −1 Li Lj = Lj Li , Li L−1 i = Li Li = 1, ±δi,j

Li Xj± L−1 i = qi

Xj± ,

Xi+ Xj− − Xj− Xi+ = δi,j Q a Ki = lj =1 Lj j i ,

Ki −Ki−1 , qi −qi−1

(1)

and the q-Serre relations: P1−aij 1 − aij r (Xi± )1−aij −r Xj± (Xi± )r = 0, i 6 = j, r=0 (−1) r q i m [m]q ! q n −q −n = [n]q ![n−m] , [n] ! = [n] . . . [1] , [n] = . where q q q q ! q−q −1 q n q The elements Xi± correspond to the simple positive (negative) root generators. We would like to show that the algebra spanned by Xi+ , i = 1, . . . , l does not admit characters which take nonvanishing values on all generators Xi+ , except for the case of Uq (sl(2)) when q-Serre relations do not appear. Suppose χ is such a character, and χ (Xi ) = ci . The q-Serre relations are homogeneous and, hence, one can put ci = 1 for all i without loss of generality. By applying the character χ to the q-Serre relations one obtains a family of identities, 1−aij

X

r

(−1)

r=0

1 − aij r

qi

= 0, i 6= j.

(2)

We claim that some of these relations fail for the quantized universal enveloping algebra UqR (g) of any simple Lie algebra g , with the exception of g = sl(2). In a more general setting, relations (2) are analysed in the following lemma. Lemma 1. The only rational solutions of equation m X m (−1)k t kc = 0, k t

(3)

k=0

where t is a complex number, 0 < |t| < 1, are of the form c = −m + 1, −m + 2, . . . , m − 2, m − 1.

(4)

Proof. According to the q-binomial theorem [6], m X k=0

(−z)k

m k

t

=

m−1 Y

(1 − t m−1−2p z).

(5)

p=0

Put z = t c in this relation. Then the l.h.s of (5) coincides with the l.h.s. of (3). Now (5) implies that c = m − 1 − 2p, p = 0, . . . , m − 1 are the only rational solutions of (3). t u

4

A. Sevostyanov

Now we return to identities (2). Any Cartan matrix contains at least one off-diagonal element equal to −1. Then, m = 1 − aij = 2 and c = ±1, and Lemma 1 implies that some of identities (2) are false for any simple Lie algebra, except for sl(2). Hence, subalgebras of UqR (g) generated by Xi+ do not possess non-singular characters. It is our goal to construct subalgebras of UqR (g) which resemble the subalgebra U (n) ⊂ U (g) and possess non-singular characters. Denote by Sl the symmetric group of l elements. To any element π ∈ Sl we associate a Coxeter element sπ by the formula sπ = sπ(1) . . . sπ(l) . For each Coxeter element sπ we define an associative algebra Fqπ generated by elements ei , i = 1, . . . l subject to the relations: 1−aij

X

π r rcij

(−1) q

r=0

1 − aij r

qi

(ei )1−aij −r ej (ei )r = 0, i 6 = j,

(6)

π = 1+sπ α , α are matrix elements of the Cayley transform of sπ in the where cij i j 1−sπ basis of simple roots. Proposition 2. The map χqπ : Fqπ → C defined on the generators by χqπ (ei ) = 1 is a character of the algebra Fqπ . To show that χqπ is a character of Fqπ it is sufficient to check that the defining relations (6) belong to the kernel of χqπ , i.e. 1−aij

X r=0

π r rcij

(−1) q

1 − aij r

qi

= 0, i 6 = j.

(7)

As a preparation for the proof of Proposition 2 we study the matrix elements of the Cayley transform of sπ which enter the definition of Fqπ . π Lemma 3. The matrix elements of 1+s 1−sπ are of the form: 1 + sπ π αi , αj = εij bij , 1 − sπ

where

  −1 π −1 (i) < π −1 (j ) π . εij = 0 i = j  1 π −1 (i) > π −1 (j )

(8)

(9)

Proof (Compare [1, Ch. V, §6 , Ex. 3]). First we calculate the matrix of the Coxeter element sπ with respect to the basis of simple roots. We obtain this matrix in the form of the Gauss decomposition of the operator sπ . Let zπ(i) = sπ απ(i) . Recall that si (αj ) = αj − aj i αi . Using this definition the elements zπ(i) may be represented as: X aπ(k)π(i) yπ(k) , zπ(i) = yπ(i) − k≥i

where yπ(i) = sπ(1) . . . sπ(i−1) απ(i) .

(10)


5

Using matrix notation we can rewrite the last formula as follows: zπ(i) = (I + V )π(k)π(i) yπ(k) , ( aπ(k)π(i) k ≥ i . where Vπ(k)π(i) = 0 k

(11)

To calculate the matrix of the operator sπ with respect to the basis of simple roots we have to express the elements yπ(i) via the simple roots. Applying the definition of simple reflections to (10) we can pull out the element απ(i) to the right: yπ(i) = απ(i) −

X

aπ(k)π(i) yπ(k) .

k

Therefore απ(i) = (I + U )π(k)π(i) yπ(k) , where Uπ(k)π(i) =

aπ(k)π(i) k < i . 0 k≥i

Thus yπ(k) = (I + U )−1 π(j )π(k) απ(j ) .

(12)

Summarizing (12) and (11) we obtain: sπ αi = (I + U )−1 (I − V ) αk . ki

(13)

This implies: 1 + sπ αi = 1 − sπ

2I + U − V U +V

ki

αk .

(14)

π . Substituting these Observe that (U + V )ki = aki and (2I + U − V )ij = −aij εij expressions into (14) we get:

1 + sπ αi , αj 1 − sπ

π api bj k = = −(a −1 )kp εpi

π π api = εij bij . − dj aj k (a −1 )kp εpi

(15) (16)

This concludes the proof of the lemma. u t Proof of Proposition 2. Identities (7) follow from Lemma 1 for t = qi , m = 1−aij , c = π a since set of solutions (4) always contains ±(m − 1). u t εij ij Motivated by relations (6) we suggest new realizations of the quantum group UqR (g), one for each Coxeter element sπ . Let Uqπ (g) be the associative algebra over C with

6

A. Sevostyanov

generators ei , fi , L±1 i i = 1, . . . l subject to the relations: −1 Li Lj = Lj Li , Li L−1 i = Li Li = 1, δ

−δi,j

i,j −1 Li ej L−1 i = qi ej , Li fj Li = qi

fj ,

Ki −Ki−1 qi −qi−1

π cij

, ei fj − q fj ei = δi,j Ql aj i Ki = j =1 Lj , π P1−aij r q rcij 1 − aij (−1) (ei )1−aij −r ej (ei )r = 0, i 6= j, r=0 r qi π P1−aij 1 − aij rcij r (fi )1−aij −r fj (fi )r = 0, i 6= j. r=0 (−1) q r q

(17)

i

It follows that the map τqπ : Fqπ → Uqπ (g); ei 7→ ei is a natural embedding of Fqπ into Uqπ (g). From now on we identify Fqπ with the subalgebra in Uqπ (g) generated by ei , i = 1, . . . l. Theorem 4. For every integer-valued solution nij ∈ Z, i, j = 1, . . . , l of equations π −di nj i + dj nij = cij

(18)

there exists an algebra isomorphism ψ{n} : Uqπ (g) → UqR (g) defined by formulas: Q n ψ{n} (ei ) = qi−nii lp=1 Lpip Xi+ , Q −n ψ{n} (fi ) = lp=1 Lp ip Xi− ,

(19)

ψ{n} (Li ) = Li . Proof. The proof is provided by direct verification of defining relations (17). The most nontrivial part is to verify deformed q-Serre relations (6). The defining relations of UqR (g) imply the following relations for ψ{n} (ei ), 1−aij

X k=0

k

(−1)

1 − aij k

qi

q −k(di nj i −dj nij ) ψ{n} (ei )1−aij −k ψ{n} ej ψ{n} (ei )k = 0,

for any i 6 = j . Now using Eq. (18) we arrive at relations (6). u t Remark 1. The general solution of Eq. (18) is given by nij =

sij 1 π ), (ε aj i + 2 ij dj

(20)

where sij = sj i . In order to show that integer solutions, nij ∈ Z, exist, we choose π + 1)a , and this is an integer. sij = bij . Then, nij = 21 (εij ji We call the algebra Uqπ (g) the Coxeter realization of the quantum group UqR (g) corresponding to the Coxeter element sπ . The subalgebra Fqπ and the character χqπ are quantum counterparts of U (n) and of the non-singular character χ0 , respectively.


7

Remark 2. Let nij be a solution of the homogeneous system that corresponds to (18), di nj i − dj nij = 0. Then the map defined by Q n Xi+ 7 → qi−nii lp=1 Lpip Xi+ , Q −n Xi− 7 → lp=1 Lp ip Xi− ,

(21)

Li 7 → Li is an automorphism of UqR (g). Therefore for given Coxeter element the isomorphism ψ{n} is defined uniquely up to automorphisms of UqR (g). g) Remark 3. Letb g be an arbitrary affine Lie algebra. Recall that the quantum group UqR (b g. If b g is defined by the same commutation relations (1), where aij is the Cartan matrix of b is not finite-dimensional the Cartan matrix aij is degenerate, and from (13) it follows that 1 is an eigenvalue of every Coxeter element sπ . Therefore the Cayley transforms of these π = ε b elements do not exist, and formula (8) no longer holds. However, putting cij ij ij in (17), where εij is a skew-symmetric matrix, εij = ±1 for i 6 = j , one can still define realizations of UqR (b g) such that the subalgebras generated by ei , i = 1, . . . , l possess non-singular characters. In the next section we shall also construct quantum counterparts of the loop algebras n((z)) ⊂ b g and of their non-singular characters for nontwisted affine quantum groups.

2. Non-Singular Characters and Affine Quantum Groups In this section we suggest new realizations of affine quantum groups labeled by Coxeter elements, similar to those described in the previous section for finite-dimensional quantum groups. ·

Let b g = g((z)) + C be the nontwisted affine Lie algebra corresponding to g and let n((z)) ⊂ b g be the loop algebra of the nilpotent Lie subalgebra n ⊂ g. Let χ be the character of n which takes value 1 on all root generators of n. χ has a unique extension to the character χ b of n((z)), such that χ b vanishes on the complement z−1 n[[z−1 ]] + zn[[z]] of n in n((z)). It is our goal to define quantum counterparts of the algebra U (n((z))) ⊂ U (b g) and of the character χ b. We start with the new Drinfeld realization of quantum affine algebras generalizing the loop realization of affine Lie algebras. ± g) be an associative algebra generated by elements Xi,r , r ∈ Z, Hi,r , r ∈ Let UqR (b c ±1 ±2 Z\{0}, Li , i = 1, . . . l , q . Put Xi± (u) =

P

± −r r∈Z Xi,r u ,

P∞ P∞ ± ∓r = K ±1 exp ±(q − q −1 ) ∓s , 8± i r=0 8i,±r u s=1 Hi,±s u i (u) = i i Q a Ki = lj =1 Lj j i .

8

A. Sevostyanov

In terms of the generating series the defining relations are [3,8]: −1 Li Lj = Lj Li , Li L−1 i = Li Li = 1, ±δi,j

Li Xj± (u)L−1 i = qi

Xj± (u),

± ± ± ± −1 ± 8± i (u)8j (v) = 8j (v)8i (u) , Li 8j (u)Li = 8j (u), c

c

c

q ± 2 are central , q 2 q − 2 = 1, − 8+ i (u)8j (v) =

c

gij ( vqu )

−c gij ( vqu

)

+ 8− j (v)8i (u), ∓ 2c

± − −1 = g ( uq ±1 ± 8− ij i (u)Xj (v)8i (u) v ) Xj (v), ∓ 2c

± + −1 = g ( vq ∓1 ± 8+ ij i (u)Xj (v)8i (u) u ) Xj (v),

(u − vq ±bij )Xi± (u)Xj± (v) = (q ±bij u − v)Xj± (v)Xi± (u), −c c c c δ (vq 2 ) − δ( uqv )8− (uq 2 ) , Xi+ (u)Xj− (v) − Xj− (v)Xi+ (u) = i,j−1 δ( uqv )8+ i i qi −qi P1−aij P 1 − aij k × π∈S1−aij k=0 (−1) k qi Xi± (zπ(1) ) . . . Xi± (zπ(k) )Xj± (w)Xi± (zπ(k+1) ) . . . Xi± (zπ(1−aij ) ) = 0, i 6= j, where gij (z) =

b

1−q ij z −bij q −b 1−q ij z

∈ C[[z]].

± , Hi,r correspond to the elements Xi± zr , Hi zr of the affine Lie algebra The generators Xi,r b g in the loop realization (here Xi± , Hi are the Chevalley generators of g). ± g) generated by Xi,r , i = 1, . . . l, r ≥ n and Let In , n > 0 be the left ideal in UqR (b ±1 by all polynomials in Hi,r , r > 0, Li of degrees greater or equal to n (deg(Hi,r ) = bR g), r, deg(L±1 i ) = 0). The algebra Uq (b

bqR (b g) = lim UqR (b g)/In (inverse limit) U ←

bqR (b g). We fix k ∈ C and denote by U g)k the is called the restricted completion of UqR (b c k ± ± R bq (b g) by the ideal generated by q 2 − q 2 . Sometimes it is convenient to quotient of U use the weight-type generators Yi,r , r Yi,r = (a r )−1 ik Hk,r , aij =

1 [raij ]qi . r

In the spirit of Theorem 4 we introduce, for any set of complex numbers n±r ij , i, j = 1, . . . l, r ∈ N and integer parameters nij , i, j = 1, . . . l, generating series of the form, {n}

ei (u) = qi−nii 80i where

{n}

{n}

8− i (u)

{n}

Xi+ (u)8+ i (u)

,

P ∞ ±r ∓r = exp Y n u , n±r j,±r r=1 ij ij ∈ C, Ql nj i = j =1 Lj , nij ∈ Z.

(22)

{n}

8± i (u) 80i

{n}

{n}

bqR (b The Fourier coefficients of ei (u) are elements of U g)k .

(23)


9 {n}

Proposition 5. The generating functions ei (u) satisfy the following commutation relations: {n}

{n}

{n}

{n}

(u − vq bij )Fj i ( uv )ei (u)ej (v) = (q bij u − v)Fij ( uv )ej (v)ei (u), # " 1 − aij P1−aij Q P zπ(q) Qk w k π∈S1−aij p 0 be the left ideal in Eq generated by ei,r , r ≥ n. Put E π b b Fq be the quotient of Eq by the two-sided ideal generated by the Fourier coefficients of the following generating series: (u − vq bij )Fj i ( uv )ei (u)ej (v) − (q bij u − v)Fij ( uv )ej (v)ei (u), P1−aij Q P zπ(q) Qk w k 1 − aij (−1) π∈S1−aij p 0 be the left ideal in Aq generated by ei,r , fi,r , r ≥ n and by all polyno+ + −1 + ±1 , Ki,r mials in Ki,r , L±1 i , r ≥ 0 of degrees greater than or equal to n (deg(Ki,r ) = b r, deg(L±1 i ) = 0). Put Aq = lim ← Aq /Jn . For Fij (z) given by (28), (29) we define the following formal power series: Mij (z) = gij (zq −k )−1 Fj i (zq k )Fj i (zq −k )−1 , Gij (z) = Mij (zq −k )Mij (zq k )−1 , Fij− (z) = Fij (zq 2k ).

12

A. Sevostyanov

bπ (b b Let U q,k g) be the quotient of Aq by the two-sided ideal generated by the Fourier coefficients of the following generating series: −1

Ki± (u)Kj± (v) − Kj± (v)Ki± (u) , Ki± (u)Ki± (u)

− 1 , Ki± (u)

−1

Ki± (u) − 1 ,

−1 Li Lj − Lj Li , Li L−1 i − 1 , Li Li − 1 , ± Li Kj± (v)L−1 i − Kj (v), Q a ± Ki,0 − ( lj =1 Lj j i )±1 ,

Ki+ (u)Kj− (v) − Gij ( uv )Kj− (v)Ki+ (u), δ

i,j Li ej (u)L−1 i − qi ej (u),

−δi,j

Li fj (u)L−1 i − qi

fj (u), k

Ki+ (u)ej (v) − Mij ( uv )ej (v)Ki+ (u), Ki+ (u)fj (v) − Mij ( vqu )−1 fj (v)Ki+ (u), Ki− (u)ej (v) − Mj i ( uv )−1 ej (v)Ki− (u), k

Ki− (u)fj (v) − Mj i ( uqv )fj (v)Ki− (u), (u − vq bij )Fj i ( uv )ei (u)ej (v) − (q bij u − v)Fij ( uv )ej (v)ei (u), (u − vq −bij )Fj−i ( uv )fi (u)fj (v) − (q −bij u − v)Fij− ( uv )fj (v)fi (u), P1−aij Q P zπ(q) Qk w k 1 − aij (−1) π∈S1−aij p 0 where E = `(` + 1)λ − t (λ). (Undefined coefficients arα for r < 0 are 0.) Hence, as a function of E, arα = arα (E) is a polynomial of degree r of the form  arα (E) = Ar E r + O(E r−1 ),

Ar = r!

 −1 1 `−r − +j  . 2

r Y j =1

(5.18)

α (E) = 0 by Eiα (i = 1, . . . , 2` + 1). The recursion Let us denote the roots of a2`+1 α α relation (5.17) implies ar (Ei ) = 0 for r ≥ 2` + 1. Hence we obtain a polynomial

32

E. K. Sklyanin, T. Takebe

solution ψ(η) = ψ(η; Eiα ) of (5.15) of the form (5.16) for each i = 1, . . . , 2` + 1, provided that t (λ) = `(` + 1)λ − Eiα .

(5.19)

Conversely, if ψ(η) ∈ V (`) is a solution of the spectral problem (5.15), then there exists certain i for each α = 1, 2, 3 such that ψ(η) = ψ(η; Eiα ). This is proved by expanding the polynomial ψ(η) as in (5.16) and tracing back the above argument. Proposition 5.1. Assume that ω2 = τ is pure imaginary and that parameters zn are all real numbers. Then all Eiα are real and the spectral problem (5.15) is non-degenerate. Namely Eiα 6 = Ejα for distinct i, j and the solutions ψ(η; Eiα ) span the space V (`) . In 1 particular Eiα (i = 1, . . . , 2` + 1) for α = 1, 2, 3 coincide up to order, and a2`+1 (E) = α α 2 3 a2`+1 (E) = a2`+1 (E). Hence we can omit the index α for Ei and a2`+1 (E). Vector ψ(η; Ei ) is an eigenvector of Uα with eigenvalue (−1)` if a`α (Ei ) 6 = 0 and (−1)`+1 if a`α (Ei ) = 0. Proof. Under the assumption τ ∈ iR, operator τˆ (u) (u ∈ R) is an hermitian operator because of (B.11), and hence it is obvious that Eiα are real and that ψ(η; Eiα ) span V (`) . In order to show non-degeneracy of the spectral problem (5.15) we have only to prove that Ei2 are distinct with each other. Define ar2 (E), r < ` + 1, 2 a˜ r (E) := r−l ` + 1 ≤ r ≤ 2` + 1. (−1) ar2 (E), Then the leading coefficient of a˜ r2 is a˜ r2 (E) = A˜ r E r + O(E r−1 ),

A˜ r = |Ar |.

(5.20)

The recursion relation (5.17) is rewritten as 2 2 (E) − kr−2 a˜ r−2 (E), cr a˜ r2 (E) = qr a˜ r−1

(5.21)

where cr = r|` +

1 2

− r|,

`(2` − 1) − 3(r − 1)(2` − r + 1) eα + E, 3 = ` − r + (2` − r + 2)(e1 − e2 )(e2 − e3 ). 2

qr = kr

(5.22)

Since e1 > e2 > e3 under the assumption of the proposition, we have cr > 0 and kr > 0. This fact together with A˜ r > 0 (see (5.20)) implies that all the roots of a˜ r2 (E) are real and distinct by Sturm’s theorem (see, e.g., Chap. IX, §§4–5, [16]). This proves the first statement of the proposition. The operators Uα and τˆ commute and each eigenspace of τˆ is one-dimensional. Hence ψ(η; Ei ) is an eigenvector of Uα . Recall that Uα has eigenvalues (−1)` with multiplicity ` + 1 and (−1)`+1 with multiplicity `. (See §B.2.) If a`α (Ei ) 6= 0, then Uα ψ(η; Ei ) = (−1)` ψ(η; Ei ) because of (B.15). Hence there are at most ` + 1 of Ei ’s such that a`α (Ei ) 6 = 0. In other words, at least ` of Ei ’s satisfy a`α (Ei ) = 0. Since a`α (E) is a polynomial of degree `, this proves the second statement of the proposition. u t

Elliptic Gaudin Model

33

5.2.2. Case ` ∈ 21 + Z. As in the case ` ∈ Z, we consider an expansion (5.16) of a solution ψ(η) of the spectral problem (5.15), but this time we consider the series which terminate at r = ` − 21 : ψ(η) =

`−1/2 X r=0

arα (η − eα )2`−r .

They are parametrized by zeros of the polynomial a α

`+ 21

(5.23)

(E), {Eiα }i=1,...,`+ 1 as in the 2

previous case: ψ(η) = ψ(η; Eiα ). Another set of solutions are obtained from this set by applying the operator Uα : Uα ψ(η; Eiα )

`− 21

=

X r=0

a α 0r (Eiα )(η − eα )r ,

(5.24)

since Uα and τˆ (u) commute. The following proposition is proved in the same manner as Proposition 5.1. Proposition 5.2. Assume that ω2 = τ is pure imaginary and that parameters zn are all real numbers. Then all Eiα are real and Eiα 6 = Ejα for distinct i, j . The solutions ψ(η; Eiα ) and Uα ψ(η; Eiα ) span the space V (`) . In particular Eiα (i = 1, . . . , `+ 21 ) for α = 1, 2, 3 coincide up to order, and a 1 1 (E) = a 2 1 (E) = a 3 1 (E). Hence we can omit the index α for Eiα and a α

`+ 21

(E).

