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THE COMPLEX TORUS

MENG-KIAT CHUAH

A

Let T be the compact real torus, and TCits complexification. Fix an integral weightα, and consider the

α-weighted T-action on TC. Ifω is a T-invariant Ka$hler form on TC, it corresponds to a pre-quantum line bundle L over TC. Let Hωbe the square-integrable holomorphic sections of L. The weighted T-action lifts

to a unitary T-representation on the Hilbert space Hω, and the multiplicity of its irreducible sub-representations is considered. It is shown that this is controlled by the image of the moment map, as well as the principle that ‘ quantization commutes with reduction ’.

1. Introduction

Let T be the compact real n-torus, and TCits complexification. Then T acts natur-ally on TC, as subgroup of TC. In [3], we study T-invariant Ka$ hler structures on TC, and the corresponding geometric quantization. The present paper follows a suggestion of V. Guillemin, and considers the more general T-actions with weights.

We write Tl Rn\Znand T

Cl Cn\Znas in [3], where

TCl oz l xjNk1[y]:x ? Rn, [ y]? Rn\Znl Tq. (1.1)

LetN be the Lie algebra of T. The notation (1.1) automatically identifies N, N*, Rn, Rn*

with one another.

Consider now a weight α l (α",…,αn) in the integral lattice Zn9 Rnl N*. We

define theα-weighted T-action on TCby

TiTC,- TC, ([tj])i(xjjNk1[yj])/- (xjjNk1[yjjtj]), (1.2)

where tj, xj, yj? R for all j l 1,…, n. In particular, if αjl 1 for all j, then (1.2) is just the standard action of subgroup T on TC. We shall always deal with Ka$ hler structures

on TCthat are invariant under this standard action, and we call them T-inŠariant. Let

Dαbe the diagonal matrix with entriesα",…,αnalong the diagonal. We shall see that a T-invariant Ka$ hler form is necessarily invariant under the weighted action (1.2), and has the expressionω l Nk1cc`F. In fact, the weighted action preserving ω is Hamiltonian, with moment mapΦ:TC,- N* given by

Φ(z) l "#Dα:Fh(x) l

1 2

0

αj

cF cxj(x)

1

for all zl xjNk1[y] ? TC.

Sinceω l Nk1cc`F, it has to be exact, and is in particular integral. We obtain a pre-quantum line bundle L over TC[5, 6]. The Chern class of L is the cohomology

class [ω] l 0, so L is a trivial bundle. It is equipped with a connection ] whose

Received 20 May 1998.

2000 Mathematics Subject Classification 53D20, 53D50, 32L10 (primary). The work was partially supported by the National Science Council of Taiwan.

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curvature isω, as well as an invariant Hermitian structure (,). We say that a smooth section s of L is holomorphic if]vsl 0 for every anti-holomorphic vector field Š. Let H(L) denote the space of all holomorphic sections. The weighted action (1.2) leads to a T-representation on H(L). Let dV be the Haar measure on TC. To obtain a unitary

representation out of H(L), let Hω be the space of all holomorphic sections s that satisfy

&

TC

(s, s) dV _. (1.3)

Then Hω is a unitary T-representation space. Its infinitesimal N-representation is written as ξ:s ? Hω, for ξ ? N, s ? Hω. The irreducible subrepresentations of Hω are 1-dimensional, and each is a subspace of

(Hω)λl os ? Hω:ξ:s l (λ, ξ) s for all ξ ? Nq,

for someλ ? Zn9 N*. A basic question in geometric quantization is to compute the

multiplicity of irreducible representations in Hω.

LetΩ be the image of the moment map. If the weight α in (1.2) contains no zero entry, then the multiplicity problem is solved by an easy generalization of [3], as follows.

T 1.1. If α has no zero entry, then the unitary representation Hω is multiplicity-free. It contains (Hω)λif and only ifλ ? Ω and λjj? Z for all j.

We shall prove Theorem 1.1 in§2. Given a unitary representation of a Lie group, we call it a model if it contains every irreducible representation once. This terminology is due to I. M. Gelfand and A. Zelevinski [4]. From Theorem 1.1, Corollary 1.1 follows.

C 1.1. Hωis a model of T if and only if the moment map is surjectiŠe and αjlp1 for all j.

The main purpose of this paper is to consider the more complicated situation where the weightα in (1.2) contains zero entries. In this case, the multiplicity of (Hω)λ is no longer determined by the image of the moment map alone. To handle this problem, we introduce symplectic reduction, a process first explored by J. Marsden and A. Weinstein [7]. In the study of Hamiltonian group actions on symplectic manifolds, two of the central aspects are geometric quantization and symplectic reduction. A unifying theme between them is given by V. Guillemin and S. Sternberg [5] and is often called ‘ quantization commutes with reduction ’. A summary of recent developments of such concepts can be found in [8]. We shall see that it helps to solve our multiplicity problem.

We now perform symplectic reduction. Suppose thatλ is in the image Ω of the moment mapΦ. Then T acts on Φ"(λ), and we call B

λl Φ−"(λ)\T the reduced space.

