# (P,P)

arXiv:dg-ga/9608008v2 29 Aug 1996 KAEHLER STRUCTURES ON KC/(P, P ) MENG-KIAT CHUAH Abstract Let K be a compact connected semi-simple Lie group, let G...

arXiv:dg-ga/9608008v2 29 Aug 1996

KAEHLER STRUCTURES ON KC/(P, P ) MENG-KIAT CHUAH Abstract Let K be a compact connected semi-simple Lie group, let G = KC , and let G = KAN be an Iwasawa decomposition. Given a K-invariant Kaehler structure ω on G/N , there corresponds a pre-quantum line bundle L on G/N . Following a suggestion of A.S. Schwarz, in a joint work with V. Guillemin, we study its holomorphic sections O(L) as a K-representation space. We define a K-invariant L2 -structure on O(L), and let Hω ⊂ O(L) denote the space of square-integrable holomorphic sections. Then Hω is a unitary K-representation space, but we find that not all unitary irreducible K-representations occur as subrepresentations of Hω . This paper serves as a continuation of that work, by generalizing the space considered. Instead of working with G/N = G/(B, B), where B is a Borel subgroup containing N , we consider G/(P, P ), for all parabolic subgroups P containing B. We carry out similar construction, and recover in Hω the unitary irreducible K-representations previously missing. As a result, we use these holomorphic sections to construct a model for K: a unitary K-representation in which every irreducible K-representation occurs with multiplicity one.

1991 Mathematics Subject Classification. Primary 53C55. Keywords: Lie group, Kaehler, line bundle.

1

INTRODUCTION

Let K be a compact connected semi-simple Lie group, let G = KC be its com1

plexification, and let G = KAN be an Iwasawa decomposition. Since G and N are complex Lie groups, G/N is a complex manifold, and G acts on G/N by left action. Let T be the centralizer of A in K, so that H = T A is a Cartan subgroup of G. Since H normalizes N, there is a right action of H on G/N. We shall often be interested in the maximal compact group action of K × T . We let g, k, h, t, a, n denote the Lie algebras of G, K, H, T, A, N respectively. The following scheme of geometric quantization was suggested by A.S. Schwarz [12]: Equip G/N with a suitable K-invariant Kaehler structure ω, and consider the pre-quantum line bundle L associated to ω ([5], [11]). The Chern class of L is [ω], and L comes with a connection ∇ whose curvature is ω, as well as an invariant Hermitian structure <, >. We denote by O(L) the space of holomorphic sections on L. The K-action on G/N lifts to a K-representation on O(L). Let µ be the K × A-invariant measure on G/N, which is unique up to non-zero constant. Given a holomorphic section s of L, we consider the integral Z

G/N

< s, s > µ .

Let Hω ⊂ O(L) denote the holomorphic sections in which this integral converges. Since µ is K-invariant, Hω becomes a unitary K-representation space. It was hoped in [12] that every irreducible K-representation occurs with multiplicity one in Hω (called a model by I.M. Gelfand [7]). By the method of highest weight, the irreducible K-representations can be labeled by the dominant integral weights in t∗ , up to isomorphism. In a joint work with V. Guillemin [4], we carry out this construction, but find that no matter how ω is chosen, the irreducibles whose highest weights lie on the wall of the Weyl chamber do not occur in the Hilbert space Hω . Therefore, not all unitary K-irreducibles occur in Hω . The present paper follows a suggestion of V. Guillemin ([4] p.192), by modifying the space G/N to more general classes of homogeneous spaces. As a result, we manage to recover the unitary K-irreducibles previously missing. Let B = HN be the Borel subgroup of G. Observe that (B, B) = N, hence 2

G/N = G/(B, B). With this in mind, we can generalize the class of homogeneous spaces considered to G/(P, P ), for P a parabolic subgroup of G containing B. Since P is a complex Lie group, so is (P, P ); hence G/(P, P ) is a complex manifold. Clearly G acts on G/(P, P ) on the left, and we shall see that a complex subgroup of H normalizes (P, P ), and hence acts on G/(P, P ) on the right. Let W ⊂ t∗ denote the open Weyl chamber, and W its closure. We say that σ ⊂ W is a cell if there exists a subset S of the positive simple roots ∆ such that (1.1)

σ = {x ∈ W ; (x, S) = 0 , (x, ∆\S) > 0},

where the pairing used is the Killing form. This way, W is a disjoint union of the cells of various dimensions. Using the Killing form and the almost complex structure, it is convenient to regard the cell σ as contained in any of the spaces h, t, a, h∗ , t∗ , a∗ , depending on the context. The cell σ defines a subalgebra hσ of h, by taking complex linear span of σ. Similarly, the subalgebras tσ , aσ are defined by intersecting hσ with t, a respectively. These subalgebras define the subgroups Hσ , Tσ , Aσ of H, T, A respectively. A bijective correspondence between the cells {σ} and the parabolic subgroups {P } containing B is given by Langlands decomposition ([10] p.132) (1.2)

P = MAσ Nσ .

Fix a parabolic subgroup P containing B, with σ its corresponding cell. Since Hσ is the normalizer of (P, P ) in H, it acts on G/(P, P ) on the right. Out of the action of the complex group G × Hσ , we shall consider the action of the maximal compact group K × Tσ on G/(P, P ). We shall show that Theorem I Let ω be a K-invariant Kaehler structure on G/(P, P ). Then ω is K × Tσ -invariant if and only if it has a potential function. Though we shall be interested mostly in Kaehler structures, Theorem I holds also for degenerate (1,1)-form ω. In the next theorem, we shall derive a necessary and 3

sufficient condition for a (1,1)-form ω to be Kaehler. Let ω be a K × Tσ -invariant (1,1)-form, so that ω=

¯ −1∂ ∂F,

for some function F on G/(P, P ). Averaging by the compact group K if necessary, we may assume that F is K-invariant. Let K σ be the centralizer of Tσ in K. It defines σ σ a compact semi-simple subgroup Kss of K, given by Kss = (K σ , K σ ). We shall show

that, as real manifolds and K × Hσ -spaces, (1.3)

σ G/(P, P ) = (K/Kss ) × Aσ .