`+ 2

`+ 2

`+ 2

Vectors ψ(η; Ei ) ± Uα ψ(η; Ei ) are eigenvectors of Uα with eigenvalues ∓i. This proposition means that each eigenvalue Ei degenerates with multiplicity two. It was Crawford [17] who first found the relation of these two solutions (one is obtained from the other by operating U2 ) by the explicit expansions of type (5.23), (5.24). See also p.578 of [13]. A. Notations We use the notation for the theta functions with characteristics as follows (see [14]): for a, b = 0, 1, X 2 eπi(n+a/2) τ +2π i(n+a/2)(u+b/2) . (A.1) θab (u; τ ) = n∈Z

Unless otherwise specified, θab (u) = θab (u; τ ). We also use abbreviations d 0 θab = θab (u). θab = θab (0), du u=0

(A.2)

Quasi-periodicity properties of theta functions: θab (u) = (−1)a θab (u + 1) = eπ iτ +2π iu θab (u + τ ).

(A.3)

Parity of thetas: θ00 (−u) = θ00 (u), θ01 (−u) = θ01 (u), θ10 (−u) = θ10 (u), θ11 (−u) = −θ11 (u).

34


A.1. Weierstrass functions. Below we fix ω1 = 1 and ω2 = τ , " # Y u 2 1 u u + 1− exp σ (u) = u , ωmn ωmn 2 ωmn

(A.4)

m,n6=0

where ωmn = mω1 + nω2 , ζ (u) =

σ 0 (u) , σ (u)

℘ (u) = −ζ 0 (u),

σ (u + ωl ) = −σ (u)eηl (2u+ωl ) , ζ (u + ωl ) = ζ (u) + 2ηl , ℘ (u + ωl ) = ℘ (u), where ηl = ζ (ωl /2) , which satisfy η1 ω2 − η2 ω1 = π i. Sigma function is expressed by theta functions as follows: θ11 (u/ω1 ) 2 , σ (u) = ω1 eη1 u /ω1 0 θ11 σ (−u) = −σ (u), ζ (−u) = −ζ (u), ℘ (−z) = ℘ (u), ζ (u) = u−1 + O(u3 ), ℘ (u) = u−2 + O(u2 ). u∼0: σ (u) = u + O(u5 ), Other sigma functions are defined as follows: ω2 σ u + ω1 + η1 2 θ (u/ω ) 2 u 00 1 = e ω1 , σ00 (u) = e−(η1 +η2 )u + ω ω θ (0) 1 2 00 σ 2 ω 1 σ u+ 2 η1 2 θ (u/ω ) u 10 1 = e ω1 , σ10 (u) = e−η1 u ω θ (0) 10 σ 21 σ u + ω22 η1 2 θ (u/ω ) u 01 1 = e ω1 , σ01 (u) = e−η2 u ω θ (0) 2 01 σ 2 which satisfy σg1 g2 (u + ωl ) = (−1)gl eηl (2u+ωl ) σg1 g2 (u), σg1 g2 (0) = 1. σg1 g2 (−u) = σg1 g2 (u), Defining e1 = ℘ (ω1 /2), e2 = ℘ ((ω1 + ω2 )/2), e3 = ℘ (ω2 /2), we have 2 (u) σ 2 (u) σ 2 (u) σ10 + e1 = 00 + e2 = 01 + e3 = ℘ (u), 2 2 σ (u) σ (u) σ 2 (u) e1 + e2 + e3 = 0, 2 2 2 π π π 4 4 θ01 (0) , e1 − e3 = θ00 (0) , e2 − e3 = θ10 (0)4 . e1 − e2 = ω1 ω1 ω1 We also use normalized Weierstraß functions: d d θ11 (u), ℘11 (u) = − ζ11 (u). (A.5) ζ11 (u) = du du


35

B. Realization of Spin ` Representations on an Elliptic Curve We recall here the following realization of the spin ` representation of the Lie algebra sl2 (C). Let e, f , h be the Chevalley generators and define S 1 = e+f , S 2 = −ie+if and S 3 = h. They satisfy the relation [S a , S b ] = 2iS c for any cyclic permutation (a, b, c) of (1, 2, 3) and represented by the Pauli matrices σ a . B.1. Spin ` representations. The representation space V (`) is realized by V (`) =

2` M

C℘ (y)k

k=0

= { even elliptic function f (y) | div(f ) ≥ −4`(Z + τ Z)}.

(B.1)

The generators S a act on this space as differential operators of first order: ρ (`) (S 1 ) =

θ10 (y)2 θ10 θ10 (2y) d + 2` , 0 θ11 θ11 (2y) dy θ11 (y)2

θ00 θ00 (2y) d θ00 (y)2 1 (`) 2 , ρ (S ) = 0 + 2` i θ11 θ11 (2y) dy θ11 (y)2 ρ (`) (S 3 ) =

θ01 (y)2 θ01 θ01 (2y) d + 2` , 0 θ11 θ11 (2y) dy θ11 (y)2

or in terms of usual Weierstraß functions, σ10 (2y) d (`) 1 + 2`(℘ (y) − e1 ) , ρ (S ) = a1 σ (2y) dy σ00 (2y) d (`) 2 + 2`(℘ (y) − e2 ) , ρ (S ) = a2 σ (2y) dy σ01 (2y) d (`) 3 + 2`(℘ (y) − e3 ) , ρ (S ) = a3 σ (2y) dy

(B.2)

(B.3)

where ea = ℘ (ωa¯ /2) (a¯ = 1, 3, 2, ω1 = 1, ω2 = τ , ω3 = 1 + τ ) for a = 1, 2, 3 respectively and 1 i 1 , a2 = √ , a3 = √ . a1 = √ √ √ √ e1 − e2 e1 − e3 e1 − e2 e2 − e3 e2 − e3 e1 − e3 (B.4) This realization is equivalent to the realization on the space of polynomials of degree ≤ 2` (or, sections of a line bundle on P1 (C)), e = x2

d − 2`x, dx

f =−

d , dx

h = 2x

d − 2`x, dx

via a coordinate transformation, x = −θ01 (y; τ/2)/θ00 (y; τ/2), and a gauge transformation: θ00 (y; τ/2) n ϕ(x(y)) ∈ V (`) . {polynomials in x} 3 ϕ(x) 7 → θ11 (y; τ )2

36


Note that this is also obtained by a gauge transformation from a quasi-classical limit of the representation of the Sklyanin algebra on theta functions [18]. The following expression is obtained from the coordinate transformation η = ℘ (y): V (`) =

2` M

Cηk ,

(B.5)

k=0

and S α acts on V (`) as d + 2`(η − eα ) . (B.6) ρ (`) (S α ) = aα ((eα − eβ )(eα − eγ ) − (η − eα )2 ) dη Let us assume that τ is a pure imaginary number. Then, as is well known (see, e.g., [13]), ea are real numbers and e1 > e2 > e3 . This implies that a1 and a3 are real, while a2 is purely imaginary. We introduce the following hermitian form in this representation space: for elliptic functions f (y), g(y) belonging to V (`) defined by (B.1), we define Z f (y¯2 ) g(y1 ) µ(y1 , y2 ), (B.7) hf, gi := C

where the 2-cycle C is defined by C := {(y1 , y2 ) ∈ (C/ 0)2 , y2 = y¯1 }, and the 2-form µ(y1 , y2 ) is defined by µ(y1 , y2 ) := (e1 − e2 )2(`+1) (e2 − e3 )2(`+1) σ (2y2 )σ (y2 )4` σ (2y1 )σ (y1 )4` dy2 ∧ dy1 (B.8) × 2(`+1) 2(`+1) 4i σ00 (y2 − y1 ) σ00 (y2 + y1 ) (℘ (y2 ) − e2 )(℘ (y1 ) − e2 ) −2(`+1) ℘ 0 (y2 )℘ 0 (y1 )dy2 ∧ dy1 . = 1+ (e1 − e2 )(e2 − e3 ) 4i This is nothing but a twisted version of the inner product introduced in [18]. If we take the description of V (`) of the form (B.5), this hermitian form is expressed as follows: Z hf, gi := f (η) ¯ g(η) µ(η, η), ¯ (B.9) C

where the 2-form µ(η, η) ¯ is defined by (η¯ − e2 )(η − e2 ) −2(`+1) d η¯ ∧ dη . µ(η, η) ¯ := 1 + (e1 − e2 )(e2 − e3 ) 2i An orthogonal basis with respect to this inner product is given by {(η − e2 )j }j =0,...,2` : h(η − e2 )j , (η − e2 )k i = 2π

(2j )!!(4` − 2j )!! (e1 − e2 )j +1 (e2 − e3 )j +1 δj k . (B.10) (4` + 2)!!

The generators S a of the Lie algebra sl2 act on the space V (`) as self-adjoint operators: hρ (`) (S a )f, gi = hf, ρ (`) (S a )gi.

(B.11)

This was first proved in [18], but we can check it directly by using formula (B.10). Hence, if u and zn are real numbers, the operator τˆ (u) defined by (2.9) and the integrals of motion Hn defined by (2.16) are hermitian operators on the Hilbert space V with respect to h·, ·i.


37

B.2. Involutions. There are involutive automorphisms of the Lie algebra sl2 defined by Xa (S b ) = (−1)1−δab S b .

(B.12)

These automorphisms are induced on the spin ` representations as Xa (S b ) = Ua−1 S b Ua , where operators Ua : V (`) → V (`) are defined by 2` ω1 ℘ (y) − e1 πi` f y+ , (U1 f )(y) = e √ √ 2 e1 − e2 e1 − e3 2` ω1 + ω2 ℘ (y) − e2 , f y+ (U2 f )(y) = e2πi` √ √ 2 e1 − e2 e2 − e3 2` ω2 ℘ (y) − e3 −πi` f y+ , (U3 f )(y) = e √ √ 2 e1 − e3 e2 − e3

(B.13)

(B.14)

for a elliptic function f (y) ∈ V (`) (cf. [18]). They satisfy commutation relations Uα2 = (−1)2` ,

Uα Uβ = (−1)2` Uβ Uα = Uγ

for any cyclic permutation (α, β, γ ) of (1, 2, 3). The action of these operators on the bases {(η − eα )j }j =0,...,2` is: U1 (η − e1 )j = eπi` (e1 − e2 )j −` (e1 − e3 )j −` (η − e1 )2`−j , U2 (η − e2 )j = eπi(2`−j ) (e1 − e2 )j −` (e2 − e3 )j −` (η − e2 )2`−j , U3 (η − e3 )j = e−πi` (e1 − e3 )j −` (e2 − e3 )j −` (η − e3 )2`−j .

(B.15)

Hence eigenvalues of Ua are (−1)` with multiplicity `+1 and (−1)`+1 with multiplicity ` if ` is an integer, and ±i both with multiplicity ` + 21 if ` is a half of an odd integer. When ω1 = 1 and ω2 is a pure imaginary number, these operators are unitary with respect to the hermitian form (B.7). Acknowledgements. One of the authors (E.S.) is grateful to the Department of Applied Mathematics, University of Leeds, UK, where the main part of the paper was written, for hospitality, and acknowledges the support of EPSRC. The other one (T.T.) is grateful to the Department of Mathematics, University of California at Berkeley, USA, where some part of the work was done, for hospitality. He also acknowledges the support of Postdoctoral Fellowship for Research abroad of JSPS. Special thanks are given to Benjamin Enriquez and Vladimir Rubtsov who gave us their article [9] before publication and explained details.

References 1. 2. 3. 4.

Gaudin, M.: Diagonalisation d’une classe d’hamiltoniens de spin. J. de Physique 37, 1087–1098 (1976) Gaudin, M.: La fonction d’onde de Bethe. Paris: Masson, 1983 Baxter, R. J.: Exactly Solved Models of Statistical Mechanics. New York: Academic Press, 1982 Sklyanin, E.K., Takebe, T.: Algebraic Bethe Ansatz for XYZ Gaudin model. Phys. Lett. A 219, 217–225 (1996) 5. Babujian, H., Lima-Santos, A., Poghossian, R. H.: Knizhnik–Zamolodchikov–Bernard equations connected with the eight-vertex model. Preprint solv-int/9804015 6. Sklyanin, E.K.: Separation of Variables in the Gaudin model. Zapiski Nauchnykh Seminarov LOMI 164, 151–169 (1987) (in Russian); J. Sov. Math. 47, 2473–2488 (1989) (English transl.)

38


7. Sklyanin, E.K.: Separation of variables. New trends. In: Quantum field theory, integrable models and beyond (Kyoto, 1994). Progr. Theoret. Phys. Suppl. 118, 35–60 (1995) 8. Frenkel, E.: Affine algebras, Langlands duality and Bethe ansatz. XIth International Congress of Mathematical Physics (Paris, 1994), Cambridge, MA: Internat. Press, 1995, pp. 606–642 9. Enriquez, B., Feigin, B., Rubtsov, V.: Separation of variables for Gaudin–Calogero systems. Compositio Math. 110, 1–16 (1998) 10. Kuroki, G., Takebe, T.: Twisted Wess–Zumino–Witten models on elliptic curves. Commun. Math. Phys. 190, 1–56 (1997) 11. Sklyanin, E. K.: On Poisson structure of periodic classical XY Z-chain. Zapiski Nauchnykh Seminarov LOMI 150, 154–180 (1986) (in Russian); J. Sov. Math. 46, 1664–1683 (1989) (English transl.). 12. Takhtajan, L.A. and Faddeev, L.D.: The quantum method of the inverse problem and the Heisenberg XYZ Model. Uspekhi Mat. Nauk 34:5, 13–63 (1979) (in Russian); Russian Math. Surveys 34:5, 11–68 (1979) (English transl.) 13. Whittaker, E.T. and Watson, G.N.: A Course of Modern Analysis. 4th ed., Cambridge: Cambridge University Press, 1927 14. Mumford, D.: Tata Lectures on Theta I. Basel–Boston: Birkhäuser, 1982 15. D’Agnolo, A., Schapira, P.: Radon–Penrose transform for D-modules. J. Funct. Anal. 139, 349–382 (1996) 16. Dickson, L. E.: Elementary theory of equations, New York: John Wiley & Sons, Inc., 1914 17. Crawford, L.: On the solution of Lamé’s equation d 2 U/du2 = U {n(n + 1)pu + B} in finite terms when 2n is an odd number. Quarterly J. Pure and Appl. Math. XXVIII, 93–98 (1895) 18. Sklyanin, E.K.: Some Algebraic Structures Connected with the Yang–Baxter Equation. Representations of Quantum Algebras. Funkts. analiz i ego Prilozh. 17-4, 34–48 (1983) (in Russian); Funct. Anal. Appl. 17, 273–284 (1984) (English transl.) Communicated by G. Felder

Commun. Math. Phys. 204, 39 – 60 (1999)

Communications in



Double Quantization on Some Orbits in the Coadjoint Representations of Simple Lie Groups J. Donin1 , D. Gurevich2 , S. Shnider1 1 Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel 2 ISTV, Université de Valenciennes, 59304 Valenciennes, France

Received: 15 August 1998 / Accepted: 13 January 1999

Abstract: Let A be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, G, with the Lie algebra g. We study one and two parameter quantizations Ah and At,h of A such that the multiplication on the quantized algebra is invariant under action of the Drinfeld–Jimbo quantum group, Uh (g). In particular, the algebra At,h specializes at h = 0 to a U (g)-invariant (G-invariant) quantization, At,0 . We prove that the Poisson bracket corresponding to Ah must be the sum of the socalled r-matrix and an invariant bracket. We classify such brackets for all semisimple orbits, M, and show that they form a dim H 2 (M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, At,h , corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-KostantSouriau bracket on M. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases.

1. Introduction Passing from classical mechanics to quantum mechanics involves replacing the commutative function algebra, A, of classical observables on the phase space, M, with a noncommutative (deformed, quantized) algebra, At , of quantum observables (see [BFFLS] where the deformation quantization scheme is developed). The algebra A is a Poisson algebra and the product in the quantized algebra At is given by a power series in the formal parameter t with leading term equal to the original commutative product and with infinitesimal generator (the linear term in t, antisymmetrized) equal to the Poisson bracket. If the classical system is invariant under a Lie group of symmetries G, the associated quantum system often retains the same group of symmetries, that is, the product

40

J. Donin, D. Gurevich, S. Shnider

on At is invariant under the action of G, or, equivalently, under the action of the universal enveloping algebra U (g). Some modern field-theories, in particular those attempting to unify gravity and quantum field theory, investigate the possibility of deforming (quantizing) the group symmetry as well as the function algebra of the phase space. This is one of the reasons for the interest in quantum groups. The quantum group, Uh (g), defined by Drinfeld and Jimbo is a deformation of U (g) as a Hopf algebra. The quantization of the symmetry group together with the phase space corresponds to defining a Uh (g) invariant deformation of the algebra At . This leads us to the problem of finding a two parameter (or double) quantization, At,h , of a Poisson algebra with a U (g) invariant Poisson bracket. In other words, the problem of two parameter quantization arises if we want to quantize a Poisson bracket (appearing as the infinitesimal generator in t) in such a way that multiplication in the quantized algebra is invariant under the quantum group action (which depends on a deformation parameter h). In the present paper we investigate the problem of one and two parameter invariant quantizations of the Poisson function algebra on a semisimple orbit in the coadjoint representation of a simple Lie group. In general, if M is a manifold on which a semisimple Lie group G acts transitively, it is not true that there exists even a one parameter Uh (g) invariant quantization of the function algebra. In [DGM] it is proven that such a quantization exists if M = G/H and the Lie algebra of H contains a maximal nilpotent subalgebra. A Uh (g) invariant quantization of the algebra of holomorphic sections of a line bundle over a flag manifold is constructed by similar methods in [DG1]. In all these cases the Poisson bracket which is quantized is the r-matrix bracket, rM , on M. This bracket is determined by the bivector field (ρ ⊗ ρ)(r) for r ∈ ∧2 g, the Drinfeld–Jimbo r-matrix (of the form (3.3)), and ρ : g → Vect(M) is the mapping defined by the action of G on M. The Sklyanin–Drinfeld (SD) bracket on G is determined by the difference of two bivector fields on G which are the right and left invariant extensions of the Drinfeld– Jimbo classical r-matrix. For details see [LW]. In [DG2] it is shown that a one parameter Uh (g) invariant quantization of the SD bracket exists for all semisimple orbits in coadjoint representation of G. In the present paper we show that for the semisimple orbits, M, the SD bracket is one of a dim H 2 M parameter family of Poisson brackets admitting Uh (g) invariant quantization. For symmetric spaces M, the r-matrix bracket satisfies the Jacobi identity and hence defines a Poisson bracket. The existence of a one parameter Uh (g) invariant quantization with the r-matrix Poisson bracket is proven in [DS1]. That paper also proves that when M is a hermitian symmetric space there exists a two parameter Uh (g) invariant quantization. The hermitian symmetric spaces form a subclass of semisimple orbits in g∗ . These orbits have a very interesting property: the one-sided invariant components of the Sklyanin–Drinfeld bracket being reduced on such an orbit become Poisson brackets separately. More precisely, one of these components being reduced on M is the r-matrix bracket and the other one becomes the Kirillov–Kostant–Souriau (KKS) bracket which is roughly speaking the restriction to the orbit of the Lie bracket in g (see [KRR,DG1]). These brackets are obviously compatible (i.e. their Schouten bracket vanishes). So, we get a Poisson pencil which is the set of linear combinations of KKS and r-matrix Poisson brackets. The two parameter quantization, At,h , for hermitian symmetric spaces constructed in [DS1] is just a quantization of such a pencil. In particular, At,0 is a U (g) invariant quantization of the KKS bracket on M.

Double Quantization on Some Orbits of Simple Lie Groups

41

The orbits in g∗ on which the r-matrix bracket is Poisson have been classified in [GP]. In particular, the only such semisimple orbits are symmetric spaces. It turns out that there exist two parameter Uh (g) invariant quantizations on some G-manifolds where the r-matrix bracket is not Poisson (does not satisfy the Jacobi identity) but one can add a G-invariant bracket to the r-matrix bracket and get a Poisson bracket. For example, in [Do] it is shown that for g = sl(n) there is a two parameter Uh (g) invariant family (Sg)t,h , where Sg is the symmetric algebra of g which can be considered as the function (polynomial) algebra on g∗ . It is proven that this family can be restricted to give a two parameter quantization on any semisimple orbit of maximal dimension (on which the r-matrix bracket may not be Poisson). In the present paper we prove the existence of a two parameter Uh (g) invariant quantization of the function algebra for some non-symmetric semisimple orbits in g∗ . Similar to the case of symmetric orbits, the quasiclassical (infinitesimal) term of this quantization is a Poisson pencil generated by the KKS bracket and another Poisson bracket which must be the sum of the r-matrix and an U (g) invariant brackets. In particular, we shall see that such pencils exist for all semisimple orbits in case g = sl(n). Note that for non-symmetric orbits both the r-matrix bracket and the SD Poisson bracket are not compatible with the KKS bracket. We give a complete classification of the orbits admitting such a Poisson pencil. Moreover, we classify all such pencils and construct the deformation quantization of some of them. We now describe the content of the paper in more detail. Let G be a simple, connected, complex Lie group, g its Lie algebra. The Drinfeld–Jimbo quantum group Uh (g) can be considered as the algebra U (g)[[h]] with undeformed multiplication but deformed noncommutative comultiplication 1h . Let M be a G-homogeneous complex manifold. It is easy to show that M is isomorphic to a semisimple orbit of G in the coadjoint representation g∗ if and only if the stabilizer, Go , of a point o ∈ M is a Levi subgroup in G (see Sect. 3 for definition). The G-invariant symplectic structure on M is not unique, but each such symplectic structure on M arises from an isomorphism of M with an orbit, Oλ , for some semisimple element λ ∈ g∗ . On Oλ there is the Kirillov–Kostant–Souriau (KKS) Poisson bracket, vλ , whose action on the restriction of linear functions to the orbit is given by the Lie bracket in g. Each Ginvariant symplectic structure on M is induced from the KKS bracket by an isomorphism of M onto a semisimple orbit (see Sect. 3). The first problem we consider is that of quantizing the algebra A of polynomial (or holomorphic) functions on M, such that the quantized algebra Ah has a Uh (g) invariant multiplication ∞ X h i µi , µh = i=0

where µ0 is the initial multiplication of functions and the µi for i ≥ 1 are bidifferential operators. Invariance means that µh satisfies the property µh 1h (x)(a ⊗ b) = xµh (a ⊗ b)

for x ∈ Uh (g), a, b ∈ A.