Let

ı:Φ−"(λ) - T

C, π:Φ"(λ) ,- Bλ

respectively denote the natural inclusion and quotient. Then Bλis equipped with a symplectic structure ωλ, such that π*ωλl ı*ω. This process is called symplectic reduction, andωλis called the reduced symplectic form. We study certain properties of the reduced space (Bλ,ωλ) in§3.

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Let k be the number of non-zero entries of the weightα, where 1  k  n. We may rearrange the indices and assume that α l (α",…,αk, 0, … , 0), where α",…,αk are non-zero. Recall that ω l Nk1cc`F. Since F can be regarded as a strictly convex function on Rn(see [3]), the subset X9 Rndefined by

Xl ox ? Rn:1

2 cF

cxj(x)lαλjjfor jl 1,…, kq

is a smooth submanifold of dimension (nkk). Let Tn−k9 T be the real subtorus,

spanned by the last (nkk) coordinates. Then XiTn−kimbeds into T

C, via

 : XiTn−k- RniT l T

C.

In§3, we prove the following theorem.

T 1.2. The reduced space (Bλ,ωλ) is symplectomorphic to the symplectic

submanifold(XiTn−k, *ω).

By Theorem 1.2, we identify the reduced space Bλwith the symplectic submanifold

XiTn−k. Since ω

λ is exact, we again have the pre-quantum line bundle over Bλ,

denoted Lλ. Here Lλ is trivial, because its Chern class is the cohomology class [ωλ]l 0. It may be regarded as the restriction of L to Bλ, due to Theorem 1.2. The space

Bλ9 TC is not complex, so there is no intrinsic polarization for an immediate

defi-nition of ‘ holomorphic ’ sections H(Lλ). We shall define H(Lλ) among the smooth sections of Lλin§4. This coincides with the usual holomorphic sections in cases where

Bλhappens to be complex.

The Haar measure of TCrestricts to a measure on Bλl XiTn−k, still denoted by

dV. We again use the Hermitian structure on Lλto define an L#-structure on H(Lλ), and let H(ωλ) denote the square-integrable sections in H(Lλ). In other words, H(ωλ) consists of all s? H(Lλ) in which

&

(s, s) dV _.

In§4, we prove that geometric quantization commutes with reduction, as stated in Theorem 1.3.

T 1.3. H(ωλ)is a Hilbert space, and(Hω)λ% H(ωλ).

Clearly, Theorem 1.1 is a special case of Theorem 1.3 : if the weightα has no zero entry, then for allλ ? Ω, the reduced space Bλis just a point. Therefore H(ωλ)l C, and Theorem 1.3 implies that (Hω)λoccurs with multiplicity 1.

If the Hilbert spaces in Theorem 1.3 are infinite-dimensional, then of course the isomorphism in question is trivial. In§5, we justify the significance of this theorem by showing that their dimensions can be any of 0, 1, 2, … ,_.

2. Geometric quantization

Letω be a T-invariant Ka$hler form on the complex torus TC. By [3],

ω l dβ l Nk1cc`F, (2.1) where β and F are T-invariant. We use the standard coordinates z l xjNk1[y]

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introduced in equation (1.1). Here F, being T-invariant, depends on the x-variables only. Since the weighted action defined in (1.2) acts along the [ y]-variables, it preserves F. Therefore, the weighted action also preservesω.

Given ξ ? N l Rn, let ξ^ be the infinitesimal vector field on T

C induced by the

weighted T-action. Hence

ξ^l 

j

αjξjcyc

j

.

In (2.1), β l "#jcF\cxjdyj. Hence the moment map Φ:TC,- N* of the weighted action is given by (Φ(z), ξ) l (β, ξ^) l

0

"# j cF cxj(x) dyj, k αkξk c cyk

1

l  j "#αj ξjcxcF j (x),

for all z? TCandξ ? N (see [1, Theorem 4.2.10]). Therefore, the moment map is

Φ(z) l1 2

0

αj

cF cxj(x)

1

.

As discussed in§1, ω corresponds to a pre-quantum line bundle L [5, 6], whose holomorphic sections are denoted by H(L). By [3], there exists a non-vanishing T-invariant holomorphic section s! satisfying

(s!,s!)le−F. (2.2)

For the rest of this section, we assume that in the weighted T-action (1.2),α has no zero entry. The weighted action lifts to a T-representation on H(L). Each irreducible subrepresentation is one-dimensional, and is of the form C(ec:z

s!) for some c? Zn

. Since s! is T-invariant, the corresponding infinitesimal N-representation is given by ξ:(ec:z s!)l(ξ:ec:z ) s! ld dt

)

!exp

0

j cj(xjjNk1[yjjαjtξj])

1

s! l  j cjαjξjec:z s! (2.3)

for allξ ? N. Therefore, if we define

H(L)λl os ? H(L): ξ:s l (λ, ξ) s for all ξ ? Nq, then (2.3) says that

H(L)λl C(ec:z

s!)λjl cjαj; jl 1,…, n. (2.4)

We consider the unitary T-representation Hωconsisting of holomorphic sections which converge under the integral (1.3). Let

(Hω)λl H(L)λEHω.