Therefore, the potential function F , being K-invariant, can be regarded as a function on Aσ . Since the exponential map identifies the vector space aσ with Aσ , F becomes a function on aσ . The almost complex structure identifies the dual spaces a∗σ ∼ = t∗σ , hence the Legendre transform of F can be written as LF : aσ −→ t∗σ . The significance of this map will become apparent shortly, when we study the moment map. We write log : Aσ −→ aσ for the inverse of the exponential map. The K-action on G/(P, P ) preserving ω is Hamiltonian: there exists a unique moment map Φ : G/(P, P ) −→ k∗ corresponding to this action. Since Φ is K-equivariant, (1.3) implies that it is deterσ mined by its value on Aσ ⊂ (K/Kss ) × Aσ , where Aσ is imbedded as its product with

σ σ . Meanwhile, since k is semi-simple, the Killing form the identity coset eKss ∈ K/Kss

on k is non-degenerate; which induces the inclusion t∗ ⊂ k∗ from t ⊂ k. Theorem II Let ω be a K × Tσ -invariant (1,1)-form on G/(P, P ). Then its

moment map Φ and its potential function F satisfy Φ(a) = 12 LF (log a) ∈ t∗σ for all √ ¯ is Kaehler if and only if: a ∈ Aσ . Further, ω = −1∂ ∂F 4

(i)

F ∈ C ∞ (aσ ) is strictly convex; and

(ii) The image of

1 L 2 F

is contained in the cell σ ⊂ t∗σ ; i.e. Φ(Aσ ) ⊂ σ.

Since a K ×Tσ -invariant Kaehler structure ω has a potential function F , it is exact. Therefore, it is in particular integral. Let L be the line bundle on G/(P, P ) whose Chern class is [ω] = 0, equipped with a connection ∇ whose curvature is ω ([5],[11]). The topology of L is trivial, but the connection ∇ gives rise to interesting geometry on the holomorphic sections of L. We recall that L is equipped with an invariant Hermitian structure <, >. Let µ be a K × Aσ -invariant measure on G/(P, P ). We consider the integral (1.4)

Z

G/(P,P )

< s, s > µ ,

for holomorphic sections s of L. As we shall see in Theorem III, convergence of this integral is determined by the image of the moment map. The K × Tσ -action on G/(P, P ) lifts to a K × Tσ -representation on O(L), the space of holomorphic sections of L. We similarly define Hω ⊂ O(L) to be the holomorphic sections in which (1.4) converges. Since µ is K-invariant, Hω becomes a unitary K-representation space. For a dominant integral weight λ, let O(L)λ be the holomorphic sections in L that transform by λ under the right Tσ -action. Since the left K-action commutes with the right Tσ -action, O(L)λ is a K-representation space. Let σ be the cell corresponding to the parabolic subgroup P , and let σ be its closure. Then

Theorem III The irreducible K-representation with highest weight λ occurs in O(L) if and only if λ ∈ σ. For λ ∈ σ, it occurs with multiplicity one, and is given by O(L)λ . Further, O(L)λ is contained in Hω if and only if λ lies in the image of the moment map.

With this result, it is now clear that in [4], the singular representations are never contained in Hω : 5

When P = B, σ becomes the open Weyl chamber W . Then Theorem II says that Φ(Aσ ) ⊂ W ; and by K-equivariance, Φ(G/(P, P )) = Ad∗K (Φ(Aσ )) does not intersect the wall W \W . Consequently, by Theorem III, the irreducible representations O(L)λ with highest weight λ ∈ W \W cannot be contained in Hω . Similarly, for general parabolic subgroup P , not all O(L)λ are contained in Hω : For λ ∈ σ\σ, Theorems II and III say that O(L)λ exists non-trivially but is not contained in Hω . We shall see that, however, for a suitable Kaehler structure ω on G/(P, P ), the image of the moment map intersects σ in all of σ. This way, by Theorem III, all the Kirreducibles O(L)λ with highest weights λ ∈ σ are contained in Hω . As an application, we provide a geometric construction of a unitary K-representation, containing all the irreducibles with multiplicity one.

Acknowledgement

The author would like to thank V. Guillemin, R. Sjamaar

and D. Vogan for many helpful suggestions. The referee has helped to clarify some definitions and notations used in this paper.

6

2

KAEHLER STRUCTURES ON G/(P, P )

The main purpose of this section is to prove Theorem I. Since K is connected and semi-simple, so is G = KC . Let P be a parabolic subgroup of G containing B, and σ the cell corresponding to P . They are related by Langlands decomposition (1.2) P = MAσ Nσ , where Aσ is the subgroup described in §1. Then Aσ ⊂ A, Nσ ⊂ N, where A, N come from Iwasawa decomposition of G. Further, Aσ normalizes Nσ , and is the centralizer of MAσ in A. Therefore, Hσ = Tσ Aσ is the normalizer of (P, P ) = (M, M)Nσ in H, which induces a natural right Hσ -action on G/(P, P ). We shall give another description of G/(P, P ), which reflects this right action better. Since G is semi-simple, the Killing form is non-degenerate. Let a⊥ σ be the orthocomplement of aσ with respect to the Killing form in a, and A⊥ σ ⊂ A the corresponding

σ ⊥ ⊥ ⊥ ⊥ subgroup induced by a⊥ σ . We construct tσ , Tσ , hσ , Hσ similarly. Let K be the sub-

group of K given by K σ = {k ∈ K ; kt = tk for all t ∈ Tσ }. σ = (K σ , K σ ) be the corresponding compact semi-simple Lie group. Then Let Kss

(2.1)

σ σ (Kss )C = Kss A⊥ σ (M ∩ N)

σ is an Iwasawa decomposition of the complexified group (Kss )C . Since N = (M ∩ N) Nσ ,

it follows from (2.1) that σ ⊥ σ Kss Aσ N = (Kss )C Nσ σ = (KC )ss Nσ

(2.2)

= (MAσ , MAσ )Nσ = (M, M)Nσ = (P, P ).

7

Then, the Iwasawa decomposition G = KAN and (2.2) imply that σ G/(P, P ) = (K/Kss ) × Aσ ,

(2.3)

as real manifolds and K × Hσ -spaces. With this description, the right action of

σ σ Hσ = Tσ Aσ is clear: Tσ acts on (K/Kss ) × Aσ simply because it commutes with Kss

σ and Aσ , while Aσ acts on (K/Kss ) × Aσ by group multiplication on itself. We shall

be concerned with the K × Tσ -action on G/(P, P ). Since N = (B, B) ⊂ (P, P ), there is a fibration (2.4)

π : G/N −→ G/(P, P ).

σ It follows from G = KAN and (2.3) that the fiber of π is Kss × A⊥ σ . Further, π

sends every right H-orbit in G/N to a right Hσ -orbit in G/(P, P ), by contracting each Hσ⊥ -coset to a point. Given a K-invariant Kaehler structure ω on G/(P, P ), we want to show that it is invariant under the right Tσ -action if and only if it has a potential function. Our strategy is to work on the (1,1)-form π ∗ ω on G/N using results in [4], then transfer this result back to ω. Let V be the orthocomplement of t in k with respect to the Killing form, so that k = t ⊕ V . The Killing form also induces t∗ ⊂ k∗ from t ⊂ k. If F is a function on A, then by the exponential map, it becomes a function on a. Using the almost complex structure, a∗ ∼ = t∗ . Therefore, the Legendre transform of F can be written as (2.5)

LF : a −→ t∗ .