The second problem we consider is the existence of a two parameter Uh (g) invariant quantization, At,h , of A such that the one parameter family At,0 is a U (g) invariant quantization of the KKS bracket on M. The infinitesimal of the one parameter quantization Ah is a Poisson bracket, whereas the infinitesimal of the two parameter quantization is a pencil of two compatible Poisson brackets.

42


In Sect. 2 we recall some facts on the Drinfeld monoidal categories of quantum group representations, used in our construction of quantization. In addition, we show that the Poisson bracket, p(a, b) = µ1 (a, b) − µ1 (b, a), corresponding to the Uh (g) invariant quantization Ah must be of a special form. Namely, let r ∈ ∧2 g be the Drinfeld–Jimbo classical r-matrix. The Schouten bracket [[r, r]] ∈ ∧3 g is the invariant element ϕ, which is unique up to a factor. Denote by rM the bracket on M determined by the bivector field (ρ ⊗ ρ)(r) where ρ : g → Vect(M) is the mapping defined by the action of G on M. We call rM an r-matrix bracket. Put also ϕM = ρ ⊗3 ϕ. Then, p(a, b) has the form p(a, b) = rM (a, b) + f (a, b),

a, b ∈ A,

(1.1)

where f (a, b) is a U (g) invariant bracket with the Schouten bracket [[f, f ]] = −ϕM .

(1.2)

Note that the r-matrix bracket is compatible with any invariant bracket, i.e. [[f, rM ]] = 0. Similarly, the two parameter quantization At,h corresponds to a pair of compatible Poisson brackets, (p, vλ ), where p = rM + f of the form (1.1) and vλ is the KKS bracket. Since rM is compatible with the invariant bracket vλ , compatibility of p with vλ is equivalent to the condition [[f, vλ ]] = 0.

(1.3)

In Sect. 3 we give a classification of all invariant brackets f satisfying condition (1.2) for all M isomorphic to semisimple orbits. We show that such brackets form a dim H 2 (M) parameter family. In the same section we show that a semisimple orbit may not have any invariant brackets satisfying conditions (1.2) and (1.3). We call an orbit a “good orbit”, if such a bracket exists and then give a classification of all good orbits. Namely, if g is of type An , all semisimple orbits are good. All orbits which are symmetric spaces are good. In cases Dn , E6 there are good orbits which are not symmetric spaces. Moreover, for the good orbits the brackets satisfying conditions (1.2) and (1.3) form a one parameter family. In fact, the property of an orbit being good depends only on its structure as a homogeneous manifold, not on the symplectic structure. The dependence is on the Lie subalgera of the stabilizer subgroup. So, if an orbit M is good, then any orbit isomorphic to M as a homogeneous manifold will be good. In Sect. 4 we consider cohomologies of the complex of invariant polyvector fields on M with differential given by the Schouten bracket with the bivector f satisfying (1.2). We show that for almost all f these cohomologies coincide with the usual de Rham cohomologies of the manifold M and then use this fact in Sect. 5 to prove the existence of an invariant quantization. In the proof we use methods of [DS1] and [DS2]. Using the same methods, we also construct the two parameter quantization for good orbits in cases Dn , E6 and brackets satisfying (1.2) and (1.3). For the case An some additional arguments are required. (See [Do] where using another method the existence of two parameter quantization for maximal orbits is proven.) In conclusion we make two remarks. Remark 1.1. Throughout this paper G is supposed to be a complex Lie group. However, one can consider the situation when G is a real simple Lie group with Lie algebra gR and M is a semisimple orbit of G in g∗R . In this case we take A = C ∞ (M), the complexvalued smooth functions. Let g be the complexification of gR . It is clear that the Lie algebra g and algebra U (g) act on C ∞ (M). Since all our results are formulated in terms of g action, they are valid in the real case as well (see also [DS1]).


43

Remark 1.2. The deformation quantization can be considered as the first step of a quantization procedure whose second step is a representation of the quantized algebra, At,h , as an operator algebra in a linear space. For some symmetric orbits in sl(n)∗ such a representation has been given in [DGR] and [DGK], but the method of [DGK] can be apparently extended to all symmetric orbits in g∗ for all simple Lie algebras g. These operator algebras have a deformed Uh (g) invariant trace which, however, is not symmetric. 2. Poisson Brackets Associated with Uh (g) Invariant Quantization We recall some facts about Drinfeld algebras and the monoidal categories determined by them. They will be used, in particular, in our construction of the quantization. Let A be a commutative algebra with unit, B a unitary A-algebra. The category of representations of B in A-modules, i.e. the category of B-modules, will be a monoidal category if the algebra B is equipped with an algebra morphism, 1 : B → B ⊗A B, called comultiplication, and an invertible element 8 ∈ B ⊗3 such that 1 and 8 satisfy the conditions (see [Dr1]) (id ⊗2

(id ⊗ 1)(1(b)) · 8 = 8 · (1 ⊗ id)(1(b)), b ∈ B, (2.1) ⊗2 ⊗ 1)(8) · (1 ⊗ id )(8) = (1 ⊗ 8) · (id ⊗ 1 ⊗ id)(8) · (8 ⊗ 1). (2.2)

Define a tensor product functor for C the category of B modules, denoted ⊗C or simply ⊗ when there can be no confusion, in the following way: given B-modules M, N, M ⊗C N = M ⊗A N as an A-module. The action of B is defined by b(m ⊗ n) = (1b)(m ⊗ n) = b1 m ⊗ b2 n,

where 1b = b1 ⊗ b2 ,

using the Sweedler convention of an implicit summation over an index. The element 8 = 81 ⊗ 82 ⊗ 83 defines the associativity constraint, aM,N,P : (M ⊗N)⊗P → M ⊗(N ⊗P ), aM,N,P ((m⊗n)⊗p) = 81 m⊗(82 n⊗83 p). Again the summation in the expression for 8 is understood. By virtue of (2.1) 8 induces an isomorphism of B-modules, and by virtue of (2.2) the pentagon identity for monoidal categories holds. We call the triple (B, 1, 8) a Drinfeld algebra. The definition is somewhat non-standard in that we do not require the existence of an antipode. The category C of B-modules for B a Drinfeld algebra becomes a monoidal category. When it becomes necessary to be more explicit we shall denote C(B, 1, 8). Let (B, 1, 8) be a Drinfeld algebra and F ∈ B ⊗2 an invertible element. Put e (b) = F1(b)F −1 , b ∈ B, 1 e = (1 ⊗ F ) · (id ⊗ 1)(F ) · 8 · (1 ⊗ id)(F −1 ) · (F ⊗ 1)−1 . 8

(2.3) (2.4)

e and 8 e satisfy (2.1) and (2.2), therefore the triple (B, 1 e, 8 e) also becomes a Then 1 e 1 e, 8 e). Drinfeld algebra which has an equivalent monoidal category of modules, C(B, Note that the equivalent categories C and Ce consist of the same objects as B-modules, and the tensor products of two objects are isomorphic as A-modules. The equivalence C → Ce is given by the pair (I d, F ), where I d : C → Ce is the identity functor of the categories (considered without the monoidal structures, but only as categories of B-modules), and F : M ⊗C N → M ⊗Ce N is defined by m ⊗ n 7 → F1 m ⊗ F2 n, where F 1 ⊗ F2 = F .

44


Assume A is a B-module with a multiplication µ : A ⊗A A → A which is a homomorphism of A-modules. We say that µ is 1 invariant if bµ(x ⊗ y) = µ1(b)(x ⊗ y) for b ∈ B, x, y ∈ A,

(2.5)

and 8 associative, if µ(81 x ⊗ µ(82 y ⊗ 83 z))) = µ(µ(x ⊗ y) ⊗ z) for x, y, z ∈ A.

(2.6)

Note that a B-module A equipped with 1 invariant and 8 associative multiplication is an associative algebra in the monoidal category C(B, 1, 8). The multiplication e µ= e e-associative and invariant in the category C. µF −1 : M ⊗A M → M will be 8 We are interested in the case when A = C[[h]], B = U (g)[[h]], where g is a complex simple Lie algebra. In this case, all tensor products over C[[h]] are completed in h-adic topology. Denote by ϕ ∈ ∧⊗3 g an invariant element (unique up to scaling for g simple) and by r ∈ ∧⊗2 g the so-called Drinfeld–Jimbo r-matrix of the form (3.3) such that the Schouten bracket of r with itself is equal to ϕ: [[r, r]] = ϕ.

(2.7)

In [Dr1], Drinfeld proved the following (see also [DS2] for the property c)). Proposition 2.1. 1. There is an invariant element 8 = 8h ∈ U (g)[[h]]⊗3 of the form 8h = 1 ⊗ 1 ⊗ 1 + h2 ϕ + · · · satisfying the following properties: a) it depends on h2 , i.e. 8h = 8−h ; b) it satisfies Eqs. (2.1) (i.e. invariant) and (2.2) with the usual 1 arising from U (g); c) it is invariant under the Cartan involution θ ; 321 321 = 8 ⊗ 8 ⊗ 8 for 8 = 8 ⊗ 8 ⊗ 8 . d) 8−1 3 2 1 1 2 3 h = 8h , where 8 2. There is an element F = Fh ∈ U (g)[[h]]⊗2 of the form Fh = 1 ⊗ 1 + (h/2)r + · · · e = 1 ⊗ 1 ⊗ 1. satisfying Eq. (2.4) with the usual 1 and with 8 This proposition implies that there are two nontrivial Drinfeld algebras: the first, (U (g)[[h]], 1, 8h ) with the usual comultiplication and 8 from Proposition 2.1, and the e , 1), where 1 e (x) = Fh 1(x)F −1 for x ∈ U (g). The pair (I d, Fh ) second, (U (g)[[h]], 1 h defines an equivalence between the corresponding monoidal categories C(U (g)[[h]], e , 1). 1, 8h ) and C(U (g)[[h]], 1 It is clear that reduction modulo h defines a functor from either of these categories to the category of representations of U (g) and the equivalence just described reduces to the identity modulo h. In fact, both categories are C[[h]]-linear extensions of the C-linear category of representations of g. Ignoring the monoidal structure the extension is a trivial e in the second case one, but the associator 8 in the first case and the comultiplication 1 make the extension non-trivial from the point of view of monoidal categories. e is denoted by Uh (g) and is The bialgebra U (g)[[h]] with comultiplication 1h = 1 isomorphic to the Drinfeld–Jimbo quantum group ([Dr1]). Let A be a U (g) invariant commutative algebra, i.e. an algebra with U (g) invariant multiplication µ in the sense of (2.5). A quantization of A is an associative algebra, Ah , which is isomorphic to A[[h]] = A ⊗ C[[h]] (completed tensor product) as a C[[h]]module, with multiplication in Ah having the form µh = µ + hµ1 + o(h). The Poisson bracket corresponding to the quantization is given by {a, b} = µ1 (a, b) − µ1 (b, a), a, b ∈ A.


45

In general, we call a skew-symmetric bilinear form A ⊗ A → A a bracket, if it satisfies the Leibniz rule in either argument when the other is fixed. The term Poisson bracket indicates that the Jacobi identity is also true. A bracket of the form {a, b}r = (r1 a)(r2 b) = µ(r1 a, r2 b)

a, b ∈ A,

(2.8)

where r = r1 ⊗ r2 (summation implicit) is the representation of the r-matrix r will be called an r-matrix bracket. Assume, Ah is a Uh (g) invariant quantization, i.e. the multiplication µh is 1h invariant. We shall show that in this case the Poisson bracket {·, ·} has a special form. Suppose f and g are two brackets on A. Then we define their Schouten bracket [[f, g]] as [[f, g]](a, b, c) = f (g(a, b), c) + g(f (a, b), c) + cyclic permutations of a, b, c.(2.9) Then [[f, g]] is a skew-symmetric map A⊗3 → A. We call f and g compatible, if [[f, g]] = 0. Proposition 2.2. Let A be a U (g) invariant commutative algebra and Ah a Uh (g) invariant quantization. Then the corresponding Poisson bracket has the form {a, b} = f (a, b) − {a, b}r ,

(2.10)

where f (a, b) is a U (g) invariant bracket. The brackets f and {·, ·}r are compatible and [[f, f ]] = −ϕA , where ϕA (a, b, c) = (ϕ1 a)(ϕ2 b)(ϕ3 c) and ϕ1 ⊗ ϕ2 ⊗ ϕ3 = ϕ ∈ ∧3 g is the invariant element given by (2.7). ⊗2 Proof. The permutation σ : A⊗2 h → Ah , a ⊗ b → b ⊗ a, is an equivariant operator in the category C(U (g)[[h]], 1, 8h ), because 1 is a cocommutative comultiplication. e , 1) implies that The equivalence of categories C(U (g)[[h]], 1, 8h ) and C(U (g)[[h]], 1 the operator e σ = F σ F −1 on A⊗2 is equivariant under the action of the quantum group Uh (g). Suppose the multiplication in Ah has the form µh (a, b) = ab + hµ1 (a, b) + o(h). It is easy to calculate that

1 σ )(a ⊗ b) = µ1 (a, b) − µ1 (b, a) + (r1 a)(r2 b) + O(h) µh (I d − e h = {a, b} + {a, b}r + O(h). But this is a Uh (g) equivariant operator A⊗2 h → Ah . Taking h = 0 we obtain that the bracket f (a, b) = {a, b} + {a, b}r must be U (g) invariant. So, we have {a, b} = f (a, b) − {a, b}r , as required. It is easy to check that any bracket of the form {a, b} = (X1 a)(X2 b) = µ(X1 a, X2 b), for X1 ⊗ X2 ∈ g ∧ g, is compatible with any invariant bracket. In particular, an r-matrix bracket is compatible with f . In addition, {·, ·} is a Poisson bracket, so its Schouten bracket with itself is equal to zero. Using this and taking into account that the Schouten bracket of the r-matrix bracket with itself is equal to ϕA , we obtain from (2.10) that t [[f, f ]] = −ϕA . u Remark 2.1. a) It is clear that if r satisfies (2.7), then −r satisfies (2.7), too, and we may fh with leading terms 1 ⊗ 1 − (h/2)r. Then, instead replace Fh in Proposition 2.1 with F (2.10) we can write {a, b} = f (a, b) + {a, b}r .

46


b) Assume that At,h is a two parameter quantization of A, i.e. a topologically free C[[t, h]]-module with a multiplication of the form µt,h (a, b) = ab + hµ1 (a, b) + tµ01 (a, b) + o(t, h). Assume, that At,h is Uh (g) invariant, so that At,0 is U (g) invariant. Then there are two compatible Poisson brackets corresponding to such a quantization: the bracket µ1 (a, b) − µ1 (b, a) of the form (2.10) and the U (g) invariant bracket v(a, b) = µ01 (a, b) − µ01 (b, a). Since v is invariant, the compatibility is equivalent to [[f, v]] = 0. e , 1), c) In view of equivalence of the categories C(U (g)[[h]], 1, 8h ) and C(U (g)[[h]], 1 the problem of quantizing the algebra A may be considered in the first category. If Ah is a Uh (g) invariant quantization with multiplication µh , then the multiplication µ¯ h = µh Fh = µ + hµ¯ 1 + o(h) will be U (g) invariant and 8h associative in the sense of (2.6). We have µ¯ 1 (a, b)− µ¯ 1 (b, a) = f (a, b), where f is from (2.10). So, we see that the invariant bracket f from (2.10) with [[f, f ]] = −ϕA plays the role of Poisson bracket for 8h associative quantization. Similarly, the two parameter quantization At,h corresponds to the U (g) invariant 8h associative quantization in C(U (g)[[h]], 1, 8h ) with a pair of compatible invariant brackets f and v, where [[f, f ]] = −ϕA and [[v, v]] = 0. Working in the category C(U (g)[[h]], 1, 8h ) can simplify the process of quantization (see [DS1] and Sect. 5). In the next section we consider the case when A is a function algebra on a semisimple orbit in the coadjoint representation of g and we give a classification of the invariant brackets f satisfying the property [[f, f ]] = −ϕA . Moreover, among such f we distinguish those which are compatible with the KKS Poisson brackets. 3. Pairs of Brackets on Semisimple Orbits Let g be a simple complex Lie algebra, h a fixed Cartan subalgebra. Let ⊂ h∗ be the system of roots corresponding to h. Select a system of positive roots, + , and denote by 5 ⊂ the subset of simple roots. Fix an element Eα ∈ g of weight α for each α ∈ + and choose E−α such that (Eα , E−α ) = 1 for the Killing form (·, ·) on g. Then, for all pairs of roots α, β such that α + β 6 = 0 we define the numbers Nα,β in the following way: [Eα , Eβ ] = Nα,β Eα+β Nα,β = 0

if α + β ∈ , if α + β ∈ / .

These numbers satisfy the following property, [He]. For the roots α, β, γ such that α + β + γ = 0 one has Nα,β = Nβ,γ = Nγ ,α .

(3.1)

Let 0 be a subset of 5. Denote by h∗0 the subspace in h∗ generated by 0. Note that ∗ h = h∗0 ⊕h∗5\0 , and one can identify h∗5\0 and h∗ /h∗0 via the projection h∗ → h∗ /h∗0 . Let 0 ⊂ h∗0 be the subsystem of roots in generated by 0, i.e. 0 = ∩ h∗0 . Denote by g0 the subalgebra of g generated by the elements {Eα , E−α }, α ∈ 0, and h. Such a subalgebra is called the Levi subalgebra. Let G be a complex connected Lie group with Lie algebra g and G0 a subgroup with Lie algebra g0 . Such a subgroup is called the Levi subgroup. It is known that G0 is a connected subgroup. Let M be a homogeneous space of G and G0 be the stabilizer of a point o ∈ M. We can identify M and the coset space G/G0 . It is known that such M is isomorphic to a semisimple orbit in g∗ . This orbit goes through an element λ ∈ g∗


47

which is just the trivial extension to all of g∗ (identifying g and g∗ via the Killing form) of a map λ : h5\0 → C such that λ(α) 6 = 0 for all α ∈ 5 \ 0. Conversely, it is easy to show that any semisimple orbit in g∗ is isomorphic to the quotient of G by a Levi subgroup. The projection π : G → M induces the map π∗ : g → To , where To is the tangent space to M at the point o. Since the ad-action of g0 on g is semisimple, there exists an ad(g0 )-invariant subspace m = m0 of g complementary to g0 , and one can identify To and m by means of π∗ . It is easy to see that the subspace m is uniquely defined and has a basis formed by the elements Eγ , E−γ , γ ∈ + \ 0 . Let v ∈ g⊗m be a tensor over g. Using the right and the left actions of G on itself, one can associate with v right and left invariant tensor fields on G denoted by v r and v l . We say that a tensor field, t, on G is right G0 invariant, if t is invariant under the right action of G0 . The G equivariant diffeomorphism between M and G/G0 implies that any right G0 invariant tensor field t on G induces a tensor field π∗ (t) on M. The field π∗ (t) will be invariant on M if, in addition, t is left invariant on G, and any invariant tensor field on M can be obtained in such a way. Let v ∈ g⊗m . For v l to be right G0 invariant it is necessary and sufficient that v be ad(g0 ) invariant. Denote π r (v) = π∗ (v r ) for any tensor v on g and π l (v) = π∗ (v l ) for any ad(g0 ) invariant tensor v on g. Note that the tensor π r (v) coincides with the image of v by the map g⊗m → Vect(M)⊗m induced by the action map g → Vect(M). Any G invariant tensor on M has the form π l (v). Moreover, v clearly can be uniquely chosen from m⊗m . Denote by [[v, w]] ∈ ∧k+l−1 g the Schouten bracket of the polyvectors v ∈ ∧k g, w ∈ ∧l g, defined by the formula X (−1)i+j [Xi , Yj ] ∧ X1 ∧ · · · Xˆ i · · · Yˆj · · · ∧ Yl , [[X1 ∧ · · · ∧ Xk , Y1 ∧ · · · ∧ Yl ]] = where [·, ·] is the bracket in g. The Schouten bracket is defined in the same way for polyvector fields on a manifold, but instead of [·, ·] one uses the Lie bracket of vector fields. We will use the same notation for the Schouten bracket on manifolds. It is easy to see that π r ([[v, w]]) = [[π r (v), π r (w)]], and the same relation is valid for π l . Denote by 0 the image of in h∗5\0 without zero. It is clear that 5\0 can be identified with a subset of 0 and each element from 0 is a linear combination of elements from 5\0 with integer coefficients which are all positive or all negative. Thus, + the subset 0 ⊂ 0 of the elements with positive coefficients is exactly the image of + . We call elements of 0 quasiroots and the images of 5 \ 0 simple quasiroots. Proposition 3.1. a) Let β and β 0 be roots from such that they give the same element in 0 . Then there exist roots α1 , . . . , αk ∈ 0 such that β + α1 + · · · + αk = β 0 and all partial sums β + α1 + · · · + αi , i = 1, . . . , k are roots. b) Let β¯1 , . . . , β¯k , β¯ be elements of 0 such that β¯ = β¯1 + · · · + β¯k . Then there exist representatives of these elements in such that β = β1 + · · · + βk . Proof. a) Let β 0 = β + γ1 + · · · + γm , where γi ∈ 0 ∪ −0. If (β 0 , β) > 0, then β 0 − β is a root, and the proposition follows. Proceed by induction on m. Since (β 0 , β 0 ) > 0, if (β 0 , β) ≤ 0, then there exist a γi , say γm , such that (β 0 , γm ) > 0, so β 0 − γm is a root and β 0 − γm = β + γ1 + · · · + γm−1 . By induction, the proposition holds for the pair β 0 − γm and β, i.e. there exist a representation β + α1 + · · · + αk−1 = β 0 − γm satisfying the proposition. Now, putting αk = γm , we obtain the required representation of β 0 . ¯ Then b) Let β10 , . . . , βk0 , β 0 be some representatives of elements β¯1 , . . . , β¯k , β¯ in . we have the equation β10 + · · · + βk0 + γ1 + · · · + γm = β 0 for some γi ∈ 0 ∪ −0,

48


i = 1, . . . , m. If (γi , β 0 ) > 0, then β 0 − γi is a root, and we can take β 0 − γi instead β 0 . After iteration, we can assume that all (γi , β 0 ) ≤ 0. Then, there exists a βi0 , say βk0 , such that (βk0 , β 0 ) > 0, so β 0 − βk0 is a root. Applying induction on k, one can suppose that there are representatives β1 , . . . , βk−1 of β¯1 , . . . , β¯k−1 , such that we have an equation β1 + · · · + βk−1 = β 0 − βk0 + γ1 + · · · + γn for some γi ∈ 0 ∪ −0, i = 1, . . . , n, and ˜ γi ) > 0, then for some βj , say β1 , (β1 , γi ) > 0, β˜ = β1 + · · · + βk−1 is a root. If (β, β1 − γi is a root, and one can replace β1 by β1 − γi . Repeating this argument we can ˜ (β 0 − β 0 )) ≥ 0, so either (β, ˜ β 0 ) > 0, and we set ˜ γi ) ≤ 0. Then (β, assume that all (β, k P P 0 0 0 0 ˜ −β ) > 0, and we set βk = β and β = β 0 + γi . In βk = βk − γi , β = β , or (β, k k ¯ u t any case, we obtain the required representatives of β¯1 , . . . , β¯k , β. Remark 3.1. It is obvious that m considered as a g0 representation space decomposes into the direct sum of subrepresentations mβ¯ , β¯ ∈ 0 , where mβ¯ is generated by all the ¯ Part a) of Proposition 3.1 elements Eβ , β ∈ , such that the projection of β is equal to β. shows that all mβ¯ are irreducible. Part b) together with part a) shows that for β¯1 , β¯2 ∈ 0 such that β¯1 + β¯2 ∈ 0 one has [mβ¯1 , mβ¯2 ] = mβ¯1 +β¯2 . Using the Killing form, it is easy to see that representations mβ¯ and m−β¯ are dual. Question. Is it true that for β¯1 , β¯2 ∈ 0 such that β¯1 + β¯2 ∈ 0 the representation mβ¯1 +β¯2 is contained in mβ¯1 ∧ mβ¯2 ⊂ ∧2 m with multiplicity one? Since g0 contains the Cartan subalgebra h, each g0 invariant tensor over m has to be of weight zero. It follows that there are no invariant vectors in m. Hence, there are no invariant vector fields on M. Consider the invariant bivector fields on M. From the above, such fields correspond from ∧2 m. Note that any h invariant bivector from ∧2 m to the g0 invariant bivectors P has to be of the form c(α)Eα ∧ E−α . Proposition 3.2. A bivector v ∈ ∧2 m is g0 invariant if and only if it has the form P v= c(α)Eα ∧ E−α , where the sum runs over α ∈ + \ 0 , and for two roots α, β which give the same element in h∗ /h∗0 one has c(α) = c(β). Proof. In view of Proposition 3.1 a), we may assume that α = β + γ , where γ ∈ 0 . Then the coefficient before Eβ+γ ∧ E−β in [[Eγ , v]] appears from the terms Eβ ∧ E−β and Eβ+γ ∧ E−β−γ in v, and is equal to Nγ ,β c(β) + Nγ ,−β−γ c(β + γ ). But from (3.1) it follows that Nγ ,β = −Nγ ,−β−γ , so if v is invariant under the action of Eγ , i.e. t [[Eγ , v]] = 0, then c(β) = c(β + γ ). u P This proposition shows that coefficients of an invariant element v = c(α)Eα ∧E−α + depend 0 , denoted α, ¯ so vPcan be written in the form P only on the image of α in c(α)E ¯ α ∧ E−α , where the v= c(α)E ¯ α ∧ E−α .Let v ∈ ∧2 m be of the form v = sum runs over α ∈ + \ 0 . Denote by θ the Cartan automorphism of g. Then, v is θ anti-invariant, i.e. θv = −v. Hence, any g0 invariant bivector is θ anti-invariant. If 2 v, Pw ∈ ∧ m are g0 invariant, then [[v, w]] is θ invariant and is of the form [[v, w]] = ¯ α+β ∧ E−α ∧ E−β , where roots α, β are both negative or both positive and e(α, ¯ β)E ¯ = −e(−α, ¯ Hence, to calculate [[v, w]] for such v and w it is sufficient to e(α, ¯ β) ¯ −β). ¯ for positive α¯ and β. ¯ calculate coefficients e(α, ¯ β)


49

P P Proposition 3.3. Let v = c(α)Eα ∧ E−α , w = d(α)Eα ∧ E−α be elements from g ∧ g. Then for any positive roots α, β, (α + β) the coefficient by the term Eα+β ∧ E−α ∧ E−β in [[v, w]] is equal to Nα,β (c(α)(d(β) − d(α + β)) + c(β)(d(α) − d(α + β)) − c(α + β)(d(α) + d(β))).

(3.2)

Proof. Direct computation, see [KRR]. u t Let r ∈ g ∧ g be the Drinfeld–Jimbo r-matrix: X Eα ∧ E−α . r=

(3.3)

α∈+

Then [[r, r]] = ϕ is an invariant element in ∧3 g. From Proposition 3.2 it follows that r reduced modulo g ∧ g0 is g0 -invariant. Hence, r and ϕ define invariant bivector and three-vector fields on M, π l (r) and π l (ϕ), which we denote by rM and ϕM . Recall, that we identify invariant tensor fields on M with invariant tensors in m. From Propositions that the Schouten bracket P3.3 and 3.1 b) it follow that the condition of the bivector v = c(α)E ¯ α ∧ E−α with itself give K 2 ϕM for a number K is ¯ ¯ = c(α)c( ¯ + K2 c(α¯ + β)(c( α) ¯ + c(β)) ¯ β)

(3.4)

for all the pairs of positive quasiroots α, ¯ β¯ such that α¯ + β¯ is a quasiroot. ¯ and assuming that c(α) ¯ 6= 0 we find that Given c(α) ¯ and c(β) ¯ + c(β) ¯ = c(α¯ + β)

¯ + K2 c(α)c( ¯ β) . ¯ c(α) ¯ + c(β)

(3.5)

¯ γ¯ are positive quasiroots such that α¯ + β, ¯ β¯ + γ¯ , α¯ + β¯ + γ¯ are also Assume α, ¯ β, quasiroots. Then the number c(α¯ + β¯ + γ¯ ) can be calculated formally (ignoring possible division by zero) in two ways, using (3.5) for the pair c(α), ¯ c(β¯ + γ¯ ) on the right-hand ¯ c(γ¯ ). But it is easy to check that these two ways give side and also for the pair c(α¯ + β), the same value of c(α¯ + β¯ + γ¯ ). In this sense the system of equations corresponding to (3.5) for all pairs is consistent. Let us consider this system represented P P more carefully. For any positive quasiroot ¯ = ai . In general the as simple quasiroots α¯ = ai α¯ i , define the height ht(α) coefficient c(α) ¯ for quasiroots of height l can be formally defined by iterating (3.5). Let α¯ = α¯ 1 + · · · + α¯ l with possible repetitions and let ci := c(α¯ i ), then P c1 c2 · · · cl + K 2 ci1 ci2 · · · cil−2 + · · · P . (3.6) c(α) ¯ =P ci1 ci2 · · · cil−1 + K 2 ci1 ci2 · · · cil−3 + · · · For K 6 = 0 the expression in the denominator can be expressed as 1 l 5i=1 (ci + K) − 5li=1 (ci − K) . K

(3.7)

50


Assumption. Let (α¯ 1 , · · · , α¯ k ) be the k-tuple of simple quasiroots. In the following we will assume that the point (c(α¯ 1 ), · · · , c(α¯ k )) ∈ Ck does not lie on any of the subvarieties defined by the expressions in the denominator of (3.6). It will be convenient to include the conditions c(α¯ i ) 6 = 0. Proposition 3.4. a) Given a k-tuple of positive numbers (c1 , . . . , ck ) := (c(α¯ 1 ), . . . , Eq. (3.6) uniquely defines ¯ for all c(α¯ k )) satisfying the Assumption, P numbers c(α) P ¯ α ∧ E−α satisfies positive quasiroots α¯ = α¯ i such that the bivector v = c(α)E the condition [[v, v]] = K 2 ϕM . b) When K = 0, the solution described in part a) defines a Poisson bracket on M and there exists a linear form λ ∈ h∗5\0 such that c(α) ¯ =

1 λ(α) ¯

(3.8)

for all quasiroots α. ¯ Proof. a) Since by assumption the denominator is never zero, Eq. (3.6) defines consis¯ satisfying (3.5) for all positive β. ¯ tently c(β) ¯ ¯ = c(α)c( ¯ and further b) When K = 0 (3.4) becomes c(α¯ + β)(c( α) ¯ + c(β)) ¯ β) Eq. (3.6) implies that c(α) ¯ 6 = 0 for all quasiroots. So setting λ(α) ¯ = 1/c(α), ¯ we find ¯ = λ(α) ¯ Thus λ is a linear that Eq. (3.4) is equivalent to the equation λ(α¯ + β) ¯ + λ(β). functional, i.e., an element of h∗5\0 , which by construction must be nonzero on all quasiroots. u t Remark 3.2. a) This proposition shows that invariant brackets v on M such that [[v, v]] = K 2 ϕM form a k-dimensional manifold, X , which equals Ck minus the subvarieties defined in the Assumption, where k is the number of elements from 5 \ 0. Further, it is known that k = dim H 2 (M), [Bo]. If K is regarded as indeterminate, then v forms a k + 1 dimensional manifold, Y = Ck × C. The submanifold Y0 corresponds to K = 0, i.e. consists of Poisson brackets. It is easy to see that all the Poisson brackets of the type c(α) ¯ = 1/λ(α) ¯ 6 = 0 are nondegenerate. Since Y is connected, it follows that almost all brackets v (except an algebraic subset in Y of lesser dimension) are nondegenerate as well. b) If v defines a Poisson bracket on M, then M is a symplectic manifold and may be realized as an orbit in g∗ passing through the element λ from (3.8) trivially extended to g∗ , with the KKS bracket. P Now we fix a Poisson bracket v = (1/λ( ¯ α ∧E−α , where λ is a fixed linear form P α))E and describe the invariant brackets f = c(α)E ¯ α ∧ E−α which satisfy the conditions for K 6= 0, [[f, f ]] = K 2 ϕM [[f, v]] = 0.

(3.9)

An ordered pair of quasiroots α, ¯ β¯ such that α¯ + β¯ is a quasiroot as well will be called an admissible pair. Substituting in (3.2) instead d(α) the coefficients of v, we obtain that the condition [[f, v]] = 0 is equivalent to the system of equations for the coefficients of f, ¯ β) ¯ 2 = c(α¯ + β)λ( ¯ α¯ + β) ¯ 2 c(α)λ( ¯ α) ¯ 2 + c(β)λ( ¯ for all admissible pairs α, ¯ β.

(3.10)


51

On the other hand, the condition [[f, f ]] = K 2 ϕM is equivalent to the system of Eqs. (3.4) for all admissible pairs of quasiroots. Substituting c(α + β) from (3.10) in (3.4) we obtain ¯ β) ¯ 2 )(c(α) ¯ = c(α)c( ¯ ¯ 2 + K 2 λ(α¯ + β) ¯ 2. ¯ + c(β)) ¯ β)(λ( α) ¯ + λ(β)) (c(α)λ( ¯ α) ¯ 2 + c(β)λ( Cancelling terms and extracting the square root, we obtain the equation ¯ β) ¯ = c(α)λ( ¯ c(β)λ( ¯ α) ¯ ± Kλ(α¯ + β).

(3.11)

¯ β) ¯ from (3.11) in (3.10), we obtain Substituting c(β)λ( ¯ α¯ + β) ¯ = c(α)λ( ¯ c(α¯ + β)λ( ¯ α) ¯ ± Kλ(β).

(3.12)

So, the conditions (3.9) on f are equivalent to the system of Eqs. (3.11, 3.12) with + the same sign before K for all admissible pairs α¯ and β¯ from 0 . ¯ We say that an ordered triple of positive quasiroots (not necessarily different) α, ¯ β, + ¯ ¯ ¯ γ¯ ∈ 0 is an admissible triple, if α¯ + β, β + γ¯ , and α¯ + β + γ¯ are quasiroots, too. ¯ γ¯ be an admissible triple of quasiroots. If c(α¯ + β)λ( ¯ α¯ + β) ¯ = Lemma 3.1. Let α, ¯ β, ¯ then c(β¯ + γ¯ )λ(β¯ + γ¯ ) = c(β)λ( ¯ β) ¯ + Kλ(γ¯ ), that is, the signs c(α)λ( ¯ α) ¯ + Kλ(β), before K in (3.12) for the admissible pairs α, β and β, γ are the same. Proof. The admissible pair α, ¯ β¯ + γ¯ gives the equation c(α)λ( ¯ α) ¯ = c(β¯ + γ¯ )λ(β¯ + γ¯ ) ± Kλ(α¯ + β¯ + γ¯ ). The first equation given in the lemma implies that the + sign appears in Eq. (3.11). These two equations imply ¯ β) ¯ − Kλ(α¯ + β) ¯ = c(β¯ + γ¯ )λ(β¯ + γ¯ ) ± Kλ(α¯ + β¯ + γ¯ ). c(β)λ( Substituting for c(β¯ + γ¯ )λ(β¯ + γ¯ ) using (3.12), we get ¯ β) ¯ − Kλ(α¯ + β) ¯ = c(β)λ( ¯ β) ¯ ± Kλ(γ¯ ) ± Kλ(α¯ + β¯ + γ¯ ), c(β)λ( where the last two ± are independent. However if ¯ β) ¯ − Kλ(γ¯ ), c(β¯ + γ¯ )λ(β¯ + γ¯ ) = c(β)λ( we have a contradiction, since either the sign in front of Kλ(α¯ + β¯ + γ¯ ) is positive and ¯ = +Kλ(α¯ + β), ¯ so 0 = λ(α¯ + β), ¯ or the sign in front of Kλ(α¯ + β¯ + γ¯ ) −Kλ(α¯ + β) is negative, implying 0 = λ(γ¯ ). We conclude that the sign in front of Kλ(γ¯ ) must be positive. u t In the situation of the lemma we can express c(γ¯ ) in terms of c(α) ¯ as ¯ + λ(β¯ + γ¯ )). c(γ¯ )λ(γ¯ ) = c(α)λ( ¯ α) ¯ + K(λ(α¯ + β)

(3.13)

Now, we consider the pair (M, λ) as an orbit in g∗ passing through λ with the KKS Poisson bracket X 1 Eα ∧ E−α . v = vλ = λ(α) ¯ + α∈

52


Definition 3.1. We call M a good orbit, if there exists on M an invariant bracket f = P c(α)E ¯ α ∧ E−α satisfying the conditions (3.9). Recall the discussion in the introduction where we explained that conditions (3.9) are necessary for the existence of a 2-parameter quantization. Proposition 3.5. The good semisimple orbits are as follows: a) For g of type An all semisimple orbits are good. b) For all other g, the orbit M is good if and only if the set 5 \ 0 consists of one or two roots which appear in representation of the maximal root with coefficient 1. Proof. a) In this case the system of quasiroots 0 looks like a system of roots of type Ak for k being equal to the number of elements of 5 \ 0. So, the simple quasiroots can be ordered in a sequence β¯1 , . . . , β¯k in such a way that all subsequences consisting of three adjacent elements are admissible. Pick an arbitrary value for c(β¯1 ) and a sign before K in (3.11) for the pair β¯1 and β¯2 . Then, due to Lemma 3.1, consistency of system (3.11, 3.12) implies that the sign before K is the same for all adjacent pairs β¯i and β¯i+1 . Using Eqs, (3.11) and (3.12) for a fixed sign before K and induction on ht(α), we find all the coefficients c(α) ¯ of f and see that the system (3.11, 3.12) is consistent. b) Let g be of type Bn or Cn . Then the maximal root has the form α1 +2α2 +· · ·+2αn , where αi ∈ 5. Denote β = α2 + · · · + αn which is a root. If 5 \ 0 does not contain α1 , + ¯ β. ¯ So, from (3.11) it follows that i.e. α¯ 1 = 0, then 0 contains the admissible pair β, ¯ ¯ ¯ ¯ ¯ ¯ c(β)λ(β) = c(β)λ(β) ± 2Kλ(β), i.e. λ(β) = 0 which is impossible. Assume that 5 \ 0 contains α1 and some roots αi for i > 1. Then both α¯ 1 and β¯ are + ¯ α¯ 1 , β. ¯ It follows from (3.13) not equal to zero, and 0 contains the admissible triple β, ¯ ¯ ¯ ¯ ¯ ¯ that c(β)λ(β) = c(β)λ(β) + 2Kλ(β + α¯ 1 ), i.e. λ(β + α¯ 1 ) = 0 which is impossible as well, because β¯ + α¯ 1 is a quasiroot. So, for consistency of system (3.11, 3.12) in cases Bn and Cn , the set 0 has to contain all the roots αi , i > 1. But in the latter case the system is trivially consistent, because in that case the set of quasiroots looks like A1 . The homogeneous space G/G0 is a symmetric space. Consider the case Dn . The maximal root has the form α1 + α2 + α3 + 2α4 + · · · + 2αn . Denote β = α4 + · · · + αn , which is a root. Consider several cases. The cases when two of α¯ i , i ≤ 3, are equal to zero and β¯ is not equal to zero lead to an inconsistency in the system (3.11, 3.12) in the same way as in the cases Bn and Cn considered above. Assume that two of α¯ i , i ≤ 3, say α¯ 1 , α¯ 3 , and β¯ are not equal to zero. Then the sequence ¯ α¯ 3 , α¯ 1 + β¯ α¯ 1 , α¯ 2 + β,

(3.14)

is a sequence of four nonzero quasiroots. It is easy to see that the subsequences α¯ 1 , α¯ 2 + ¯ α¯ 3 , α¯ 1 + β¯ form admissible triples in + ¯ α¯ 3 and α¯ 2 + β, β, 0 . From Lemma 3.1 it follows that the sign before K must be the same in (3.11) for all adjacent pairs. Taking, for example, the sign plus and applying (3.13) to the second triple, we obtain the equation ¯ α¯ 1 + β) ¯ = c(α¯ 2 + β)λ( ¯ α¯ 2 + β) ¯ + K(λ(α¯ 2 + β¯ + α¯ 3 ) c(α¯ 1 + β)λ( ¯ + λ(α¯ 3 + α¯ 1 + β)).