Given ξ ∈ k, we let ξ ♯ denote its infinitesimal vector field on G/N induced by the K-action. Let J be the almost complex structure on G/N. For η = Jξ ∈ a, where

ξ ∈ t, we define η ♯ to be the vector field Jξ ♯ . Let a ∈ A ⊂ KA = G/N. Then its tangent space is Ta (G/N) = h♯a ⊕ Va♯ . We recall the following result from [4]: Proposition 2.1 [4]

Let ω be a K × T -invariant (1,1)-form on G/N. Then 8

ω=

¯ , where F ∈ C ∞ (A) by K-invariance. It satisfies ω(h♯ , V ♯ )a = 0. The −1∂ ∂F

K-action is Hamiltonian, with moment map Φ : G/N −→ k∗ satisfying (i) (ii)

Φ(a) ∈ t∗ for all a ∈ A ⊂ KA = G/N;

Φ : A −→ t∗ is given by Φ(a) = 12 LF (log a).

Let m = dim σ, n = dim t. Let {λ1 , ..., λr } be the positive roots of g, where {λ1 , ..., λn } are simple. Here m ≤ n ≤ r. Then dim V = 2r, and dim k = n + 2r. In the following proposition, we give a useful decomposition of V . Recall that we define the cell σ in (1.1) using a subset S of the positive simple roots ∆. By switching the roles of S and ∆\S, we can define another cell σ ′ , with dimension n − m. We call σ ′ the complementary cell to σ. Let J be the almost complex structure on k ⊕ a = g/n. Recall that, V is the orthocomplement of t in k.

Proposition 2.2 Let σ, σ ′ be complementary cells of dimensions m, n − m respectively, where m ≤ n ≤ r =

1 2

dim V . There exists a decomposition V = ⊕r1 Vi into

two dimensional subspaces Vi . Each Vi is preserved by J and satisfies [Vi , Vi ] ⊂ t. Further, (i) (ii)

m t⊥ σ′ = ⊕1 [Vi , Vi ] ,

n t⊥ σ = ⊕m+1 [Vi , Vi ] .

If ω is a K × T -invariant (1,1)-form on G/N, then ω(Vi♯ , Vj♯ )a = 0 for all i 6= j, a ∈ A ⊂ KA = G/N. Proof:

Let {λ1 , ..., λr } be the positive roots of g, indexed such that the first n of

them are simple. Further, we can require that (λi , σ) > 0 , (λi , σ ′ ) = 0 ; i = 1, ..., m, and (λi , σ) = 0 , (λi , σ ′ ) > 0 ; i = m + 1, ..., n, where the pairing taken is the Killing form. 9

Let g±i be the root spaces corresponding to ±λi . Then there exist ξ±i ∈ g±i such that (2.6)

{ ζi = ξi − ξ−i ,

γi =

−1(ξi + ξ−i ) }i=1,...,r

form a basis of V ([8] p.421). Here {ζi, γi } are orthogonal to t because the root spaces gi are orthogonal to h. Further, {ξ±i } can be chosen such that [ζi , γi] ∈ t, and is dual to λi ∈ t∗ with respect to the Killing form. We define Vi = R(ζi , γi). Then [Vi , Vi ] ⊂ t. Let J be the almost complex structure on k ⊕ a = g/n. ¿From (2.6), it follows that J sends ζi to γi, and sends γi to −ζi . Therefore, each Vi is preserved by J. For i = 1, ..., m, (λi , σ ′ ) = 0. Since [ζi , γi ] is dual to λi , it follows that [ζi , γi ] ∈ t⊥ σ′ . Hence [Vi , Vi ] ⊂ t⊥ σ′ for i = 1, ..., m. But the dual vectors of λ1 , ..., λm form a basis of m ⊥ t⊥ σ′ , hence tσ′ = ⊕1 [Vi , Vi ].

n For i = m + 1, ..., n, (λi , σ) = 0. By similar argument, t⊥ σ = ⊕m+1 [Vi , Vi ].

Let ω be a K × T -invariant (1,1)-form on G/N. Suppose that i 6= j; we want to

show that ω(Vi♯ , Vj♯ )a = 0 for a ∈ A ⊂ KA = G/N. Let p : k −→ t be the orthogonal projection, annihilating V . Let ξ ∈ Vi , η ∈ Vj . From (2.6), it follows that [ξ, η] is either 0 or in Vk , depending on whether λi + λj is some positive root λk . In any case, (2.7)

p[ξ, η] = 0 ; ξ ∈ Vi , η ∈ Vj .

Let Φ : G/N −→ k∗ be the moment map corresponding to the K-action preserving ω. Then Φ(a) ∈ t∗ , by Proposition 2.1. Consequently, ω(ξ ♯, η ♯ )a = (Φ(a), [ξ, η]) = (Φ(a), p[ξ, η])

since Φ(a) ∈ t∗

= 0. Therefore, ω(Vi♯ , Vj♯ )a = 0 for i 6= j. This proves the proposition.

10

2

Let ω be a K ×Tσ -invariant Kaehler structure on G/(P, P ). Let π be the fibration in (2.4). Then π ∗ ω is a K × T A⊥ σ -invariant (1,1)-form on G/N. By Proposition 2.1, it has the form π∗ω =

¯ −1∂ ∂f,

where f is a K-invariant function on G/N. Since G/N = KA, f ∈ C ∞ (A). We shall show that f can be replaced with another function F which is in the image of π ∗ : C ∞ (G/(P, P )) −→ C ∞ (G/N), so that we get a potential function for ω. Let σ be the cell which corresponds to P by (1.2), and σ ′ its complementary cell. Then σ ′ defines subgroups Hσ′ , Tσ′ , Aσ′ of H, T, A respectively. By taking the orthocomplements of the Lie algebras hσ′ , tσ′ , aσ′ , we construct the subgroups Hσ⊥′ , Tσ⊥′ , A⊥ σ′ ∞ ⊥ as before. Note in particular that A = A⊥ σ Aσ′ . Define F ∈ C (A) by

F = ρ∗ f , ρ : A −→ A⊥ σ′ −→ A ;

(2.8)

⊥ where ρ is the composite function of the submersion A −→ A⊥ σ′ annihilating Aσ ,

followed by the inclusion A⊥ σ′ −→ A. By G/N = KA, F extends uniquely to be a ∗ K × T A⊥ σ -invariant function on G/N. Note that F is in the image of π . We define

the K × T A⊥ σ -invariant (1,1)-form Ω=

¯ −1∂ ∂F.