(3.15)


53

From (3.11) applied to the pair α¯ 1 , α¯ 2 + β¯ we obtain ¯ α¯ 2 + β) ¯ = c(α¯ 1 )λ(α¯ 1 ) + Kλ(α¯ 1 + α¯ 2 + β). ¯ c(α¯ 2 + β)λ(

(3.16)

Putting c(α¯ 1 ) from (3.16) in (3.15) and taking into account linearity of λ, we obtain the equality ¯ α¯ 1 + β) ¯ = c(α¯ 1 )λ(α¯ 1 ) + 2Kλ(α¯ 1 + α¯ 2 + α¯ 3 + β) ¯ + Kλ(β). ¯ (3.17) c(α¯ 1 + β)λ( ¯ in terms of c(α¯ 1 ) from (3.12), we obtain On the other hand, expressing c(α¯ 1 + β) ¯ α¯ 1 + β) ¯ = c(α¯ 1 )λ(α¯ 1 ) ± Kλ(β). ¯ c(α¯ 1 + β)λ(

(3.18)

Now, comparing (3.17) and (3.18), we see that if we take plus before K in (3.18), then ¯ = 0, if we take minus, then λ(α¯ 1 + α¯ 2 + α¯ 3 + β) ¯ = 0. But both λ(α¯ 1 + α¯ 2 + α¯ 3 + 2β) of the cases are impossible, since λ is not equal to zero on quasiroots. Next, assume that β¯ = 0 but α¯ i 6 = 0 for i = 1, 2, 3. In this case the sequence (3.14) becomes the sequence α¯ 1 , α¯ 2 , α¯ 3 , α¯ 1 . Using the above arguments, we obtain that in this case λ(α¯ 1 + α¯ 2 + α¯ 3 ) = 0 must be true, which is impossible, since α¯ 1 + α¯ 2 + α¯ 3 is a quasiroot. In the case when 5 \ 0 contains only one or two roots of αi , i = 1, 2, 3, system (3.11, 3.12) is consistent, because in these cases the set of quasiroots 0 looks like the system of roots of type A1 or A2 . So, the proposition is proved for the classical g. For the exceptional g and a semisimple orbit which is not a symmetric space the system of quasi-roots reduces to one of the t cases which we have just excluded for g equal to Bn , Cn or Dn . u Remark 3.3. a) Note that Proposition 3.5 may be reformulated in the following way: An orbit M is good if and only if the corresponding system of quasiroots 0 is isomorphic to a system of roots of type Ak . We say in this case that M is of type Ak . Orbits of type A1 are exactly the orbits which are symmetric spaces. For such orbits ϕM = 0, and we may take f = 0. Symmetric orbits exist for all classical g and also for g of types E6 , E7 . Orbits of type A2 exist for g of type An , Dn , and E6 . Orbits of type Ak , k > 2, exist only in case g = sl(n). Moreover, in this case all semisimple orbits have the type Ak for k ≥ 1. b) From the proof of Proposition 3.5 it follows that the bracket f satisfying (3.9) is defined on good orbits by the value of its coefficient c(α) for a fixed simple root and the choice of a sign before K. On the other hand, if a fixed f0 satisfies (3.9), then the family ±f0 + sv for arbitrary numbers s also satisfies these conditions. So, this family consists of all invariant brackets satisfying (3.9). Almost all brackets from this family (except a finite number) are nondegenerate, since v is nondegenerate and, therefore, for large s0 the bracket ±f0 + s0 v is nondegenerate as well. For symmetric orbits 5 \ 0 consists of one element, there is one quasiroot and f0 is a multiple of v. Note that in [GP] a classification of all orbits in coadjoint representation (not necessarily semisimple) is given for which ϕM = 0. In particular, if we take K = 1 and find f0 such that [[f0 , f0 ]] = ϕM , then the family ±if0 + sv gives all the brackets satisfying [[f0 , f0 ]] = −ϕM and compatible with the KKS bracket on M. Note that if f is a bracket satisfying [[f, f ]] = −ϕM and {·, ·}r is the r-matrix bracket (2.8), then f ± {·, ·}r is a Poisson bracket on M compatible with KKS bracket.

54


4. Cohomologies Defined by Invariant Brackets In the next section we prove the existence of a Uh (g) invariant quantization of the Poisson brackets described above using the methods of [DS1]. This requires us to consider the 3-cohomology of the complex (3• (g/g0 ))g0 = (3• m)g0 of g0 invariants with differential given by the Schouten bracket with the bivector v ∈ (32 m)g0 from Proposition 3.4 a), for u ∈ (3• m)g0 . δv : u 7 → [[v, u]] The condition δv2 = 0 follows from the Jacobi identity for the Schouten bracket together with the fact that [[v, v]] = K 2 ϕM . The latter equation is equivalent to [[v, v]] = ϕ modulo g0 ∧ g ∧ g, hence [[v, v]] is invariant modulo g0 ∧ g ∧ g. Denote these cohomologies by H k (M, δv ), whereas the usual de Rham cohomologies are denoted by H k (M). Recall, Remark 3.2 a), that the brackets v satisfying [[v, v]] = K 2 ϕ form a connected manifold Y = X × C. Proposition 4.1. For almost all v ∈ Y (except an algebraic subset of lesser dimension) one has H k (M, δv ) = H k (M) for all k. In particular, H k (M, δv ) = 0 for odd k. Proof. First, let v be a Poisson bracket, i.e. v ∈ Y0 . Then the complex of polyvector fields on M, 2• , with the differential δv is well defined. Denote by • the de Rham complex on M. Since none of the coefficients c(α) ¯ of v are zero, v is a nondegenerate bivector field, and therefore it defines an A-linear isomorphism v˜ : 1 → 21 , ω 7 → v(ω, ·), which can be extended up to the isomorphism v˜ : k → 2k of k-forms onto k-vector fields for all k. Using the Jacobi identity for v and invariance of v, one can show that v˜ gives a G invariant isomorphism of these complexes, so their cohomologies are the same. Since g is simple, the subcomplex of g invariants, (• )g , splits off as a subcomplex of • . In addition, g acts trivially on cohomologies, since for any g ∈ G the map X → X, x 7 → gx, is homotopic to the identity map. (G is assumed connected.) It follows that cohomologies of complexes (• )g and • coincide. But v˜ gives an isomorphism of complexes (• )g and (2• )g = ((3• m)g0 , δv ). So, cohomologies of the latter complex coincide with de Rham cohomologies, which proves the proposition when v is a Poisson brackets. Now, consider the family of complexes ((3• m)g0 , δv ), v ∈ Y. It is clear that δv depends algebraically on v. It follows from the uppersemicontinuity of dim H k (M, δv ) and the fact that H k (M) = 0 for odd k, [Bo], that H k (M, δv ) = 0 for odd k and almost all v ∈ Y. Using the uppersemicontinuity again and the fact that the number P k k k k (−1) dim H (M, δv ) is the same for all v ∈ Y, we conclude that dim H (M, δv ) = k dim H (M) for even k and almost all v. u t Remark 4.1. Call v ∈ Y admissible, if it satisfies Proposition 4.1. From the proof of the proposition it follows that the subset D such that Y \ D consists of admissible brackets does not intersect with the subset Y0 consisting of Poisson brackets. Let M be a good orbit and f0 + sv the family from Remark 3.3 b) satisfying (3.9) for a fixed K. Then for almost all numbers s this bracket is admissible. Indeed, this family is


55

contained in the two parameter family tf0 + sv. For t = 0, s 6 = 0 we obtain admissible brackets. So, there exist t0 6 = 0 and s0 such that the bracket t0 f0 + s0 v is admissible. It follows that the bracket f0 + (s0 /t0 )v is admissible, too. So, in the family f0 + sv there is an admissible bracket, and we conclude that almost all brackets in this family (except a finite number) are admissible. Question. Is it true that the set of admissible brackets contains all the nondegenerate brackets? For the proof of existence of two parameter quantization for the cases Dn and E6 in the next section we will use the following result on invariant three-vector fields. Denote by θ the Cartan automorphism of g. Lemma 4.1. For either Dn or E6 and one of the subsets, 0, of simple roots such that G0 defines a good orbit, any g0 and θ invariant element v in 33 m is a multiple of ϕM , that is, g0 ∼ 33 (m = hϕM i. Proof. Let g be a simple Lie algebra of type Dn or E6 and {α1 , . . . , αn } a system of simple roots. Changing notation slightly from Sect. 3, we assume that for g = Dn , (αi , αi+1 ) = −1 for i = 1, . . . , n − 2, (αn−2 , αn ) = −1 with all other inner products of distinct simple roots are zero, and for g = E6 , the non-zero products are (αi , αi+1 ) = −1 for i = 1, 2, 3, 4 and (α3 , α6 ) = −1. For g = Dn , 0 is one of the subsets of simple roots, 01 = {α1 , . . . , αn−2 }, 02 = {α2 , . . . , αn−1 }, or 03 = {α2 , . . . , αn−2 , αn }. For g = E6 , 0 = {α2 , α3 , α4 , α6 }. The positive quasiroots consist of three elements, α, ¯ α¯ 0 , and α¯ + α¯ 0 . Since a θ invariant element has the form w + θw for w ∈ mα¯ ⊗ mα¯ 0 ⊗ m−(α+ ¯ α¯ 0 ) , it is sufficient to show that the space of invariants in mα¯ ⊗ mα¯ 0 ⊗ m−(α+ ¯ α¯ 0 ) has dimension one. We know from Remark 3.1 that the subspaces mα¯ , mα¯ 0 , mα+ ¯ α¯ 0 are irreducible representations of g0 and that mα¯ and m−α¯ are dual. Therefore the dimension of the space of invariants in mα¯ ⊗ mα¯ 0 ⊗ m−(α+ ¯ α¯ 0 ) is the multiplicity of the representation mα+ ¯ α¯ 0 in the tensor product mα¯ ⊗ mα¯ 0 . For Dn and any of 0i the algebra g0 ∼ = An−2 . For 01 , α = αn−1 and α 0 = αn , the representations mα¯ and mα¯ 0 are both isomorphic to the dual vector representation for An−2 , that is, the contragredient representation to the representation for the fundamental weight λn−2 , mα¯ n−1 ∼ = mα¯ n ∼ = (V λn−2 )∗ ∼ = V λ1 . To see that this is so, note first of all that mα¯ n−1 is a lowest weight representation because it has a cyclic vector Eαn−1 and all negative simple root vectors of g0 annihilate Eαn−1 . The corresponding weight of An−2 is −λn−2 because (αn−1 , αj ) = −(λn−2 , αj ) if 1 ≤ j ≤ n − 2. The irreducible lowest weight representations with lowest weight −λn−2 is (V λn−2 )∗ and An−2 , (V λj )∗ ∼ = V λn−1−j . Since the subspaces mα¯ and mα¯ 0 of g have zero intersection, the wedge product mα¯ ∧ mα¯ 0 projects isomorphically onto the tensor product which contains the representation ∼ 2λn−2 )∗ ∼ mα+ = V 2λ1 ¯ α¯ 0 = (V with multiplicity one. In the cases 02 or 03 the representation mα¯ 1 is the contragredient representation to the vector representation (V λ1 )∗ ∼ = V λn−2 . The representations mα 0 = mαn for the case

56


02 and mα 0 = mαn−1 for 03 are (V λn−3 )∗ ∼ = V λ2 . From the elementary representation theory of the Lie algebra sl(n − 1) (type An−2 ) we see that the tensor product mα¯ ⊗ mα¯ 0 contains mα+ ¯ α¯ 0 with multiplicity one. In the case of E6 , g0 ∼ = D4 ∼ = so(8). The representations mα¯ , mα¯ 0 and m−(α+ ¯ α¯ 0 ) are the three inequivalent irreducible eight dimensional representations, the two spinor representations and the vector representation. It is well known that there is a one diment sional space of invariants in the tensor product mα¯ ⊗ mα¯ 0 ⊗ mα+ ¯ α¯ 0 . u The proof of the lemma gives a positive answer on the Question from Remark 3.1 for the particular case. Note that in case of symmetric orbits the space of invariant three-vector fields is equal to zero. 5. Uh (g) Invariant Quantizations in One and Two Parameters In this section we prove the existence of two types of Uh (g) invariant quantization of the function algebra A on M = G/G0 . The first is a one parameter quantization µh (a, b) =

X n≥0

hn µn (a, b) = ab+

X

hn µn (a, b), µ1 (a, b) =

n≥1

1 (f (a, b)−{a, b}r ), 2

as described in Sect. 2, where f is one of the invariant brackets found above, f ∈ (32 m)g , [[f, f ]] = −ϕM . The second is a two parameter quantization µt,h (a, b) = ab + hµ1 (a, b) + tµ01 (a, b) +

X

hk t l µk,l (a, b).

k,l≥1

Recall that in this case there are two compatible Poisson brackets corresponding to such a quantization: the bracket µ1 (a, b)−µ1 (b, a) is skew-symmetric of the form (2.10) and µ01 (a, b) − µ01 (b, a) is a U (g) invariant bracket v(a, b), which we will assume to be the KKS bracket defined by identifying G/G0 with an orbit of the coadjoint representation. We remind the reader of the method in [DS1]. The first step is to construct a U (g) invariant quantization in the category C(U (g)[[h]], 1, 8h ). Then we use the equivalence given by the pair (I d, Fh ) between the monoidal categories C(U (g)[[h]], 1, 8h ) and e , 1) to define a Uh (g) invariant quantization, either µh F −1 in the one C(U (g)[[h]], 1 h parameter case or µt,h Fh−1 in the two parameter case (see Sect. 2). In the first step we used the fact that (33 m)g0 = 0 for symmetric spaces. In the examples considered in this paper, (33 m)g0 does not necessarily vanish, and we modify the proof using a method from [DS2] (see also [NV]). In the case of An any semisimple orbit is a good orbit and a different method is required (see [Do] where the existence of two parameter quantization for maximal orbits is proven). For the cases Bn and Cn , the only good orbits are symmetric spaces and the quantization was dealt with in [DS1]. In the remaining cases g = Dn or E6 , we proved in Lemma 4.1 that (33 m)g0 ,θ = hϕA i, and a suitable modification of the proof still applies.


57

Theorem 5.1. For almost all (in sense of Proposition 4.1) g invariant brackets satisfying [[f, f ]] = −ϕM , there exists a multiplication µh on A, X hn µn (a, b), µh (a, b) = ab + (h/2)f (a, b) + n≥2

which is g invariant (Eq. 2.5)) and 8 associative (Eq. (2.6)). Proof. To begin, consider the multiplication µ(1) (a, b) = ab + (h/2)f (a, b). The corresponding obstruction cocycle is given by obs2 =

1 (1) (1) (µ (µ ⊗ id) − µ(1) (id ⊗ µ(1) )8) h2

considered modulo terms of order h. No h1 terms appear because f is a biderivation and, therefore, a Hochschild cocycle. The fact that the presence of 8 does not interfere with the cocyle condition and that this equation defines a Hochschild 3-cocycle was demonstrated in the proof of Proposition 4 in [DS1]. It is well known that if we restrict to the subcomplex of cochains given by differential operators, the differential Hochschild cohomology of A in dimension p is the space of polyvector fields on M. Since g is reductive, the subspace of g invariants splits off as a subcomplex and has cohomology given by (3p m)g0 . The complete antisymmetrization of a p-tensor projects the space of invariant differential p-cocycles onto the subspace (3p m)g0 representing the cohomology. The equation [[f, f ]] + ϕM = 0 implies that obstruction cocycle is a coboundary, and we can find a 2-cochain µ2 so that µ(2) = µ(1) + h2 µ2 satisfies µ(2) (µ(2) ⊗ id) − µ(2) (id ⊗ µ(2) )8 = 0 mod h2 . Assume we have defined the deformation µ(n) to order hn such that 8 associativity holds modulo hn , then we define the (n + 1)st obstruction cocycle by obsn+1 =

1 (µ(n) (µ(n) ⊗ id) − µ(n) (id ⊗ µ(n) )8) mod h. hn+1

In [DS1] (Proposition 4) we showed that the usual proof that the obstruction cochain satisfies the cocycle condition carries through to the 8 associative case. The coboundary of obsn+1 appears as the hn+1 coefficient of the signed sum of the compositions of µ(n+1) with obsn+1 . The fact that 8 = 1 mod h2 together with the pentagon identity implies that the sum vanishes identically, and thus all coefficients vanish, including the coboundary 0 ∈ (33 m)g0 be the projection of obsn+1 on the totally skew in question. Let obsn+1 symmetric part, which represents the cohomology class of the obstruction cocycle. The coefficient of hn+2 in the same signed sum, when projected on the skew symmetric part is 0 0 ]] which is the coboundary of obsn+1 in the complex (3• m)g0 , δf = [[f, .]]). [[f, obsn+1 0 Thus obsn+1 is a δf cocycle. We have shown in Sect. 4 that this complex has zero cohomology. Now we modify µ(n+1) by adding a term hn µn with µn ∈ (32 m)g0 and consider the (n + 1)st obstruction cocycle for µ0(n+1) = µ(n+1) + hn µn . Since the term we added at degree hn is a Hochschild cocyle we do not introduce a hn term in the calculation of µ(n) (µ(n) ⊗ id) − µ(n) (id ⊗ µ(n) )8 and the totally skew symmetric projection hn+1 term has been modified by [[f, µn ]]. By choosing µn appropriately we can make the (n + 1)st obstruction cocycle represent the zero cohomology class, and we are able to continue the recursive construction of the desired deformation. u t

58


Now we prove the existence of a two parameter deformation for good orbits in the cases Dn and E6 . Theorem 5.2. Given a pair of g invariant brackets, f, v, on a good orbit in Dn or E6 satisfying [[f, f ]] = −ϕM , [[f, v]] = [[v, v]] = 0, there exists a multiplication µh,t on A, X hk t l µk,l (a, b), µt,h (a, b) = ab + (h/2)f (a, b) + (t/2)v(a, b) + k,l≥1

which is g invariant (Eq. 2.5)) and 8 associative (Eq. (2.6)). Proof. The existence of a multiplication which is 8 associative up to and including h2 terms is nearly identical to the previous proof. Both f and v are anti-invariant under the Cartan involution θ. We shall look for a multiplication µt,h such that µk,l is θ antiinvariant and skew-symmetric for odd k + l and θ invariant and symmetric for even k + 1. So suppose we have a multiplication defined to order n, X hk t l µk,l (a, b), µt,h (a, b) = ab + hµ1 (a, b) + tµ01 (a, b) + k+l≤n

with the above mentioned invariance properties and 8 associative to order hn . Using properties c) and d) for 8 from Proposition 2.1, direct computation shows that the obstruction cochain, X hk t n+1−k βk , obsn+1 = k=0,... ,n+1

has the following properties: obsn+1 is θ invariant and obsn+1 (a, b, c) = −obsn+1 (c, b, a) for odd n, and obsn+1 is θ anti-invariant and obsn+1 (a, b, c) = obsn+1 (c, b, a) for even n. Hence, the projection of obsn+1 on (33 m)g0 is equal to zero for even n. It follows that all the βk are Hochschild coboundaries, and the standard argument implies that the multiplication can be extended up to order n + 1 with the required properties. For odd n, Lemma 4.1 shows that the projection on (33 m)g0 has the form   X ak hk t n+1−k  ϕM . obsn+1 =  k=0,... ,n+1

The KKS bracket is given by the two-vector v=

X α∈+ \0

Setting w=

X

1 Eα ∧ E−α . λ(α) ¯

λ(α)E ¯ α ∧ E−α ,

α∈+ \0

gives [[v, w]] = −3ϕM .


Defining µ0(n) = µ(n) + the new obstruction cohomology class is X 0 =( obsn+1

59

a0 n t w, 3

ak hk t n+1−k )ϕM .

k=1,... ,n+1

Finally we define µ00(n) = µ0(n) +

X

ak hk−1 t n+1−k )f

k=1,... ,n

and get an obstruction cocycle which is zero in cohomology. Now the standard argument implies that the deformation can be extended to give a 8 associative invariant multiplication with the required properties of order n + 1. So, we are able to continue the recursive construction of the desired multiplication. t u Remark 5.1. This theorem is also true for semisimple orbits in An , but the proof, which requires different techniques, will be deferred to a forthcoming paper. Remark 5.2. Using the 8h associative multiplications µh and µt,h from Propositions 5.1 and 5.2 and the equivalence between the monoidal categories C(U (g)[[h]], 1, 8h ) e , 1) given by the pair (I d, Fh ) (see Sect. 2), one can define Uh (g) and C(U (g)[[h]], 1 invariant multiplications, either µh Fh−1 in the one parameter case or µt,h Fh−1 in the two parameter case. Acknowledgements. We thank the referee for bringing to our attention some as yet unpublished work dealing with related topics. In “Manin pairs and moment maps" (Ecole Polytechnique preprint, 1998) A. Alekseev and Y. Kosmann–Schwarzbach consider bivectors with Schouten bracket [[f, f ]] = ϕ in the framework of moment map theory. J.-H. Lu has given a classification of Poisson brackets on G/H which are Poisson-invariant relative to the Poisson–Lie action of G.

Note added in proof. The answer to the question after Remark 3.1 is yes. See Bourbaki: Groupes et algèbres de Lie, Chap. 8, § 9, Ex. 14. References [BFFLS] Bauen, F., Flato, M., Fronsdal, C., Lichnerovicz, A., Sternheimer, D.: Deformation theory and quantization, I. Deformations of symplectic structures. Ann. Physics 111, 61–110 (1978) [Bo] Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. Math. 57, 115–207 (1953) [Dr1] Drinfeld, V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1, 1419–1457 (1990) [Do] Donin, J.: Double quantization on the coadjoint representation of sl(n)∗ . Czechoslovak J. of Phys. 47 No 11, 1115–1122 (1997), q-alg/9707031 [DG1] Donin, J. and Gurevich, D.: Quasi-Hopf Algebras and R-Matrix Structure in Line Bundles over Flag Manifolds. Selecta Math. Sovietica, 12, No 1, 37–48 (1993) [DG2] Donin, J. and Gurevich, D.: Some Poisson structures associated to Drinfeld–Jimbo R-matrices and their quantization. Israel Math. J. 92, No 1, 23–32 (1995) [DGK] Donin, J., Gurevich, D. and Khoroshkin, S.: Double quantization of CP n type orbits by generalized Verma modules. J. of Geom. and Phys. To appear

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[DGM] [DGR]

[DS1] [DS2] [GP] [He] [KRR] [LW] [NV]


Donin, J., Gurevich, D. and Majid, S.: R-matrix brackets and their quantization. Ann. de l’Institut d’Henri Poincaré, Phys. Theor. 58, No 2, 235–246 (1993) Donin, J., Gurevich, D., Rubtsov, V.: Quantum hyberboloid and braided modules. In: Algebre non commutative, Groupes Quantiques et Invariants, Societé Mathématique de France, Collection Séminaires et Congres, no. 2, 103–118 (1997) Donin, J. and Shnider, S.: Quantum symmetric spaces. J. of Pure and Appl. Algebra 100, 103–115 (1995) Donin, J. and Shnider, S.: Cohomological construction of quantum groups. To appear Gurevich, D. and Panyushev, D.: On Poisson pairs associated to modified R-matrices. Duke Math. J. 73, n. 1 (1994) Helgason, S.: Differential geometry, Lie groups, and symmetric spaces, London–New York: Academic Press, 1978 Khoroshkin, S., Radul, A. and Rubtsov, V.: A family of Poisson structures on compact Hermitian symmetric spaces. Commun. Math. Phys. 152, 299–316 (1993) Lu, J.H. and Weinstein, A.: Poisson–Lie groups, dressing transformations, and Bruhat decompositions. J. of Diff. Geom. 31, 501–526 (1990) Neroslavsky, O.M. and Vlasov, A.T.: Éxistence de produits * sur une variété. C.R. Acad. Sci. Paris, 292 I, 71–76 (1981)

Communicated by G. Felder

Commun. Math. Phys. 204, 61 – 84 (1999)

Communications in



A Functional-Analytic Theory of Vertex (Operator) Algebras, I Yi-Zhi Huang Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA. E-mail: [email protected] Received: 15 August 1998 / Accepted: 13 January 1999

Abstract: This paper is the first in a series of papers developing a functional-analytic theory of vertex (operator) algebras and their representations. For an arbitrary Z-graded finitely-generated vertex algebra (V , Y, 1) satisfying the standard grading-restriction axioms, a locally convex topological completion H of V is constructed. By the geometric interpretation of vertex (operator) algebras, there is a canonical linear map from V ⊗ V to V (the algebraic completion of V ) realizing linearly the conformal equivalence class of a genus-zero Riemann surface with analytically parametrized boundary obtained by deleting two ordered disjoint disks from the unit disk and by giving the obvious parametrizations to the boundary components. We extend such a linear map ˜ (⊗ ˜ being the completed tensor product) to H , and prove to a linear map from H ⊗H the continuity of the extension. For any finitely-generated C-graded V -module (W, YW ) satisfying the standard grading-restriction axioms, the same method also gives a topo˜ W to H W logical completion H W of W and gives the continuous extensions from H ⊗H of the linear maps from V ⊗ W to W realizing linearly the above conformal equivalence classes of the genus-zero Riemann surfaces with analytically parametrized boundaries. 0. Introduction We begin a systematic study of the functional-analytic structure of vertex (operator) algebras and their representations in this paper. Vertex (operator) algebras were introduced rigorously in mathematics by Borcherds and by Frenkel, Lepowsky and Meurman (see [B,FLM] and [FHL]). Incorporating modules and intertwining operators, the author introduced intertwining operator algebras in [H5]. The original definition of vertex (operator) algebra is purely algebraic, but a geometric interpretation of vertex operators motivated by the path integral picture in string theory was soon observed mathematically by Frenkel [F]. In [H1]–[H7], the author developed a geometric theory of vertex operator algebras and intertwining operator algebras; the hard parts deal with the Virasoro algebra, the central charge, the interaction

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between the Virasoro algebra and vertex operators (or intertwining operators), and the monodromies. But in this geometric theory, the linear maps associated to elements of certain moduli spaces of punctured spheres with local coordinates (or to elements of the vector bundles forming partial operads called “genus-zero modular functors” over these moduli spaces) are from the tensor powers of a vertex operator algebra (or an intertwining operator algebra) to the algebraic completion of the algebra. To be more specific, let (V , Y, 1) be a Z-graded vertex algebra. For any nonzero complex number z, let P (z) be the the conformal equivalence class of the sphere C∪{∞} with the negatively oriented puncture ∞, the ordered positively oriented punctures z and 0, and with the standard local coordinates. Then associated to P (z) is the linear map Y (·, z)· : V ⊗ V → V given by the vertex operator map. Note that the image is in the algebraic completion V of V , not V itself. One of the main goals of the theory of vertex operator algebras is to construct conformal field theories in the sense of Segal [S1,S2] from vertex operator algebras and their representations, and to study geometry using the theory of vertex operator algebras. It is therefore necessary to construct locally convex topological completions of the underlying vector spaces of vertex (operator) algebras and their modules, such that associated to any Riemann surface with boundary, one can construct continuous linear maps between the tensor powers of these completions. In particular, it is necessary to construct a locally convex completion H of the underlying vector space V of a vertex algebra (V , Y, 1) such that associated to the conformal equivalence class of a disk with two smaller ordered disks deleted and with the obvious boundary parametrization, there ˜ (⊗ ˜ being the completed tensor product) to is a canonical continuous map from H ⊗H H . Though in some algebraic applications of conformal field theories, only the algebraic structure of conformal field theories is needed, in many geometric applications, it is necessary to have a complete locally convex topological space and continuous linear maps, associated to Riemann surfaces with boundaries, between tensor powers of the space. In the present paper (Part I), we construct a locally convex completion H V of an arbitrary finitely-generated Z-graded vertex algebra (V , Y, 1) satisfying the standard grading-restriction axioms. Since V is fixed in the present paper, we shall denote H V simply by H . The completion H is the strict inductive limit of a sequence of complete locally convex spaces constructed from the correlation functions of the generators. The strong topology, rather than the weak-∗ topology, on the topological dual spaces of certain function spaces is needed in the construction. For any positive numbers r1 , r2 and nonzero complex number z satisfying r2 +2r1 < 1 and r2 < |z| < 1, there is a unique genus-zero Riemann surface with analytically parametrized boundary given by deleting two ordered disjoint disks, the first centered at z with radius r1 and the second centered at 0 with radius r2 , from the unit disk and by giving the obvious parametrizations to the boundary components. Associated to this genus-zero Riemann surface with analytically L(0) L(0) parametrized boundary is a linear map Y (r1 ·, z)r2 · : V ⊗ V → V . We extend this ˜ (⊗ ˜ being the completed tensor product) to H , linear map to a linear map from H ⊗H and prove the continuity of the extension. It is clear that H is linearly isomorphic to a L(0) L(0) subspace of V containing both V and the image of Y (r1 ·, z)r2 ·. For any finitely-generated C-graded module (W, YW ) satisfying the standard gradingrestriction axioms for such a finitely-generated vertex algebra, we also construct a locally ˜ W to H W convex completion H W of W , and construct continuous extensions from H ⊗H L(0) L(0) of the linear map YW (r1 ·, z)r2 · : V ⊗ W → W .

Functional-Analytic Theory of Vertex (Operator) Algebras, I

63

Our method depends only on the axiomatic properties or “world-sheet geometry” (mainly the duality properties) of the vertex algebra. Since our construction does not use any additional structure, we expect that the locally convex completions constructed in the present paper will be useful in solving purely algebraic problems in the representation theory of finitely-generated vertex (operator) algebras. This paper is organized as follows: In Sect. 2, we construct a locally convex completion H of a finitely-generated Z-graded vertex algebra (V , Y, 1) satisfying the standard L(0) L(0) grading-restriction axioms. In Sect. 3, we extend Y (r1 ·, z)r2 · : V ⊗ V → V to ˜ continuous linear maps from H ⊗H to H . In Sect. 4, we present the corresponding results for modules. In this paper, we assume that the reader is familiar with the basic definitions and results in the theory of vertex algebras. The material in [B,FLM] and [FHL] should be enough. We also assume that the reader is familiar with the basic definitions, constructions and results in the theory of locally convex topological vector spaces. The reader can find this material in, for example, [K1] and [K2]. We shall denote the set of integers, the set of real numbers and the set of complex numbers by the usual notations Z, R and C, respectively. We shall use i, j, k, l, m, n, p, q to denote integers. In particular, when we write, say, k > 0 (or k ≥ 0), we mean that k is a positive integer (or a nonnegative integer). For a graded vector space V , we use V 0 , V ∗ and V to denote the graded dual space, the dual space and the algebraic completion of V , respectively. For a topological vector space E, we use E ∗ to denote the topological dual space of E. The symbol ⊗ always denotes the vector space tensor product. The bifunctor given by completing the vector space tensor product of two topological vector ˜ spaces with the tensor product topology is denoted by ⊗. 1. A Locally Convex Completion of a Finitely-Generated Vertex Algebra In this section, we use “correlation functions” to construct the locally convex completion of a finitely-generated vertex algebra. For any k ≥ 0, let Rk be the space of rational functions in the complex variables z1 , . . . , zk with the only possible poles zi = zj for i 6= j and zi = 0, ∞ (i, j = 1, . . . , k). Let M k = {(z1 , . . . , zk ) ∈ Ck | zi 6 = zj for i 6 = j ; zi 6= 0 (i, j = 1, . . . , k)} (k)

and let {Kn }n>0 , be a sequence of compact subsets of M k satisfying (k)

Kn(k) ⊂ Kn+1 , n > 0, and

Mk =

[ n>0

Kn(k) .

For any n > 0, we define a map k · kRk ,n : Rk → [0, ∞) by kf kRk ,n =

max

(k)

(z1 ,...,zk )∈Kn

|f (z1 , . . . , zk )|

for f ∈ Rk . Then it is clear that k · kRk ,n is a norm on Rk . Using this sequence of norms, we obtain a locally convex topology on Rk . Note that a sequence in Rk is convergent if

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Y.-Z. Huang

and only if this sequence of functions is uniformly convergent on any compact subset (k) of M k . Clearly, this topology is independent of the choice of the sequence {Kn }n>0 . Let V be a Z-graded vertex algebra satisfying the standard grading-restriction axioms, that is, a V(n) , V = n∈Z

dim V(n) < ∞ for n ∈ Z and V(n) = 0 for n sufficiently small. By the duality properties of V , for any v 0 ∈ V 0 , any u1 , . . . , uk , v ∈ V, hv 0 , Y (u1 , z1 ) · · · Y (uk , zk )vi is absolutely convergent in the region |z1 | > · · · > |zk | > 0 and can be analytically extended to an element R(hv 0 , Y (u1 , z1 ) · · · Y (uk , zk )vi) of Rk . For any u1 , . . . , uk , v ∈ V and any (z1 , . . . , zk ) ∈ M k , we have an element Q(u1 , . . . , uk , v; z1 , . . . , zk ) ∈ V defined by hv 0 , Q(z1 , . . . , zk , v; z1 , . . . , zk )i = R(hv 0 , Y (u1 , z1 ) · · · Y (uk , zk )vi) ˜ be for v 0 ∈ V 0 . We denote the projections from V to V(n) , n ∈ Z, by Pn . Let G ∗ the subspace of V consisting of linear functionals λ on V such that for any k ≥ 0, u1 , . . . , uk , v ∈ V , X λ(Pn (Q(u1 , . . . , uk , v; z1 , . . . , zk ))) (1.1) n∈Z

is absolutely convergent for any z1 , . . . , zk in the region k = {(z1 , . . . , zk ) ∈ M k | |z1 |, . . . , |zk | < 1}. M 1 such that the Laurent series (z1 , . . . , zk ) ∈ M 0, with the surface r = 0 being N i is obtained by periodically identifying the coordinate u with some period P ∈ R. Thus, in particular, the components of ψi∗ gab and ψi∗ Fab in these coordinates are periodic functions of u with period P . (iii) We have, writing in the following for convenience gab

Rigidity Theorem for Spacetimes with a Stationary Event Horizon

697

and Fab instead of ψi∗ gab and ψi∗ Fab , f |Nei = −2κ◦ and FuA |Nei = 0,

(3.1)

ei , the r-derivatives of the with κ◦ ∈ R, where f is defined in Appendix A. (iv) On N metric and Maxwell field tensor components up to any order are u-independent, i.e., in the notation of Appendix A, n ∂ ∂ = 0, {f, } h , g ; F , F , F , F (3.2) A AB ur uA rA AB n ∂u ∂r ei N for all n ∈ N ∪ {0}. ei the vector field k a = (∂/∂u)a Remark 3.1. Along the null geodesic generators of N (which we take to be future directed) satisfies the equation k a ∇a k b = κ◦ k b .

(3.3)

ei are past incomplete but Thus, if κ◦ > 0, then all of the null geodesic generators of N ei are past complete future complete. If κ◦ < 0 then all of the null geodesic generators of N but future incomplete. Similarly, if κ◦ = 0 (usually referred to as the “degenerate case”), ei are complete then u is an affine parameter and all of the null geodesic generators of N in both the past and future directions. Remark 3.2. In the analytic case, Eq. (3.2) directly implies that k a = (∂/∂u)a is a ei . Since the projection map ψ ei is obtained Killing vector field in a neighborhood of N by periodically identifying the coordinate u, it follows immediately that k a projects to bi . Appealing then to the argument of a Killing vector field b k a in a neighborhood of N [12], it can be shown that the b k a further projects to a Killing vector field under the action ei ◦ ψ bi projects k a = (∂/∂u)a to a well-defined bi , so that the map ψi = ψ of the map ψ Killing field in a neighborhood of Ni . The arguments of [11,12] then establish that the local Killing fields obtained for each fibered neighborhood can be patched together to produce a global Killing field on a neighborhood of N . In the next section, we shall generalize the result of Remark 3.2 to the smooth case. However, to do so we will need to impose the additional restriction that κ◦ 6= 0, and we will prove existence only on a one-sided neighborhood of the horizon. 4. Existence of a Killing Vector Field The main difficulty encounted when one attempts to generalize the Isenberg–Moncrief theorem to the smooth case is that suitable detailed information about the spacetime metric and Maxwell field is known only on N (see Eq. (3.2) above). If a Killing field k a exists, it is determined uniquely by the data ka , ∇[a kb] at one point of N , because Eqs. (B.4), (B.6) imply a system of ODE’s for the tetrad components kj , ∇[i kj ] along each C 1 curve. But the existence of a Killing field cannot be shown this way. Thus we will construct the Killing field as a solution to a PDE problem. However, N is a null surface, and thus, by itself, it does not comprise a suitable initial data surface for the relevant hyperbolic equations. We now remedy this difficulty by performing a ei which is covered by the Gaussian null suitable local extension of a neighborhood of N coordinates of Theorem 3.1. This is achieved via the following proposition:

698

H. Friedrich, I. Rácz, R. M. Wald

Proposition 4.1. Let (Oi , gab |Oi ) be an elementary spacetime region associated with an electrovac spacetime of class B such that κ◦ > 0 (see Eq. (3.1) above). Then, there ei in Oi such that (O00 , gab |O00 , Fab |O00 ) can be exists an open neighborhood, Oi00 , of N i i i ∗ , F ∗ ), that possesses a bifurcate extended to a smooth electrovac spacetime, (O∗ , gab ab e∗ is the union of two null hypersurfaces, N ∗ and N ∗ , which e ∗ – i.e., N null surface, N 1 2 ei corresponds to the intersect on a 2-dimensional spacelike surface, S – such that N ei ]. Furthermore, the portion of N1∗ that lies to the future of S and I + [S] = Oi00 ∩ I + [N expansion and shear of both N1∗ and N2∗ vanish. Proof. It follows from Eq. (3.2) that in Oi0 , the spacetime metric gab can be decomposed as (0)

gab = gab + γab ,

(4.1) (0)

(0)

where, in the Gaussian null coordinates of Theorem 3.1, the components, gµν , of gab are independent of u, whereas the components, γµν , of γab and all of their derivatives ei ). Furthermore, taking account of the with respect to r vanish at r = 0 (i.e., on N periodicity of γµν in u (so that, in effect, the coordinates (u, x 3 , x 4 ) have a compact range of variation), we see that for all integers j ≥ 0 we have throughout Oi0 |γµν | < Cj |r|j

(4.2)

for some constants Cj . Similar relations hold for all partial derivatives of γµν . ei in the spacetime It follows from Eq. (4.2) that there is an open neighborhood of N (0) (0) (Oi0 , gab ) such that gab defines a Lorentz metric. It is obvious that in this neighborhood, ei is a Killing horizon of g (0) with respect to the Killing field k a = (∂/∂u)a . ConseN ab ei in quently, by the results of [15,16], we may extend an open neighborhood, Oi00 , of N (0)∗ ∗ e∗ , Oi to a smooth spacetime (O , gab ), that possesses a bifurcate Killing horizon, N (0)∗ (0)∗ e ∗ with respect to gab . Furthermore, with respect to the metric gab , N automatically satisfies all of the properties stated in the proposition. In addition, by Theorem 4.2 of (0)∗ [16], the extension can be chosen so that k a extends to a Killing field, k ∗a , of gab in (0)∗ O∗ , and (O∗ , gab ) possesses a “wedge reflection” isometry (see [16]); we assume that such a choice of extension has been made. (0) Let (u0 , r0 , x03 , x04 ) denote the Gaussian null coordinates in Oi00 associated with gab , 3 4 3 4 ei we have r0 = r = 0, u0 = u, x = x , x = x . Since γab is smooth in such that on N 0 0 00 Oi and is periodic in u, it follows that each of the coordinates (u0 , r0 , x03 , x04 ) are smooth functions of (u, r, x 3 , x 4 ) which are periodic in u. It further follows that the Jacobian matrix of the transformation between (u0 , r0 , x03 , x04 ) and (u, r, x 3 , x 4 ) is uniformly bounded in Oi00 , and that, in addition, there exists a constant, c such that |r| ≤ c|r0 | in Oi00 . Consequently, the components, γµ0 ν0 of γab in the Gaussian null coordinates (0) associated with gab satisfy for all integers j ≥ 0, |γµ0 ν0 | < Cj0 |r0 |j .

(4.3) (0)

Let (U, V ) denote the generalized Kruskal coordinates with respect to gab introduced in [15,16]. In terms of these coordinates, Oi00 corresponds to the portion of O∗ satisfying U > 0 and the wedge reflection isometry mentioned above is given by U → −U, V →


699

−V . The null hypersurfaces N1∗ and N2∗ that comprise the bifurcate Killing horizon, e ∗ , of g (0)∗ correspond to the hypersurfaces defined by V = 0 and U = 0, respectively. N ab It follows from Eq. (23) of [15] that within Oi00 we have |r0 | < C|U V |

(4.4)

for some constant C. Hence, we obtain for all j , |γµ0 ν0 | < Cj00 |U V |j

(4.5)

with similar relations holding for all of the derivatives of γµ0 ν0 with respect to the coordinates (u0 , r0 , x03 , x04 ). Taking account of the transformation between the Gaussian null coordinates and the generalized Kruskal coordinates (see Eqs. (24) and (25) of [15]), we see that the Kruskal components of γab and all of their Kruskal coordinate derivatives also go to zero uniformly on compact subsets of (V , x 3 , x 4 ) in the limit as U → 0. It follows that the tensor field γab on Oi00 extends smoothly to U = 0 – i.e., the null hypersurface N2∗ of O∗ – such that γab and all of its derivatives vanish on N2∗ . We ∗ now further extend γab to the region U < 0 – thereby defining a smooth tensor field γab on all of O∗ – by requiring it to be invariant under the above wedge reflection isometry. In O∗ , we define (0)∗

∗ ∗ = gab + γab . gab

(4.6)

∗ is smooth in O ∗ and is invariant under the wedge reflection isometry. FurThen gab ∗ vanishes on N e∗ , it follows that N e∗ is a bifurcate null surface with thermore, since γab ∗ ∗a ∗ e . In addition, on N e∗ we have Lk ∗ g (0)∗ = 0 respect to gab and that k is normal to N ab (0)∗ ∗ = 0 (since γ ∗ and its derivatives vanish (since k ∗a is a Killing field of gab ) and Lk ∗ γab ab e∗ . By Lemma B.1, it follows that the e ∗ ). Therefore, we have Lk ∗ g ∗ = 0 on N on N ab expansion and shear of both N1∗ and N2∗ vanish. Finally, by a similar construction (using the fact that FuA |Nei = 0; see Eq. (3.1)), we ∗ in O ∗ which can extend the Maxwell field Fab in Oi00 to a smooth Maxwell field Fab ∗ ∗ ) satisfies is invariant under the wedge reflection isometry. By hypothesis, (gab , Fab the Einstein-Maxwell equations in the region U > 0. By invariance under the wedge ∗ , F ∗ ) also satisfies the Einstein-Maxwell equations in the region reflection isometry, (gab ab U < 0. By continuity, the Einstein-Maxwell equations also are satisfied for U = 0, so ∗ , F ∗ ) is a solution throughout O ∗ . u t (gab ab

Remark 4.1. By Remark 3.1, the hypothesis that κ◦ > 0 is equivalent to the condition ei are past incomplete. Therefore, it is clear that that the null geodesic generators of N Proposition 4.1 also holds for κ◦ < 0 if we interchange futures and pasts. However, no analog of Proposition 4.1 holds for the “degenerate case” κ◦ = 0. We are now prepared to state and prove our main theorem: Theorem 4.1. Let (M, gab ) be a smooth electrovac spacetime of class B for which the generators of the null hypersurface N are past incomplete. Then there exists an open neighborhood, V of N such that in J + [N ] ∩ V there exists a smooth Killing vector field k a which is normal to N . Furthermore, in J + [N ] ∩ V the electromagnetic field, Fab , satisfies Lk Fab = 0.