We shall show that (2.9)

Ω = π ∗ ω.

⊥ ⊥ Here both Ω and π ∗ ω are K × T A⊥ σ -invariant. Since G/N = KAσ′ Aσ , we only

♯ ♯ have to compare them at a ∈ A⊥ σ′ . Also, Proposition 2.1 says that ha and Va are

complementary with respect to both Ωa and π ∗ ωa . Therefore, (2.9) will follow if we can show that (2.10)

Ω(ξ ♯ , η ♯ )a = π ∗ ω(ξ ♯ , η ♯ )a ; ξ, η ∈ h or ξ, η ∈ V , a ∈ A⊥ σ′ . 11

This will be checked by the following two lemmas. Recall that LF , Lf : a −→ t∗ are the Legendre transforms of F and f , described in (2.5).

Lemma 2.3 Ω(ξ ♯ , η ♯ )a = π ∗ ω(ξ ♯, η ♯ )a for all ξ, η ∈ V, a ∈ A⊥ σ′ . Proof:

By Proposition 2.2, the spaces (V1 )♯a , ..., (Vr )♯a are pairwise complementary

with respect to Ωa and π ∗ ωa , a ∈ A⊥ σ′ . Therefore, to prove the statement in this lemma, we may consider ξ, η ∈ Vi for each component Vi seperately. Since each Vi is two dimensional, it suffices to consider ξ = ζi , η = γi . Let ΦF , Φf : G/N −→ k∗ be the moment maps of the K-actions preserving Ω, π ∗ ω respectively. We recall from Proposition 2.1 that ΦF (a) = 21 LF (log a), Φf (a) = 21 Lf (log a). We follow the indices i = 1, ..., r used in Proposition 2.2, as well as the cells σ, σ ′ of dimensions m, n − m respectively. In what follows, we break up our arguments into three cases, according to the different values of the index i. Case 1: i = 1, ..., m. Ω(ζi♯ , γi♯ )a = (ΦF (a), [ζi, γi ]) = ( 12 LF (log a), [ζi , γi ]). By Proposition 2.2, [ζi , γi ] ∈ t⊥ σ′ , for i = 1, ..., m. By (2.8), LF (log a) and Lf (log a) ⊥ agree on t⊥ σ′ , for a ∈ Aσ′ . Therefore, the last expression is

( 21 Lf (log a), [ζi , γi ]) = (Φf (a), [ζi , γi]) = π ∗ ω(ζi♯, γi♯ )a . Case 2: i = m + 1, ..., n. We recall (2.6), which implies that (2.11)

[v, ζi] =

√ −1(λi , v)γi , [v, γi ] = − −1(λi , v)ζi

for all v ∈ t. Therefore, the Lie algebra kσ of K σ is given by kσ = {ξ ∈ k ; [ξ, σ] = 0} = t ⊕(λi ,σ)=0 Vi . 12

The center of this Lie algebra is tσ , hence the semi-simple Lie algebra kσss is given by kσss = t⊥ σ ⊕(λi ,σ)=0 Vi .

(2.12)

σ For i = m + 1, ..., n, (λi , σ) = 0; hence ζi , γi ∈ kσss . But Kss is in the fiber of π, so

ı(ξ ♯ )π ∗ ωa = 0 for all ξ ∈ Vi . We shall show that

ı(ξ ♯ )Ωa = 0 for all ξ ∈ Vi . Since each Vi is two dimensional, this will follow if we can show that Ω(ζi♯ , γi♯ )a = 0, for i = m + 1, ..., n. But

1 Ω(ζi♯ , γi♯ )a = ( LF (log a), [ζi, γi ]) = 0, 2 since [ζi , γi] ∈ t⊥ σ and by (2.8), LF (log a) vanishes there. Case 3: i = n + 1, ..., r. ⊥ ¿From Cases 1, 2, we see that LF (log a), Lf (log a) ∈ t∗ agree on the spaces t⊥ σ , tσ′ . ∗ ⊥ Since t = t⊥ σ ⊕ tσ′ , it follows that LF (log a) = Lf (log a) ∈ t . Therefore,

Ω(ζi♯ , γi♯ )a = (ΦF (a), [ζi , γi]) = ( 21 LF (log a), [ζi , γi]) = ( 12 Lf (log a), [ζi, γi ]) = (Φf (a), [ζi , γi]) = π ∗ ω(ζi♯, γi♯ )a . This proves Lemma 2.3.

2

Lemma 2.4 Ω(ξ ♯ , η ♯ )a = π ∗ ω(ξ ♯, η ♯ )a for all ξ, η ∈ h, a ∈ A⊥ σ′ . Proof:

Let hσ , hσ′ denote the subalgebras of h, by taking the complex linear spans

⊥ of σ, σ ′ respectively. Let h⊥ σ , hσ′ denote their orthocomplements with respect to the ⊥ Killing form. Then h = h⊥ σ ⊕ hσ ′ .

Case 1: ξ, η ∈ h⊥ σ′ . 13

Let ι : Hσ⊥′ −→ H denote the inclusion. ¿From (2.8), we get √

¯ ∗F ) = −1∂ ∂(ι

¯ ∗ f ), −1∂ ∂(ι

⊥ where ∂, ∂¯ are Dolbeault operators on Hσ⊥′ here. Therefore, given a ∈ A⊥ σ ′ ⊂ Hσ ′ ,

Ω(ξ ♯ , η ♯ )a = (ι∗ Ω)(ξ ♯ , η ♯ )a √ ¯ ∗ F ))(ξ ♯, η ♯ )a = ( −1∂ ∂(ι √ ¯ ∗ f ))(ξ ♯, η ♯ )a = ( −1∂ ∂(ι = (ι∗ π ∗ ω)(ξ ♯, η ♯ )a

= π ∗ ω(ξ ♯, η ♯ )a . Case 2: ξ ∈ h⊥ σ. We shall show that ı(ξ ♯ )π ∗ ωa = ı(ξ ♯ )Ωa = 0,

(2.13)

which completes the proof of this lemma. Since π ∗ ω and Ω are (1,1)-forms, it suffices to check (2.13) for ξ ∈ t⊥ σ.

σ ⊥ The fiber of π is Kss × A⊥ σ , which contains Hσ . Therefore,

ı(ξ ♯ )π ∗ ωa = 0. We observe that, as complex manifolds, H = Cn /Zn , Hσ⊥ = Cn−m /Zn−m , Hσ⊥′ = Cm /Zm , and H = Hσ⊥ Hσ⊥′ . We introduce complex coordinates {z1 , ..., zm } on Hσ⊥′ as well as √ {zm+1 , ..., zn } on Hσ⊥ ; so that H adopts the product coordinates. Let z = x + −1y, and we let x, y be the coordinates on T, A respectively. ¿From H = T A, G/N = KA and T ⊂ K, we get a natural holomorphic imbedding ι : H −→ G/N. Then ι∗ F , being T -invariant, is a function on y only. For simplicity we still denote it as F . It follows from (2.8) that ∂F = 0 for i = m + 1, ..., n. ∂yi 14

Therefore, for a ∈ A⊥ σ′ , (2.14)

√ ¯ )a ı(ξ ♯ )(ι∗ Ω)a = ı(ξ ♯ )( −1∂ ∂F = ı(ξ ♯ )( 12

= ı(ξ ♯ )( 21

Pn

∂2F j,k=1 ∂yj ∂yk dxj

Pm

∂2F

j,k=1 ∂yj ∂yk dxj

∧ dyk ) ∧ dyk ).

♯ On the other hand, since ξ ∈ t⊥ σ , the vector field ξ on H is of the form

ξ♯ =

n X

m+1

ci

∂ . ∂xi

This, together with (2.14), imply that ı(ξ ♯ )Ωa = 0. This proves (2.13). Combining the results in Cases 1,2, we have proved Lemma 2.4. 2

Lemmas 2.3 and 2.4 imply (2.10), and hence (2.9). Namely, we have shown that given a K × Tσ -invariant Kaehler structure ω on G/(P, P ), there exists a function F , which is in the image of π ∗ by virtue of (2.8), such that π∗ω =

¯ −1∂ ∂F.

Since F is in the image of π ∗ , and since π ∗ is injective, it follows that ω itself has a potential function. Conversely, suppose that a K-invariant Kaehler structure ω on G/(P, P ) has a potential function F . Averaging by the compact group K if necessary, we may assume that F is K-invariant. But by (2.3), this means that F is just a function on Aσ , and is automatically K × Tσ -invariant. Then ω is also K × Tσ -invariant. This proves Theorem I. We note that our arguments do not require ω to be positive definite. Namely, Theorem I holds even if ω is merely a K-invariant (1,1)-form. In the next section, we use the moment map to derive a necessary and sufficient condition for a K × Tσ invariant (1,1)-form to be Kaehler. 15

3

MOMENT MAP

Let ω be a K × Tσ -invariant (1,1)-form on G/(P, P ), with moment map Φ : G/(P, P ) −→ k∗ corresponding to the Hamiltonian action of K on G/(P, P ) preserving ω. It is easy to see that this action is Hamiltonian; either from the semi-simplicity of K ([6], §26), √ ¯ implies ω = dβ for some K-invariant real 1-form or from the fact that ω = −1∂ ∂F β ([1], Theorem 4.2.10). We shall study the moment map Φ, and derive a necessary and sufficient condition for ω to be Kaehler. Suppose now that ω is a K × Tσ -invariant Kaehler structure. We want to derive the two conditions stated in Theorem II. By Theorem I, ω has a potential function F . Averaging by K if necessary, we may assume that F is K-invariant. By (2.3), σ G/(P, P ) = (K/Kss ) × Aσ ; so the K-invariant function F is just a function on Aσ .

Let π be the fibration in (2.4). Then Φ ◦ π : G/N −→ k∗ is the moment map corresponding to the K-action on (G/N, π ∗ω). Recall that P σ corresponds to a cell σ via (1.2). Also, G/N = KA and G/(P, P ) = (K/Kss ) × Aσ

induce the inclusions σ σ ) × Aσ = G/(P, P ) . A ֒→ {e} × A ⊂ KA = G/N , Aσ ֒→ {eKss } × Aσ ⊂ (K/Kss

Therefore, we can regard A and Aσ as contained in G/N and G/(P, P ) respectively. Note that π(A) = Aσ . ¿From Proposition 2.1, we see that (Φ ◦ π)(A) ⊂ t∗ . Since the fibration π sends A to Aσ , it follows that Φ(Aσ ) ⊂ t∗ . By K-equivariance of Φ, Φ|Aσ determines Φ entirely. The exponential map from aσ to Aσ is a diffeomorphism, and we let log be its inverse. This way, the potential function F becomes a 16

function on aσ . Then, by the almost complex structure, a∗σ ∼ = t∗σ . Consequently, the Legendre transform of F is LF : aσ −→ t∗σ . We shall show that Φ : Aσ −→ t∗ is given by Φ(a) = 21 LF (log a) for all a ∈ Aσ . Let ı : Hσ −→ G/(P, P ) be the natural holomorphic imbedding of Hσ = Tσ Aσ . Then ı∗ ω is a Tσ -invariant Kaehler structure on Tσ Aσ , with potential function ı∗ F . For simplicity, we still write ı∗ F as F . Let m be the dimension of the cell σ. Then, as a complex manifold, Hσ = Cm /Zm . Therefore, we can introduce complex coordinates {z1 , ..., zm } on Hσ , where (3.1)

Hσ = Cm /Zm = {z1 , ..., zm } , Tσ = Rm /Zm = {x1 , ..., xm } , √ Aσ = Rm = {y1 , ..., ym } , zi = xi + −1yi .

Since F is Tσ -invariant, it is a function on y only. Then ı∗ ω becomes (here ∂, ∂¯ are Dolbeault operators on Hσ ) (3.2)

ı∗ ω =

¯ = −1∂ ∂F

m ∂2F 1 X dxj ∧ dyk , 2 j,k=1 ∂yj ∂yk

where F ∈ C ∞ (Rm ). Since ω is Kaehler, so is ı∗ ω; and (3.2) says that ı∗ ω is Kaehler if and only if the Hessian matrix of F is positive definite, i.e. F is strictly convex. The moment map Φ of the K-action on (G/(P, P ), ω) restricts to be the moment map Φ′ of the Tσ -action on (Tσ Aσ , ı∗ ω). Let β=−

m 1X ∂F dxj 2 j=1 ∂yj

be a Tσ -invariant 1-form on Tσ Aσ . From (3.2), it follows that dβ = ı∗ ω, so the moment

17

map Φ′ of the Tσ -action is (Φ′ (ta), ξ) = −(β, ξ ♯ )(ta) = ( 21

=

1 2

Pm ∂ ∂F k=1 ξk ∂xk )(ta) j=1 ∂yj dxj ,

Pm

Pm

∂F j=1 ∂yj (a)ξj

= 12 (LF (a), ξ) , where ta ∈ Tσ Aσ , ξ ∈ t = Rm . Our computation identifies a with A by the exponential

map, so in fact Φ′ (ta) = 12 LF (log a) for all ta ∈ Tσ Aσ . But Φ and Φ′ agree on Aσ , so Φ(a) = 21 LF (log a). Hence Φ(Aσ ) ⊂ t∗σ . We claim further that Φ(Aσ ) ⊂ σ:

Let Vi ⊂ V ⊂ k be the subspaces constructed in Proposition 2.2, and let {ζi , γi} ∈ Vi be the vectors in (2.6). Recall that these indices are made with respect to the σ ) × Aσ , the infinitesimal vector fields positive roots {λi }. Since G/(P, P ) = (K/Kss

ζi♯ , γi♯ on G/(P, P ) are non-zero if and only if ζi, γi 6 ∈kσss . By (2.12), this is equivalent

to (λi , σ) > 0. Let J be the almost complex structure in G/(P, P ), a ∈ Aσ , and (λi , σ) > 0 so that ζi♯ , γi♯ 6= 0. By (2.6), Jζi = γi . Since ω is Kaehler, 0 < ω(ζi♯, Jζi♯ )a (3.3)

= ω(ζi♯, γi♯ )a = (Φ(a), [ζi , γi ]) = (Φ(a), λi ).

We have shown that, for all a ∈ Aσ , (Φ(a), λi ) > 0 whenever λi is a positive root satisfying (λi , σ) > 0. This, together with Φ(Aσ ) ⊂ t∗σ , imply that Φ(Aσ ) ⊂ σ, as claimed. We have shown that if ω is Kaehler, then the two conditions stated in Theorem II have to be satisfied. We next show that, conversely, these two conditions are sufficient for ω to be Kaehler. Recall that the infinitesimal vector field ξ ♯ on G/(P, P ) vanishes if ξ ∈ kσss . Hence

♯ ♯ the tangent space at a ∈ Aσ ⊂ G/(P, P ) is spanned by (kσ⊥ ss )a , (aσ )a . Here we define

η ♯ for η = Jξ ∈ aσ by η ♯ = Jξ ♯ , where ξ ∈ tσ . However, it follows from (2.12) that kσ⊥ ss = tσ ⊕(λi ,σ)>0 Vi , 18

where Vi is the space described in Proposition 2.2. Here the distinct Vi are orthogonal to one another, due to the orthogonality of the root spaces gi ([8] p.166). Consequently, the tangent space at a ∈ Aσ ⊂ G/(P, P ) is (3.4)

Ta (G/(P, P )) = (hσ )♯a ⊕(λi ,σ)>0 (Vi )♯a .

We claim that ω(h♯σ , Vi♯ )a = ω(Vi♯ , Vj♯ )a = 0, for i 6= j: Since J preserves hσ and Vi , and ω is a (1,1)-form, the first part follows if we can show that ω(t♯σ , Vi♯ )a = 0. Let p : k −→ t be the orthogonal projection, annihilating V . Let ξ ∈ tσ , η ∈ Vi . Then p[ξ, η] = 0, by (2.11). Since Φ(a) ∈ t∗ for a ∈ A, ω(ξ ♯, η ♯ )a = (Φ(a), [ξ, η]) = (Φ(a), p[ξ, η]) = 0. Hence ω(h♯σ , Vi♯ )a = 0. For i 6= j, it follows from (2.7) that p[Vi , Vj ] = 0. So, by similar argument, ω(Vi♯ , Vj♯ )a = 0 as claimed.

Therefore, by K-invariance of ω and (3.4), the positive definite condition of ω follows if we can check that (3.5)

ω(ξ ♯ , Jξ ♯)a > 0 ; ξ ∈ hσ or ξ ∈ Vi , (λi , σ) > 0, a ∈ Aσ .

But they follow from the two conditions of Theorem II: Condition (i) of Theorem II implies that the expression in (3.2) is positive definite and hence (3.5) holds for ξ ∈ hσ . Condition (ii) of Theorem II implies that (Φ(a), λi ) > 0 whenever (λi , σ) > 0, so it follows from (3.3) that (3.5) holds for ξ ∈ Vi . This proves Theorem II.

19

4

LINE BUNDLE

Fix a K × Tσ -invariant Kaehler structure ω on G/(P, P ). By Theorem I, ω has a potential function F . Recall that P determines the subgroup Aσ by (1.2). By K-invariance and (2.3), we can regard F as a function on Aσ . In particular, the √ ¯ also implies that ω is exact. Hence ω is integral, and there expression ω = −1∂ ∂F exists a complex line bundle L on G/(P, P ) whose Chern class is [ω] = 0, equipped with a connection ∇ whose curvature is ω, as well as an invariant Hermitian structure <, > ([5], [11]). The line bundle L is trivial since [ω] = 0, but the connection ∇ gives rise to interesting geometry. We say that a section s is holomorphic if ∇s annihilates anti-holomorphic vector fields on G/(P, P ). We shall show that the K × Tσ -action on G/(P, P ) lifts to a K × Tσ -representation on the space of holomorphic sections of L. To do this, we shall construct a global trivialization of L. The following topological property of G/(P, P ) is useful in this construction:

Lemma 4.1 Proof:

H 1 (G/(P, P ), C) = 0.

σ By (2.3), G/(P, P ) = (K/Kss ) × Aσ . Since Aσ is Euclidean, it suffices to

σ , C) = 0. show that H 1 (K/Kss

σ The fibration K −→ K/Kss induces a long exact sequence of homotopy groups,

(4.1)

σ σ ... −→ π1 (K) −→ π1 (K/Kss ) −→ π0 (Kss ) −→ ...

However, by ([2] p.223), π1 (K) ∼ = ker(exp : t → T )/Z(roots of k). σ σ Therefore, since K is semi-simple, π1 (K) is finite. By compactness of Kss , π0 (Kss ) is σ finite. Hence π1 (K/Kss ), being caught in the middle in (4.1), is also finite. It follows

that σ σ ), C) = 0, H 1 (K/Kss , C) ∼ = Hom(π1 (K/Kss

which proves the lemma.