700


Proof. As explained in Sect. 3, we can cover N by a finite number of fibered neighborhoods, Ni . Let Oi denote the elementary spacetime region obtained by “unwrapping” a neighborhood of Ni , as explained in Sect. 3. By Remark 3.1, the past incompleteness of the null geodesic generators of N implies that κ◦ > 0, so Proposition 4.1 holds. We now apply Proposition B.1 to the extended spacetime O∗ to obtain existence of a e∗ = N ∗ ∪ N ∗ . By restriction to Killing vector field in the domain of dependence of N 1 2 a Oi , we thereby obtain a Killing field K (which also Lie derives the Maxwell field) on a ei ] ∩ V ei , where V ei is an open neighborei of the form J + [N one-sided neighborhood of N ei . Both K a and k a = (∂/∂u)a are tangent to the null geodesic generators of hood of N ei we clearly have K a = ϕk a for some function ϕ. Furthermore, on N ei , we ei , so on N N have LK gab = 0 (since K a is a Killing field) and Lk gab = 0 (as noted in the proof of ei , so we may rescale K a so Proposition 4.1). It follows immediately that ∇a ϕ = 0 on N a a a e e that K = k on Ni . Since the construction of k off of Ni (as described in Appendix A) is identical to that which must be satisfied by a Killing field (as described in Remark B.1 below), it follows that K a = k a within their common domain of definition. Thus, the vector field k a = (∂/∂u)a – which previously had been shown to be a Killing field ei ] ∩ V ei . By in the analytic case – also is a Killing field in the smooth case in J + [N exactly the same arguments as given in [11,12] (see Remark 3.2 above) it then follows ei ◦ ψ bi projects k a = (∂/∂u)a to a well-defined Killing field in a that the map ψi = ψ one-sided neighborhood of Ni , and that the local Killing fields obtained for each fibered neighborhood can be patched together to produce a global Killing field on a one-sided t neighborhood of N of the form J + [N ] ∩ V, where V = ∪i ψi [Vi ]. u In view of Proposition 3.1, we have the following corollary Corollary 4.1. Let (M, gab ) be a smooth electrovac spacetime of class A for which the generators of the event horizon N are past incomplete. Then there exists an open neighborhood, V of N such that in J + [N ] ∩ V there exists a smooth Killing vector field k a which is normal to N . Furthermore, in J + [N ] ∩ V the electromagnetic field, Fab , satisfies Lk Fab = 0. A. Gaussian Null Coordinate Systems In this appendix the construction of a local Gaussian null coordinate system will be recalled. Let (M, gab ) be a spacetime, let N be a smooth null hypersurface, and let ς be a ς smooth spacelike 2-surface lying in N . Let x 3 , x 4 be coordinates on an open subset e of ς. On a neighborhood of e ς in N , let k a be a smooth, non-vanishing normal vector field to N , so that the integral curves of k a are the null geodesic generators of N . Without loss of generality, we may assume that k a is future directed. On a sufficiently small open neighborhood, S, of e ς × {0} in e ς × R, let ψ : S → N be the map which takes (q, u) into the point of N lying at parameter value u along the integral curve of k a starting at q. Then, ψ is C ∞ , and it follows from the inverse function theorem that ψ is 1 : 1 and onto from an open neighborhood of e ς × {0} onto an open e e, of ς ς onto neighborhood, N e in N . Extend the functions x 3 , x 4 from e N by keeping their values constant along the integral curves of k a . Then u, x 3 , x 4 are coordinates e. on N e satisfying l a ka = 1 and e let l a be the unique null vector field on N At each p ∈ N a a a e and satisfy X ∇a u = 0. On a sufficiently l Xa = 0 for all X which are tangent to N


701

e × {0} in N e × R, let 9 : Q → M be the map which small open neighborhood, Q, of N takes (p, r) ∈ Q into the point of M lying at affine parameter value r along the null geodesic starting at p with tangent l a . Then 9 is C ∞ and it follows from the inverse e × {0} onto function theorem that 9 is 1 : 1 and onto from an open neighborhood of N e to O e in M. We extend the functions u, x 3 , x 4 from N an open neighborhood, O, of N by requiringtheir values to be constant along each null geodesic determined by l a . Then u, r, x 3 , x 4 yields a coordinate system on O which will be referred to as Gaussian null e we have k a = (∂/∂u)a . coordinate system. Note that on N Since by construction the vector field l a = (∂/∂r)a is everywhere tangent to null geodesics we have that grr = 0 throughout O. Furthermore, we have that the metric functions gru , gr3 , gr4 are independent of r, i.e. gru = 1, gr3 = gr4 = 0 throughout O. e. In addition, as a direct consequence of the above construction, guu and guA vanish on N Hence, within O, there exist smooth functions f and hA , with f |Ne = (∂guu /∂r) |r=0 and hA |Ne = (∂guA /∂r) |r=0 , so that the spacetime metric in O takes the form ds 2 = r · f du2 + 2drdu + 2r · hA dudx A + gAB dx A dx B ,

(A.1)

where gAB are smooth functions of u, r, x 3 , x 4 in O such that gAB is a negative definite 2 × 2 matrix, and the uppercase Latin indices take the values 3, 4. B. The Existence of a Killing Field Tangent to the Horizon The purpose of this section is to prove the following fact. Proposition B.1. Suppose that (M, g, F ) is a time oriented solution to the EinsteinMaxwell equations with Maxwell field F (without sources). Let N1 , N2 be smooth null hypersurfaces with connected space-like boundary Z, smoothly embedded in M, which are generated by the future directed null geodesics orthogonal to Z. Assume that N1 ∪N2 is achronal. Then there exists on the future domain of dependence, D + , of N1 ∪ N2 a non-trivial Killing field K which is tangent to the null generators of N1 and N2 if and only if these null hypersurfaces are expansion and shear free. If it exists, the Killing field is unique up to a constant factor and we have LK F = 0. Remark B.1. We shall prove Proposition B.1 by first deducing the form that K must have on N1 ∪ N2 , then defining K on D + by evolution from N1 ∪ N2 of a wave equation that must be satisfied by a Killing field, and, finally, proving that the resulting K is indeed a Killing field. An alternative, more geometric, approach to the proof of Proposition B.1 would be to proceed as follows. Suppose that the Killing field K of Proposition B.1 exists. Denote by ψt the local 1-parameter group of isometries associated with K. Since ψt maps N1 ∪ N2 into itself, geodesics into geodesics, and preserves affine parameterization, we can describe for given t the action of ψt on a neighborhood of N1 ∪ N2 in D + in terms of its action on geodesics passing through N1 ∪ N2 into D + and affine parameters which vanish on N1 ∪ N2 . There are various possibilities, one could employ e.g. the future directed time-like geodesics starting on Z or the null geodesics which generate double null coordinates adapted to N1 ∪ N2 . Changing the point of view, one could try to use such a description to define maps ψt , show that they define a local group of isometries, and then define K as the corresponding Killing field. To show that ψt∗ g = g, one would prove this relation on N1 ∪ N2 and then invoke the uniqueness for the characteristic initial value problem for the Einstein-Maxwell equations to show that this relation holds also in a neighborhood of N1 ∪ N2 in D + . For this to work we would

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need to show that ψt∗ g has a certain smoothness (C 2 say). This is not so difficult away from N1 ∪ N2 but it is delicate near the initial hypersurface. The discussion would need to take into account the properties of the underlying space-time exhibited in Lemmas B.1, B.2 below and would become quite tedious. For this reason we have chosen not to proceed in this manner. It will be convenient to use the formalism, notation, and conventions of [30] in a gauge adapted to our geometrical situation. Since we will be using the tetrad formalism, throughout this Appendix we shall omit all abstract indices a, b, c . . . on tensors and use the indices i, j, k . . . to denote the components of tensors in our tetrad. We begin by choosing smooth coordinates u = x 1 , r = x 2 , x A , A = 3, 4, and a smooth tetrad field z1 = l, z2 = n z3 = m, z4 = m, with gik = g(zi , zk ) such that g12 = g21 = 1, g34 = g43 = −1 are the only nonvanishing scalar products. Let x A be coordinates on a connected open subset ζ of Z on which also m, m can be introduced such that they are tangent to ζ . On ζ we set x 1 = 0, x 2 = 0 and assume that l is tangent to N1 , n is tangent to N2 , and both are future directed. The possible choices which can be made above will represent the remaining freedom in our gauge. We assume ∇z2 z2 = 0,

< z2 , d x µ > = δ µ 1 , on N20 ,

and set ζc = {x 1 = c} ⊂ N20 for c ≥ 0, where N20 denotes the subset of N2 generated by the null geodesics starting on ζ . We assume that m, m are tangent to ζc . From the transformation law of the spin coefficient γ under rotations m → ei φ m we find that we can always assume that γ = γ on N20 . With this assumption m, whence also l, will be fixed uniquely on N20 and we have γ = 0,

ν = 0 on N20 .

The coordinates and the frame are extended off N20 such that ∇z1 zi = 0,

< z1 , d x µ > = δ µ 2 .

On a certain neighborhood, D, of N10 ∪N20 in D + (where N10 is the subset of N1 generated by the null geodesics starting on ζ ), we obtain by this procedure a smooth coordinate system and a smooth frame field which has in these coordinates the local expression lµ = δ 1 µ ,

lµ = δµ 2 ,

nµ = δ µ 1 + U δ µ 2 + XA δ µ A ,

mµ = ω δ µ 2 + ξ A δ µ A .

We have N10 = {x 1 = u = 0}, N20 = {x 2 = r = 0} and κ = 0, = 0, π = 0, τ = α + β on D,

U = 0, XA = 0, ω = 0 on N20 .

We shall use alternatively the Ricci rotation coefficients defined by ∇i zj ≡ ∇zi zj = γj l i zl or their representation in terms of spin coefficients as given in [30]. The gauge above will be used in many local considerations whose results extend immediately to all of N1 and N2 . We shall then always state the extended result. We begin by showing the necessity of the conditions on the null hypersurfaces in Proposition B.1 and some of their consequences.


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Lemma B.1. Let N be a smooth null hypersurface of the space-time (M, g) and X a smooth vector field on M which is tangent to the null generators of N and does not vanish there. If g 0 denotes the pull back of g to N , then LX g 0 = 0 on N if and only if the null generators of N are expansion and shear free. Proof. We can assume that N coincides with the hypersurface N1 . Then we have in our gauge X = X 1 z1 on N with X1 6 = 0, and LX g 0 = 0 translates into 0 = ∇(i Xj ) = z(i (X 1 ) δ 2 j ) − γ(j 2 i) X 1 on N with i, j 6 = 2. By our gauge this is equivalent to 0 = γ(A 2 B) = −σ δ 3 A δ 3 B − Re ρ δ 3 (A δ 4 B) − σ¯ δ 4 A δ 4 B . Since ρ is real on the hypersurface N , the assertion follows. u t Lemma B.2. If the null hypersurfaces N1 , N2 are expansion and shear free, then the frame coefficients, the spin coefficients, the components 9i of the conformal Weyl spinor field, the components φk of the Maxwell spinor field, and the components 8ik = k φi φ¯ k of the Ricci spinor field are uniquely determined in our gauge on N10 and N20 by the field equations and the data φ1 ,

τ,

ξ A , A = 3, 4, on Z.

(B.1)

In particular, we have 90 = 0, 91 = 0, φ0 = 0, 80k = 8k0 = 0, D φ1 = 0, D φ2 = δ φ1 , φ2 = r δ φ1 , ω = −r τ, µ = r 92 , on N10 ,

(B.2)

94 = 0, 93 = 0, φ2 = 0, 8i2 = 82i = 0, 1 φ1 = 0, 1 φ0 = δ φ1 − 2 τ φ1 , φ0 = u (δ φ1 − 2 τ φ1 ), ρ = u (δ τ − 2 α τ − 92 ) on N20 .

(B.3)

Proof. In our gauge the relations 90 = 0, 800 = k φ0 φ 0 = 0 on N10 are an immediate consequence of the NP equations and our assumption that ρ = 0, σ = 0 on N10 . Similarly, the assumptions µ = 0, λ = 0 on N20 imply 94 = 0, 822 = k φ2 φ 2 = 0 on N20 . The relation 8ij = k φi φ j implies the other statements on the Ricci spinor on N10 , N20 and it allows us to determine 811 on Z from the data (B.1). The NP equations involving only the operators δ, δ, the data (B.1), and our gauge conditions allow us to calculate the functions α, β, 91 = 0, 92 , 93 = 0 on Z. Then all metric coefficients, spin coefficients, and the Weyl, Ricci, and Maxwell spinor fields are known on Z. The remaining assertions follow by integrating in the appropriate order the NP equations (cf. also the appendix of [30]) involving the operator D on N10 and the t equations involving the operator 1 on N20 . u Lemma B.3. A Killing field K as considered in Proposition B.1 satisfies, up to a constant factor, K = r z1 on N1 ,

K = −u z2 on N2 .

704


Proof. We note, first, that the statement above is reasonable, because the vector fields z1 , z2 can be defined globally on N1 and N2 respectively and the form of K given above is preserved under rescalings consistent with our gauge freedom on Z. Writing K = K i zi , we have by our assumptions K = K 1 z1 on N1 , K = K 2 z2 on N2 for some smooth functions K 1 , K 2 which vanish on Z. To determine their explicit form we use, in addition to the Killing equation LK gij = ∇i Kj + ∇j Ki = 0

(B.4)

the identity ∇i ∇j Kl + Km R m ilj =

1 ∇i (LK glj ) + ∇j (LK gli ) − ∇l (LK gij ) 2

(B.5)

which holds for arbitrary smooth vector field K and metric g and which implies, together with Eq. (B.4), the integrability condition ∇i ∇j Kl + Km R m ilj = 0.

(B.6)

The restriction of Eq. (B.4) to ζ gives ∇i Kj = 2 h δ 1 [i δ 2 j ] with h = z1 (K 1 ) = −z2 (K 2 ). Using this expression to evaluate Eq. (B.6) on ζ for i = A = 3, 4, and observing that ζ is connected we get zA (h) = 0, whence h = const. on ζ . Since Z is connected the same expression for ∇i Kj will be obtained everywhere on Z with the same constant h. If h were zero, K would vanish identically by Eqs. (B.4) and (B.6). Since K is assumed to be non-trivial we have h 6 = 0 and can rescale K to achieve h = 1. Equations (B.4), (B.6) imply in our gauge z1 (K2 ) = −∇2 K1 ,

z1 (∇2 K1 ) = 0 on N10 ,

z2 (K1 ) = −∇1 K2 ,

z2 (∇1 K2 ) = 0 on N20 ,

t which, together with the value of ∇i Kj , i, j = 1, 2, on Z entail our assertion. u Taking into account ρ = 0, σ = 0 on N1 , µ = 0, λ = 0 on N2 , and in particular Eq. (B.2), we immediately get the following. Lemma B.4. By calculations which involve only inner derivatives on the respective null hypersurface one obtains from Eq. (B.3), ∇ i Kj = δ 1 i δ 2 j ,

i 6= 2, on N1 ,

∇i Kj = −δ 2 i δ 1 j + u τ δ 3 i δ 1 j + u τ δ 4 i δ 1 j ,

i 6 = 1, on N2 .

(B.7) (B.8)

Equation (B.6) implies the hyperbolic system ∇i ∇ i Kl − Km R m l = 0,

(B.9)

and the initial data for the Killing field we wish to construct are given by Lemma B.3. Both have an invariant meaning. Lemma B.5. There exists a unique smooth solution, K, of Eq. (B.9) on D + which takes on N1 ∪ N2 the values given in Lemma B.3.


705

Proof. The uniqueness of the solution is an immediate consequence of standard energy estimates. The results in [31] or [32] entail the existence of a unique smooth solution of Eq. (B.9) for the data given in Lemma B.3 on an open neighborhood of ζ in D ∩ D + (N10 ∪ N20 ). These local solutions can be patched together to yield a solution in some neighborhood of Z in D. Because of the linearity of Eq. (B.9) this solution can be t extended (e.g. by a patching procedure) to all of D + . u Lemma B.6. The vector field K of Lemma B.5 satisfies LK g = 0 and LK F = 0 on D+. Proof. The equations above need to be deduced from the structure of the data in Lemma B.3 and from Eq. (B.9). Applying ∇j to (B.9) and commuting derivatives we get ∇i ∇ i (LK gj l ) = 2 LK Rj l + 2 R i j l k (LK gik ) − 2 R i (j (LK gl)i ). The Einstein equations give 1 LK Rij = k 0 2 F(i l (LK Fj )l ) − gij (LK Fkl ) F kl − (LK gkl ) Fi k Fj 2

(B.10)

l

(B.11)

1 1 kl km l − (LK gij ) Fkl F + gij (LK gkl ) F F m . 4 2 The identity d LK F = d (iK d F + d iK F ) together with Maxwell’s equations implies ∇[i (LK Fj l] ) = 0.

(B.12)

Applying LK to the second part of Maxwell’s equations and using the identity (B.5) as well as the fact that K solves Eq. (B.9), we get ∇ i (LK Fik ) = F j l ∇j (LK glk ) + (LK gj l ) ∇ j F l k .

(B.13)

Substituting Eq. (B.11) in Eq. (B.10), we can view the system (B.10), (B.12), (B.13) as a homogeneous linear system for the unknowns LK g, LK F . This system implies a linear symmetric hyperbolic system for the unknowns LK g, ∇ LK g, LK F (cf. [33]). We shall show now that these unknowns vanish on N1 ∪ N2 . The standard energy estimates for symmetric hyperbolic systems then imply that the fields vanish in fact on D + , which will prove our lemma and thus Proposition B.1. Equation (B.9) restricted to N10 reads 0 = ∇i ∇ i Kl − Km R m l = 2 (∇1 ∇2 Kl − ∇3 ∇4 Kl ) − Km (R m l + R m l21 − R m l43 ). Using this equation together with Eqs. (B.2) and (B.7) and our gauge conditions, we obtain by a direct calculation a system of ODE’s of the form ∇1 (∇(i Kj ) ) = Hij (∇(k Kl) ), on the null generators of N10 . Here Hij is a linear function of the indicated argument (suppressing the dependence on the points of N10 ). Since ∇(k Kl) = 0 on Z, we conclude that LK g = 0 on N1 . An analogous argument involving Eqs. (B.3) and (B.8) shows that LK g = 0 on N2 . It follows in particular that ∇ LK g = 0 on Z.

706


Writing (LK F )AA0 BB 0 = A0 B 0 pAB + AB pA0 B 0 and using (B.3) we find on N10 in NP notation p0 = r D φ0 + φ0 ,

p1 = r D φ1 ,

p2 = r D φ2 − φ2 .

It follows from Eq. (B.2) that pAB , whence LK F , vanishes on N1 . An analogous argument involving Eq. (B.3) shows that LK F vanishes on N2 . Observing in Eqs. (B.11) and (B.10) that LK F = 0, LK g = 0, whence also ∇l (∇(i Kj ) ) = 0 for l 6 = 2 on N10 , we obtain there 0 = ∇k ∇ k (∇(i Kj ) ) = 2 ∇1 (∇2 (∇(i Kj ) )). We conclude that ∇ LK g, which vanishes on Z, vanishes on N1 . In a similar way it t follows that ∇ LK g = 0 on N2 . This completes the proof. u Acknowledgements. This research was supported in part by Monbusho Grant-in-aid No. 96369 and by NSF grant PHY 95-14726 to the University of Chicago. We wish to thank Piotr Chrusciel and James Isenberg for reading the manuscript. One of us (IR) wishes to thank the Albert Einstein Institute and the Physics Department of the Tokyo Institute of Technology for their kind hospitality during part of the work on the subject of the present paper.

References 1. Hawking, S.W.: Black holes in general relativity. Commun. Math. Phys. 25, 152–166 (1972) 2. Hawking, S.W. and Ellis, G.F.R.: The large scale structure of space-time. Cambridge: Cambridge University Press, 1973 3. Chru´sciel, P.T.: On rigidity of analytic black holes. Commun. Math. Phys. 189, 1–7 (1997) 4. Chru´sciel, P.T.: Uniqueness of stationary, electro-vacuum black holes revisited. Helv. Phys. Acta 69, 529–552 (1996) 5. Israel, W.: Event horizons in static vacuum space-times. Phys. Rev. 164, 1776–1779 (1967) 6. Israel, W.: Event horizons in static electrovac space-times. Commun. Math. Phys. 8, 245–260 (1968) 7. Carter, B.: Axisymmetric black hole has only two degrees of freedom. Phys. Rev. Lett. 26, 331–333 (1971) 8. Carter, B.: Black hole equilibrium states. In: Black Holes, C. de Witt and B. de Witt (eds.), New York, London, Paris: Gordon and Breach, 1973 9. Mazur, P.O.: Proof of uniqueness of the Kerr–Newman black hole solutions. J. Phys. A: Math. Gen. 15, 3173–3180 (1982) 10. Bunting, G.L.: Proof of the uniqueness conjecture for black holes. Ph. D. Thesis, University of New England, Admirale (1987) 11. Moncrief, V. and Isenberg, J.: Symmetries of cosmological Cauchy horizons. Commun. Math. Phys. 89, 387–413 (1983) 12. Isenberg, J. and Moncrief, V.: Symmetries of cosmological Cauchy horizons with exceptional orbits. J. Math. Phys. 26, 1024–1027 (1985) 13. Penrose, R.: Singularities an time asymmetry. In: General relativity; An Einstein centenary survey, eds. S.W. Hawking, W. Israel, Cambridge: Cambridge University Press, 1979 14. Rácz, I.: On further generalization of the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon. In preparation 15. Rácz, I. and Wald, R.M.: Extension of spacetimes with Killing horizon. Class. Quant. Grav. 9, 2643–2656 (1992) 16. Rácz, I. and Wald, R.M.: Global extensions of spacetimes describing asymptotic final states of black holes. Class. Quant. Grav. 13, 539–553 (1996) 17. Wald, R.M.: General relativity. Chicago: University of Chicago Press, 1984 18. Chru´sciel, P.T. and Wald, R.M.: Maximal hypersurfaces in asymptotically flat spacetimes. Commun. Math. Phys. 163, 561–604 (1994) 19. Friedman, J.L., Schleich, K. and Witt, D.M.: Topological censorship. Phys. Rev. Lett. 71, 1486–1489 (1993) 20. Chru´sciel, P.T. and Wald, R.M.: On the topology of stationary black holes. Class. Quant. Grav. 11, L147– L152 (1994)


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21. Galloway, G.J.: On the topology of the domain of outer communication. Class. Quant. Grav. 12, L99–L101 (1995) 22. Galloway, G.J.: A “finite infinity” version of the FSW topological censorship. Class. Quant. Grav. 13, 1471–1478 (1996) 23. Geroch, R.: A method for constructing solutions of Einstein’s equations. J. Math. Phys. 12, 918–924 (1971) 24. Wald, R.M.: Quantum field theory on curved spacetimes. Chicago: University of Chicago Press, 1994 25. Miller, J.G.: Global analysis of the Kerr–Taub-NUT metric. J. Math. Phys. 14, 486–494 (1973) 26. Moncrief, V.: Infinite-dimensional family of vacuum cosmological models with Taub-NUT (Newman– Unti–Tamburino-type extensions. Phys. Rev. D. 23, 312–315 (1981) 27. Moncrief, V.: Neighborhoods of Cauchy horizons in cosmological spacetimes with one Killing field. Ann. of Phys. 141, 83–103 (1982) 28. Seifert, H.: Topologie dreidimensionaler gefaserter Räume. Acta. Math. 60, 147–238 (1933); English translation in: H. Seifert, W. Threlfall: A Textbook of Topology, New York: Academic Press, 1980 29. Epstein, D.B.A.: Periodic flows on three-manifolds. Ann. Math. 95, 66–81 (1972) 30. Newman, E., Penrose, R.: An Approach to Gravitational Radiation by a Method of Spin coefficients. J. Math. Phys. 3, 566–578 (1962), 4, 998 (1963) 31. Müller zum Hagen, H.: Characteristic initial value problem for hyperbolic systems of second order differential systems. Ann. Inst. Henri Poincaré 53, 159–216 (1990) 32. Rendall, A.D.: Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations. Proc. R. Soc. Lond. A 427, 221–239 (1990) 33. Friedrich, H.: On the Global Existence and the Asymptotic Behaviour of Solutions to the Einstein– Maxwell–Yang–Mills Equations. J. Diff. Geom. 34, 275–345 (1991) Communicated by H. Nicolai

Commun. Math. Phys. 204, 709 – 729 (1999)

Communications in



Spacing Between Phase Shifts in a Simple Scattering Problem Steve Zelditch1 , Maciej Zworski2 1 Department of Mathematics, The Johns Hopkins University, Baltimore, MD 21218, USA.