2 20

We return to our pre-quantum line bundle L on G/(P, P ), corresponding to the √ K × Tσ -invariant Kaehler structure ω. Let β be the 1-form − −1∂F , so dβ = ω. We claim that

Proposition 4.2 There exists a non-vanishing section so on L , with the property (4.2)

1 ∇so β=√ . −1 so

This section is unique up to a non-zero constant multiple, and is holomorphic. Up to a non-zero constant, < so , so >= e−F . Proof: Since [ω] = 0, L is a trivial bundle; so there exists a nowhere zero section s1 of L. Let 1 ∇s1 α= √ . −1 s1

By the definition of the curvature form on L, dα = ω; so d(β − α) = 0. Since

H 1 (G/(P, P ), C) = 0, there exists a complex-valued function f such that β = α + df . √ Let so = (exp −1f )s1 . Then 1 ∇so 1 ∇s1 √ =√ + df = β. −1 so −1 s1 This proves the existence of a holomorphic section so satisfying (4.2). Suppose that s1 and s2 are two sections satisfying this formula. Let h =

s2 . s1

Then

1 ∇s2 1 ∇s1 1 √ =√ + √ d log h, −1 s2 −1 s1 −1 which implies that h is a constant. Hence, up to a constant, the solution of (4.2) is unique. If v is an anti-holomorphic vector field, then 1 ∇v s o √ = ι(v)β = 0, −1 so 21

as β is a form of type (1,0). Hence so is holomorphic. Since β is K × Tσ -invariant, so induces a K × Tσ -representation on the space of holomorphic sections on L, where so is K × Tσ -invariant. Namely, given a holomorphic section f so of L (note so is non-vanishing), K × Tσ acts by L∗k Rt∗ (f so ) = (L∗k Rt∗ f )so ; k ∈ K, t ∈ Tσ ,

(4.3)

where L∗k Rt∗ f denotes the standard action on the holomorphic functions lifted from the K × Tσ -action on G/(P, P ). Hence so defines a K × Tσ -equivariant trivialization. For this section so , we now show that < so , so >= e−F . By K-invariance, it suffices to show that this is the case when restricted to Aσ . Let σ be the cell corresponding to the parabolic subgroup P , and let m be the dimension of σ. We write Tσ Aσ = Cm /Zm = {z1 , ..., zm } as in (3.1), so that F , being Tσ -invariant, is a function on y only. Let ı : Tσ Aσ −→ G/(P, P ) be the natural inclusion. Then m √ 1X ∂F ı∗ β = − −1∂F = dzi . 2 1 ∂yi

(4.4) Let ∇i = ∇

∂ ∂yi

. Then ∂ < so , so >= < ∇i so , so > + < so , ∇i so > . ∂yi

However, by (4.2) and (4.4), ∂ 1 ∂F ∇i s o √ = −1(β, )=− so ∂yi 2 ∂yi so ∂ ∂F log < so , so >= − . ∂yi ∂yi Therefore, up to a non-zero constant multiple, < so , so >= e−F . This proves the proposition.

2

22

Let O(L) denote the space of holomorphic sections of the line bundle L on G/(P, P ). By Proposition 4.2, so induces a K × Tσ -representation on O(L), given by (4.3), where so is K × Tσ -invariant. Let λ ∈ t∗σ be a dominant integral weight, and let O(L)λ denote the holomorphic sections that transform by λ under the right Tσ -action. Since this action commutes with the left K action, O(L)λ is a K-subrepresentation of O(L). We now show that the K-finite vectors in O(L) decompose into {O(L)λ ; λ ∈ σ} as irreducible K-representations with highest weights λ. Using the holomorphic section so of Proposition 4.2, it suffices to consider the holomorphic functions O(G/(P, P )); since O(G/(P, P )) ⊗ so = O(L) is a K × Tσ -equivariant trivialization. Recall that W is the closure of the Weyl chamber W , and σ ⊂ W is the cell corresponding to P . Let σ denote its closure in W . For a dominant integral weight λ ∈ t∗ , let Oλ ⊂ O(G/(P, P )) denote the holomorphic functions that transform by λ under the right Tσ -action. Since the right Tσ -action commutes with the left K-action, each Oλ is a K-representation space. Proposition 4.3 The irreducible K-representation with highest weight λ occurs in O(G/(P, P )) if and only if λ ∈ σ. For λ ∈ σ, it occurs with multiplicity one, and is given by Oλ . Proof: The fibration π of (2.4) induces an injection of holomorphic functions, π ∗ : O(G/(P, P )) −→ O(G/N). This map intertwines with the K × Tσ -action. Let λ be a dominant integral weight, but suppose that λ 6 ∈σ. We shall show that the K-irreducible with highest weight λ does not occur in O(G/(P, P )). By the Borel-Weil theorem, the K-irreducible with highest weight λ occurs in O(G/N) with multiplicity one, and can be taken as the holomorphic functions in G/N that 23

transform by λ under the right T -action. We denote this space by Vλ ⊂ O(G/N). Since π ∗ is injective, it suffices to show that (4.5)

π ∗ O(G/(P, P )) ∩ Vλ = 0.

∗ Since λ 6 ∈σ, (λ, ξ) 6= 0 for some ξ ∈ t⊥ σ . Let 0 6= f ∈ Vλ . Then the right action Rξ on

Vλ satisfies (4.6)

Rξ∗ f = (λ, ξ)f 6= 0.

Since Tσ⊥ is in the fiber of π, the image of π ∗ is Tσ⊥ -invariant. Therefore, (4.6) says that f cannot be in the image of π ∗ . This proves (4.5). Conversely, suppose that λ ∈ σ is a dominant integral weight. We again let Vλ ⊂ O(G/N) be the holomorphic functions that transform by λ. By the Borel-Weil theorem, Vλ is an irreducible representation with highest weight λ, and such irreducible occurs with multiplicity one. Therefore, to complete the proof of Proposition 4.3, we need to show (4.7)

Vλ ⊂ π ∗ O(G/(P, P )) , λ ∈ σ.

σ σ Recall from (2.2) that the fiber of π is (Kss )C /(M ∩ N) = Kss × A⊥ σ . Choose a

fiber of π, and let σ ⊥ ı : Kss Aσ ֒→ G/N σ be a holomorphic Kss × A⊥ σ -equivariant imbedding as this fiber of π. Let f ∈ Vλ . We

claim that f is constant on this fiber: σ σ )C /(M ∩ N) = Kss × A⊥ By applying the Borel-Weil theorem on (Kss σ , we see

that ı∗ f , which is right Tσ⊥ -invariant since (λ, t⊥ σ ) = 0, has to be a constant function. Hence f is constant on that fiber, as claimed. Since our argument is independent of the choice of fiber and element of Vλ , we conclude that every element of Vλ is constant on every fiber of π. This implies (4.7), and Proposition 4.3 is now proved.

2

24

We have shown that the irreducible K-representation with highest weight λ occurs in O(L) if and only if λ ∈ σ. For λ ∈ σ, it occurs with multiplicity one, and is given by O(L)λ . We shall decide which of these irreducible K-representations are squareintegrable, in the following sense. σ ¿From the description G/(P, P ) = (K/Kss ) × Aσ , we see that there is a K × Aσ -

invariant measure µ on G/(P, P ), which is unique up to non-zero constant. Given a holomorphic section s of L, we consider the integral Z

G/(P,P )

< s, s > µ .

Let Hω ⊂ O(L) be the holomorphic sections in which this integral converges. Since the Hermitian structure <, > and µ are K-invariant, Hω becomes a unitary Krepresentation space. The next proposition shows which irreducible K-representations occur in Hω . Let λ ∈ σ be a dominant integral weight. Let Φ : G/(P, P ) −→ k∗ be the moment map of the K-action on (G/(P, P ), ω). Recall that Oλ and O(L)λ are respectively the holomorphic functions and sections that transform by λ ∈ t∗σ under the right Tσ -action.

Proposition 4.4 Let s ∈ O(L)λ . Then s ∈ Hω if and only if λ is in the image of the moment map. Proof:

Let so be the unique holomorphic section of Proposition 4.2. Therefore,

< so , so >= e−F , where F is the potential function of ω. Since so is non-vanishing and K × Tσ -invariant, O(L)λ = Oλ ⊗ so . Therefore, we are reduced to showing that f ∈ Oλ satisfies (4.8)

Z

G/(P,P )

|f |2e−F µ < ∞ 25

if and only if λ is in the image of Φ. σ Here µ is the product of a K-invariant measure dk on K/Kss and a Haar measure

da on Aσ . By the exponential map, the measure da on Aσ can be identified with the Lebesgue measure dy on Rm , where m = dim σ. Given k ∈ K, the left K-action on Oλ , L∗k : Oλ −→ Oλ , is

(L∗k f )(p) = f (kp).

Let f1 , ..., fN be a basis of Oλ which is orthonormal with respect to the (unique) K-invariant inner product on Oλ . Given an element f = f (ky) = (L∗k f )(y) =

X

P

ci fi of Oλ ,

ci air (k)fr (y),

where air (k) is the irth matrix coefficient of the K-representation on Oλ with respect to the basis above. Thus Z

|f (ky)|2 dk =

X

Z

ci cj ( air (k)ajs (k)dk)fr (y)fs(y),

σ where the integrals are taken over K/Kss . However, by Peter-Weyl the inner integral

is equal to 1 δij δrs , N ([3] p.186) so the integral (4.8) reduces to (4.9)

1 kf k2 N

Z

Rm

X

|fr (y)|2 e−F (y) dy,

where kf k is the norm of f with respect to the given K-invariant inner product structure on Oλ . However, each of the functions fr (y) transforms under the infinitesimal tσ -action according to the character λ ∈ t∗σ , and therefore, being holomorphic, transforms under the action of hσ = (tσ )C according to the complexified character λC ∈ h∗σ . In particular, |fr (y)|2 is a constant multiple of e2λ(y) . Hence if f 6= 0, (4.9) is a constant multiple of the integral Z

Rm

e−F (y)+2λ(y) dy.

26

However, this integral converges if and only if 2λ is in the image of the Legendre transform of F ([4], Appendix); or equivalently if and only if λ is in the image of the moment map. This proves the proposition.

2

With this result, Theorem III follows. We see from Theorems II, III that not all irreducibles are contained in Hω : The irreducible representation O(L)λ with highest weight λ satisfies O(L)λ ⊂ Hω if and only if λ ∈

1 L (aσ ) 2 F

⊂ σ. This necessarily

excludes λ ∈ σ\σ. However, in the next section, we shall see that the potential function F can be constructed such that 12 LF (aσ ) = σ, and hence O(L)λ ⊂ Hω for

all λ ∈ σ.

27

5

CONSTRUCTION OF A MODEL

Let P be a parabolic subgroup of G, and let σ its corresponding cell of dimension m, given in (1.2). There exist dominant fundamental weights α1 , ..., αm ∈ a∗σ ([9] p.498) such that σ={

m X

yi αi ; yi > 0}.

1

Let FP : aσ −→ R be defined by (5.1)

FP (v) =

m X

eαi (v) .

1

Then FP ∈ C ∞ (aσ ) is strictly convex, and the image of its Legendre transform is exactly σ. Therefore, the moment map Φ satisfies Φ(Aσ ) = σ. Extend FP to G/(P, P ) by K-invariance, and it follows from Theorem II that ωP =

¯ P −1∂ ∂F

is a Kaehler structure on G/(P, P ). Let LP be the corresponding line bundle, described before. For a dominant integral weight λ, we let O(LP )λ denote the holomorphic sections of LP that transform by λ under the right Tσ -action. Let HωP be the holomorphic sections that are square-integrable under (1.4), so that it is a unitary K-representation space. By Theorem III, O(LP )λ is an irreducible K-representation with highest weight λ, whenever λ ∈ σ. Further, since Φ(Aσ ) = σ, O(LP )λ ⊂ HωP whenever λ ∈ σ. We repeat this geometric construction among all the parabolic subgroups P containing the fixed Borel subgroup B = HN. In each case, we use FP in (5.1) as the potential function for the Kaehler structure ωP on G/(P, P ). Then the direct sum ⊕B⊂P HωP is a model in the sense of I.M. Gelfand [7]: a unitary K-representation where all irreducibles occur with multiplicity one.

28

References [1] R. Abraham and J. Marsden, Foundations of Mechanics, 2nd. ed., AddisonWesley, 1985. [2] T. Brocker and T. Dieck, Representations of compact Lie groups,

Springer-

Verlag, N.Y. 1985. [3] C. Chevalley, Theory of Lie Groups, Princeton U. Press, Princeton 1946. [4] M.K. Chuah and V. Guillemin, Kaehler structures on KC /N, Contemporary Math. 154 : The Penrose transform and analytic cohomology in representation theory (1993), 181-195. [5] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515-538. [6] V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge U. Press, Cambridge 1991. [7] I.M. Gelfand and A. Zelevinski, Models of representations of classical groups and their hidden symmetries, Funct. Anal. Appl. 18 (1984), 183-198. [8] S. Helgason, Differential Geometry, Lie groups, and symmetric spaces, Academic Press, 1978. [9] S. Helgason, Groups and Geometric Analysis, Academic Press, 1984. [10] A. Knapp, Representation Theory of Semisimple Groups, Princeton U. Press, Princeton 1986. [11] B. Kostant, Quantization and unitary representations, Lecture Notes in Math. 170, Springer 1970, 87-208. [12] H.S. La, P. Nelson, A.S. Schwarz, Virasoro Model Space, Comm. Math. Phys. 134 (1990), 539-554. 29

DEPARTMENT OF APPLIED MATHEMATICS, NATIONAL CHIAO TUNG UNIVERSITY, HSINCHU, TAIWAN. E-mail address: [email protected]

30

 Date Published 1996-08-29