E-mail: [email protected]

2 Mathematics Department, University of California, Berkeley, CA 94720, USA.

E-mail: [email protected] Received: 15 August 1998 / Accepted: 15 February 1999

Abstract: We prove a scattering theoretical version of the Berry–Tabor conjecture: for an almost every surface in a class of cylindrical surfaces of revolution, the large energy limit of the pair correlation measure of the quantum phase shifts is Poisson, that is, it is given by the uniform measure. 1. Introduction and Statement of the Result The Berry–Tabor conjecture [2] for quantum integrable systems with discrete spectra asserts that the spacings between normalized eigenvalues of a quantum integrable system should exhibit Poisson statistics in the semi-classical limit. In particular, when the eigenvalues are scaled to have unit mean level spacing, the distribution of their differences should be uniform. This conjecture has been verified numerically in many cases [4] and has been rigorously proved in an almost everywhere sense for flat 2tori (Sarnak [14]), flat 4-tori (Vanderkam [17]), deterministically for almost all flat tori (Eskin–Margulis–Mozes [5]), and for certain integrable quantum maps in one degree of freedom (Rudnick–Sarnak [12], Zelditch [1]). Smilansky [16] has more recently posed an analogous conjecture for scattering systems with continuous spectra. Since the scattering matrix S(E) at energy level E is, at least heuristically, the quantization of the classical scattering map, he argues that when the scattering map is integrable the eigenvalues of S(E) ( known as phase shifts) should exhibit Poisson statistics. In particular, he proposed that the pair correlation function of scaled phase shifts should be uniform for surfaces of revolution with a cylindrical end (see Fig. 1). The purpose of this paper is to prove (a somewhat modified form of) this conjecture for almost all surfaces in an infinite dimensional family of (pairs of) such surfaces. To explain the modifications and state our results, we need to introduce some notation. The surfaces we consider are topological discs X on which S 1 acts freely except for a unique fixed point m. The metrics g we consider are invariant under the S 1 action and in geodesic polar coordinates centered at m have the form g = dr 2 + a(r)2 dθ 2 , where

710

S. Zelditch, M. Zworski

a(r) defines a short range cylindrical end metric (see Sect. 2). For technical reasons, we are only able to analyse the phase shifts at this time in the case where g has a conic singularity at m.

θ

Fig. 1. A surface of revolution with a conic singularity and a cylindrical end

To define the pair correlation measure, we recall that at energy λ2 the scattering matrix for a surface of revolution with a cylindrical end is given by a diagonal (2[λ] + 1) × (2[λ] + 1) matrix with entries exp(2πiδk (λ)), |k| ≤ [λ] – see Sect. 3 for a detailed presentation. The phase shifts are given by δk (λ) and are well defined modulo Z. The parameter k corresponds to the angular momentum or in other words to the eigenvalues of the Laplacian on the cross-section (the circle in our case). When |k| is close to λ we expect no scattering phenomena as the classical motion is close to the bounded motion along the cross-sections (see Fig. 2). At the opposite extreme, when |k|/|λ| is close to 0, the classical motion is along geodesics approaching the singularity on the surface. Since the properties of the pair correlation measure are supposed to correspond to the properties of smooth classical motion it is natural, at least at this early stage, to delete the angular momenta corresponding to the neighbourhoods of the singularities. Based on this discussion we define for any > 0 the following measure ρλ ([a, b]) =

def

1 1 ] {(l, m, k) : l, m, k ∈ Z, < |l/λ| , |m/λ| < 1−, (1−2) 2λ+1 (1.1) (2λ + 1)(1 − 2)(δl (λ) − δm (λ) + k) ∈ [a, b] } .

In other words for f ∈ S(R), Z f (x)ρλ (dx) =

1 X 1 (1 − 2) 2λ + 1 k∈Z

X

f ((1 − 2)(1 + 2λ)(δl (λ) − δm (λ) + k)) .

m,l∈Z 0.

(2.4)

712


The metric can be extended to a smooth metric on X (endowed with a natural C ∞ structure coming from polar coordinates, (r, θ) ) if and only if a 0 (0) = 1, a 2p (0) = 0 , p ≥ 0,

(2.5)

see for instance [3]. We will not assume (2.5) and consequently we allow bullet like surfaces shown in Fig.1. The classical dynamics is given by the Hamiltonian flow of the metric: p = |ξ |2gx = ρ 2 + a(r)−2 t 2 ,

(2.6)

where we parametrized T ∗ (X \ {m}) by (x, ξ ) = (r, θ; ρ, t), with ρ and t dual to r and θ respectively. As is well known this flow is completely integrable: {p, t} = 0, and t = ξ(∂θ ) is called the Clairaut integral. Abstractly, ∂θ is the vector field generating the S1 action on X \ {m}. As in the case of compact simple surfaces of revolution (see [1,6,3]) we have a stronger statement: Proposition 1. For (X\{m}, g) with the metric g satisfying (2.1)–(2.4) there exist global action angle variables on T ∗ (X \ {m}). Although it plays no part in the proof, this is worth presenting here as the global action variables are closely related to the asymptotics of the phase shifts. Proof. The moment map P T ∗ (X \ {m}) 3 (x, ξ ) 7 −→ |ξ |gx , ξ(∂θ ) ∈ R+ × R has the range given by the open set B = {(b1 , b2 ) : |b2 | < b1 }. For any (b1 , b2 ) ∈ B, P −1 (b1 , b2 ) consists of a R × S1 orbit of a single geodesic in T ∗ (X \ {m} (the R-action corresponds to the geodesic flow and the S1 -action to the θ -rotation). In the case of a simple surface of revolution, the global action variables, (I1 , I2 ), are defined by Z 1 α, α = ξ · dx, (2.7) Ij (b) = 2π γj (b) where (γ1 (b), γ2 (b)) is a global trivialization of the bundle H1 (P −1 (b), Z) of the homology groups along the fibers of P . When γ1 (b) is chosen as the orbit of the S1 -action, then I1 = ξ(∂θ ). In the case of non-compact surfaces discussed here the fibers are given by R × S1 and not by S1 × S1 (except for the degenerate case of the meridians, t = 0, where the fiber is (R \ {0}) × S1 , where 0 corresponds to the point m). Consequently the integral for I2 given by (2.7) diverges (for γ1 we can still take the compact orbit of the S1 -action). Hence we have to normalize the integral using the fact that the surface is asymptotic to a cylinder with a(r) ≡ 1. If we take γ2 (b) to correspond to a geodesic in P −1 (b), then outside of the turning point ρ = 0 (or r = 0 for the degenerate case of the meridians) it can be parametrized by r. Then ξ · dx becomes ρdr and we can put 1 lim I2 (b) = π R→∞

Z

R 0

b12

b2 2 − a(r)2

21 +

Z dr − 0

R

1

(b12 − b22 ) 2 dr,

(2.8)

that is we normalize by subtracting the “free” ρdr defined by ρ 2 + b22 = b12 . From this we find the angle variables as in [1,6]. u t

Spacing Between Phase Shifts

713

3. Review of Scattering Theory There are many ways of introducing the scattering matrix on a manifold of the type we consider. Since we only assume (2.2), X is not a b-manifold in the sense of Melrose – see [9,8]. It is a manifold with a cusp metric at one end and a conic metric at the other – see [9]. We shall not however use this point of view here. Instead we will proceed more classically and we will define the scattering matrix using the wave operators – see [8] for an indication of the relation between the two approaches. As in the proof of Proposition 1 we need a free reference problem X0 ' R × S1 , g0 = dr 2 + dθ 2 .

(3.1)

On X and X0 we define the wave groups, U (t) and U0 (t): U (t) : Cc∞ (X) × Cc∞ (X) 3 (u0 , u1 ) 7−→ (u(t), Dt u(t)) , where (Dt2 − 1g )u = 0 , ut=0 = u0 , Dt ut=0 = u1 . The operators U (t) extend as a unitary group to the energy space, H(X), obtained by taking the closure of Cc∞ (X) × Cc∞ (X) with respect to the norm k(u0 , u1 )k2E = k∇u0 k2L2 + ku1 k2L2 . The definition and properties of U0 (t) are analogous. We then define the Møller wave operators W± : H(X0 ) −→ H(X), by W± [w] = lim U (−t)χ(r)U0 (t)w, w ∈ H(X0 ), t→±∞

where χ ∈ C ∞ ([0, ∞); [0, 1]), χ(r) ≡ 0 for r < 1 and χ(r) ≡ 1 for r > 2, and where for r > 1 we used the obvious identification of the corresponding subsets of X and X0 . In the situation we consider the existence of W± is quite straightforward and we choose the wave rather than the Schrödinger picture just for variety. The scattering operator is S = W−∗ W+ : H(X0 ) −→ H(X0 ) def

(3.2)

and, as we will see below, it is a unitary operator. When there is no pure point spectrum then the wave operators W± are themselves unitary. In all situations they are partial isometries and W±∗ = limt→±∞ U0 (−t)χ(r)U (t). The null space of W±∗ is the span of the L2 eigenfunctions of 1. Under our assumptions there could only be finitely many such eigenfunctions. The wave operators have the intertwining properties: 0 I 0 I 0 I = W± H⇒ S, = 0. W± 1g0 0 1g0 0 1g 0

714


Since all operators commute with the generator of the S1 action, ∂θ , we decompose S using the spectral decompositions of 1g0 and of ∂θ . It is easy to check that Z ∞ 0 I = λdEλ0 , 1g0 0 −∞ where the Schwartz kernel of dEλ0 is given by 1 sgn(λ) X in(θ−θ 0 ) 0 I 2 2 2 0 −1 e eisgn(λ)(λ −n ) (r−r ) (λ2 − n2 )+ 2 dλ. dEλ0 (r, θ; r 0 , θ 0 ) = 2 2 λ 0 (2π) n∈Z

Because S commutes with the generator of the free propagator, U0 (t), we obtain the scattering matrix at fixed energy using the above spectral decomposition: Z S = S(λ)dEλ0 , and then the decomposition corresponding to the eigenvalues of ∂θ : S(λ) =

1 X 0 Sn (λ)ein(θ −θ ) . 2π n∈Z

From the structure of dEλ0 it is clear that Sn (λ) ≡ 0 for |n| > |λ|. For |λ| = |n| we follow [8] and put Sn (λ) = lim Sn (sgn(λ)τ ). τ →|λ|+

We also note that Sn (λ) = S−n (λ). For |n| ≤ |λ|, Sn (λ) is a unitary operator, that is, it is given by multiplication by a complex number of unit length: Sn (λ) = e2π iδn (λ) ,

(3.3)

and the number δn (λ) is the nth phase shift at energy λ2 . Another way to think about S(λ) is as a diagonal unitary (2n + 1) × (2n + 1) matrix, where n = [|λ|]: . S(λ) = e2πiδk (λ) δkj −n≤k,j ≤n

A more “down-to-earth” definition, following the traditional way of introducing phase shifts in one dimensional scattering, is given through asymptotic expansions in (3.6) below. The uniform behaviour as k and λ go to infinity and k λ is a well understood semi-classical problem. To describe it we separate variables in the eigen-equation of the Laplacian. We remark that this procedure can also provide direct proofs of the general scattering theoretical statements above. The Laplace operator is given by 1g = Dr2 − i

a 0 (r) 1 Dr + D2 a(r) a(r)2 θ


715

and on the eigenspaces of Dθ it acts as a 0 (r) 1 n2 Dr + a(r) a(r)2 1 1 n2 2a 00 (r)a(r) − (a 0 (r))2 = a(r)− 2 Dr2 + − a(r) 2 . a(r)2 4a(r)2

1n = Dr2 − i

(3.4)

The reduced operator appearing in brackets in the second line above has a self-adjoint realization on L2 ((0, ∞)r ) and for large λ it can be considered semi-classically: 1n − λ2 = λ2 a(r)− 2 P (x, h)a(r) 2 , h = 1

1

1 n , x= , |λ| λ

P (x, h) = (hDr )2 + V (r; x, h) − 1, V (r; x, h) =

x2 a(r)2

− h2

2a 00 (r)a(r) − (a 0 (r))2 4a(r)2

(3.5) def

, V0 (r; x) = V (r; x, 0).

The principal symbol of P (x, h) is given by p = ρ 2 +x 2 /a(r)2 −1 and the natural range of x for which semi-classical methods are applicable is given by 0 < < |x| < 1 − . In fact, since a(r) is one at infinity, we approach zero energy when x 2 is close to 1. On the other hand when x → 0 the characteristic variety of p has a singular limit – see Fig.3. A detailed analysis of the x → 0 limit has to involve the lower order terms in V (r; x, h). In particular, miraculous cancellations in the expansions due to the interaction between the leading and lower order terms occur when we have product type conic singularities since we can then use the theory of Bessel functions. The general situation is, at least to the authors, unclear at the moment. What is quite clear is that we have a uniform expansion in h/x. ρ p(0) = 0 1

p(x) = 0, x = 0 r

-1 p(0) = 0 T * IR + Fig. 3. The characteristic variety of P (x, h)

716


The phase shifts δn (λ) are related to the semi-classical phase shifts of the operator P (x, h), ψ(x, h), which are defined by asymptotics of solutions: i

P (x, h)u = 0, u(r) = e h

√

i

1−x 2 r

i

√

+ e h ψ(x,h) e− h 1−x n 1 |λ| ψ , . δn (λ) = 2π λ |λ|

2r

+O

1 , r → ∞, r

(3.6)

We recall now the essentially standard asymptotic properties of ψ – see [10], Chapter 6 and for a more microlocal discussion [11]. Proposition 2. As h → 0, ψ(x, h) defined by (3.6) has an asymptotic expansion uniform in < |x| < 1 − for any fixed > 0: ψ(x, h) ∼ ψ(x) + h

π + h2 ψ2 (x) + · · · , 2

(3.7)

where ∞

Z ψ(x) = 0

1

1

(1 − V0 (r, x))+2 − (1 − x 2 ) 2

dr.

(3.8)

We remark that when we translate the asymptotics to the coordinates on T ∗ (X \ {m}): x = t/λ, λ2 = ρ 2 + t 2 /a(r)2 we obtain the second action variable defined in the proof of Proposition 1. As mentioned in the introduction, we can describe this connection between the phase shifts and action variables by saying that S(λ) is a quantum map on G(X, g), the space of geodesics. We now digress to explain this statement in more detail. For simplicity we will consider S at integral values of λ and denote them by N. Since it is not needed in the calculation of the limit pair correlation function we give a somewhat sketchy discussion and refer to [18] for background on Toeplitz quantization. See also [15] for a related discussion from a physicist’s point of view. ∗ (X ) of incoming vectors at the parallel We can identify G(X, g) with the set Sin r0 def

Xr0 = X ∩ {r = r0 }. As in Proposition 2 we have to delete -neighborhoods of the singular set, given by {|t/λ| < }, where t = I1 = ξ(∂θ ) is the first action variable and λ2 = ρ 2 + t 2 a(r)−1 is the energy. We denote the deleted space of geodesics by G (X, g) and identify it with the set ∗ (X ) of incoming vectors at X with incoming angle satisfying |θ | > , |θ −π | > Sin, r0 r0 and |θ − π2 | > . If we consider the deleted space of geodesics as a phase space then, on the quantum level, it corresponds to the sequence of truncated Hilbert spaces HN, spanned by the eigenfunctions {einθ } of the quantum action N1 Iˆ1 with < | Nn | < 1 − , where Iˆ1 = −i∂θ . Here, 1/N plays the role of the Planck constant and we restrict it to integral values. Since HN, is invariant under S(N ) we may restrict the latter to a unitary scattering matrix S (N) on HN, . We now state the somewhat informal:


717

Proposition 3. The sequence {S (N)} is a semiclassical quantum map over G (X, g) associated to the classical scattering map ∗ ∗ (Xr0 ) → Sin, (Xr0 ), β : Sin,

where β(x, ξ ) is obtained by following the geodesic γ(x,ξ ) through (x, ξ ) until it intersects Xr0 for the last time and reflecting the outgoing tangent vector inward. Proof. From the explicit formula S (N ) = (e

2πiδk (N )

δkj )≤|k/N|,|j/N|≤(1−) ,

|N | ψ δk (N ) = 2π

k 1 , N N

(3.9)

we see that S (N) is the exponential of N times the Hamiltonian bN, = χ H

b b I1 1 I1 ψ , N N N

on HN, where χ is a smooth cutoff function defining the truncated Hilbert space. The truncated phase space G (X, g) is symplectically equivalent to a truncated S 2 , equipped with its standard area form, with neighbourhoods of the poles and of the equator deleted. Indeed, the equivalence is defined by the identity map between global actionangle charts on the surfaces. This map intertwines the obvious S 1 actions which rotate the spaces. The quantization of this chart then defines a unitary equivalence on the quantum level which intertwines the operators ∂θ on cylinder and sphere (they can be considered as the angular momentum operators). The equivalence is specified up to a choice of 2N + 1 phases by mapping the spherical harmonic of degree N which transforms under rotation by θ on S 2 by eikθ to the exponential eikθ with k ∈ [−N, . . . , N]. The map is completely specified by requiring that the spherical harmonic be real valued along θ = 0. bN, with a Hamiltonian over the compact phase space S 2 . Thus we may identify H Since it is a function of the (Toeplitz) action operator Ib1 it is necessarily a semiclassical Toeplitz operator of order zero with principal symbol χ (I1 /E)ψ(I1 /E) on S 2 . The semiclassical parameter N is identified in the Toeplitz theory with a first order positive elliptic Toeplitz operator with eigenvalue N in HN – see [18] and references given there. bN, is a first order Toeplitz operator of real principal type. As in the essentially Hence N H analogous case of pseudodifferential operators, the exponential of a first order Toeplitz operator of real principal type is a Fourier integral Toeplitz operator whose underlying classical map is the Hamilton flow generated by ψ. We now wish to identify this map at time one with the classical scattering (or billiard ∗ (X ), that is, we wish to prove that β = exp 4 , where 4 is the ball) map on Sin, r0 ψ ψ Hamilton vector field of ψ. Indeed, let us work in the symplectic action-angle coordinates (θ, I1 ), where θ is the angle along Xr0 . The Hamilton flow of ψ then takes the form exp t4ψ (θ, I1 ) = (θ + tω, I1 ),

ω = ∂I1 ψ.

(3.10)

At time t = 1 the angle along the parallel Xr0 changes by ω. We claim that ω is also the change in angle along the incoming geodesic through θ ∈ Xr0 in the direction I1 as it scatters in the bullet head before exiting again along Xr0 .

718


To see this, we use Proposition 2 which shows that ψ is closely related to the second action variable: in the notation of the proof of Proposition 1, b2 , I1 (b1 , b2 ) = b2 . I2 (b1 , b2 ) = b1 ψ b1 Since b1 = λ is preserved by the flow we can fix it at λ = 1 and then Z ∂I1 ψ(I1 ) = 2I1

dr a(r)2

I2 1− 1 2 a(r)

!− 1 2

.

(3.11)

+

On the other hand the equations of motion show that θ˙ 2ta(r)−2 I1 dθ = = = dr r˙ 2ρ a(r)2

I2 1− 1 2 a(r)

!− 1 2

.

(3.12)

It follows that ω(I1 ) is twice the change in angle as the radial distance changes from r0 to its minimum along the geodesic. The piece of the geodesic lying in the bullet-head consists of two segments: the initial segment beginning on Sr0 and ending upon its tangential intersection with the parallel Xr− (I1 ) closest to m, and the segment beginning at this intersection and ending on Xr0 . The change in θ-angle along both segments is the same, so that the total change in angle during the scattering is given by the integral (3.12) above. This shows that β and exp 4ψ have precisely the same formula in action-angle variables and completes the proof of the proposition. u t 4. Exponential Sums Following [18] we will reduce the study of (1.2) to a study of certain exponential sums. We first remark that because of symmetries of δk (λ) we can study a slightly simpler expression ρ˜λ (f ) =

X 1 (1 − 2)λ

m∈Z

X

f (1 − )λ(δj (λ) − δk (λ) + m)

<j/λ

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