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Proof of Theorem1.1. By (2.4), we only need to consider the cases withλjj? Z for all j. For such cases, let cjl λjj? Z. Consider s l ec:z

s!?H(L)λ, where s! is the

holomorphic section in (2.2). Define G? C_(Rn) by G(x)l F(x)k2c:x. Then

&

TC (ec:z s!,ec:z s!)dVl

&

T C e#c:xe−FdVl

&

TC e−GdV. (2.5)

Since F and G have the same Hessian and F is strictly convex, so is G. According to Proposition 3.3 of [3], the integral (2.5) converges if and only if G has a global minimum. This is equivalent to 2c being contained in the image of the gradient function Fh. Recall that Dαis the diagonal matrix with entriesα",…,αn, so that the moment map isΦ l "#Dα:Fh. It follows that

ec:z s!?Hω 0 ? Image(Gh)  2c ? Image(Fh)  c ? Image("#Fh)  Dαc? Image("#Dα:Fh)  λ ? Image(Φ). By (2.4), C(ec:z

s!)lH(L)λ, which completes the proof of Theorem 1.1.

Since F? C_(Rn) is a strictly convex function [3], the image of"#Fh is a convex set

in Rn. Thus the image of the moment mapΦ l "#D

α:Fh is convex, and it includes all

λ ? Znexactly whenΦ is surjective. Therefore, by Theorem 1.1, (H

ω)λ 0 for all λ ? Zn

if and only ifΦ is surjective and αjlp1 for all j. This proves Corollary 1.1.

3. Symplectic reduction

Letω be a T-invariant Ka$hler form on TC, preserved by theα-weighted T-action

(1.2). From now on, we consider the more interesting case whereα has zero entries, which is the main purpose of this paper. The square-integrable holomorphic sections

Hωnow have a more complicated multiplicity problem. It turns out that symplectic reduction [7] can handle this problem. In this section, we describe the process of symplectic reduction, and prove Theorem 1.2.

The torus T has dimension n. Let k be the number of non-zero entries of the weight α, where 1  k  n. We may arrange the indices so that the first k entries

α",…,αkare non-zero. We identify Rkwith the subspace of Rn spanned by the first

k variables. Intuitively, we can think of it as being ‘ horizontal ’. In this way, the horizontal k-dimensional affine subspaces Hv9 Rnare defined by

Hvl RkjŠ l o(x",…,x

k, 0, … , 0) : xj? RqjŠ, Š ? Rn. (3.1)

Similarly, we may regard Rn−k9 Rn as the subspace spanned by the last (n

kk)-coordinates, and define the ‘ vertical ’ affine (nkk)-subspaces Vc9 Rn by

Vcl cjRn−kl cjo(0,…, 0, x

k+", … , xn) : xj? Rq, c ? Rn. (3.2)

Recall thatω has potential function F. Let Ω be the image of the moment map, and let Dαbe the diagonal matrix with entriesα",…,αn. Fixλ ? Ω, and consider

Xl ("#Dα:Fh)" (λ) l ox ? Rn:1

2 cF

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The space X will play an important role in our study of symplectic reduction. If we let X` denote the closure of X in Rn, then the boundary of X is defined by

cX l X`BX. The following proposition gives some properties of X.

P 3.1. The space X l ("#Dα:Fh)"(λ) is a closed, unbounded

(nkk)-dimensional submanifold of Rn, and cX l 6. For each Š ? Rn, the horizontal affine

k-space Hvintersects X at most once.

Proof. Since F? C_(Rn) is strictly convex [3], "#Fh maps Rn diffeomorphically

onto a domain U9 Rn. Then D

αmaps U onto the imageΩ of the moment map. Since

Dαis a diagonal matrix whose last (nkk) entries vanish, λ ? Ω may be written as λ l (λ",…,λk, 0, … , 0).

Let cl (λ"\α",…,λkk, 0, … , 0), and let Vc9 Rn be the vertical affine (n

kk)-space defined in (3.2). Then D−"

α (λ)EU l VcEU is (nkk)-dimensional. However, "#Fh

is a diffeomorphism between X and VcEU, so X is an (nkk)-dimensional manifold.

Since VcEU is closed in U, we conclude from the diffeomorphism "#Fh that X is

closed in Rn. Since VcEU is not compact, neither is X. Hence X, being non-compact

and closed in Rn, is unbounded. Also, X equals its closure X` simply because X is

closed, so the boundarycX is empty.

ForŠ ? Rn, let Hvbe the horizontal affine k-space defined in (3.1). It remains to

show that X intersects each Hvat most once. Suppose that, for someŠ ? Rn, there exist

distinct p, q? XEHv. Let S9 Hv9 Rn be the straight line joining p and q. Let

f? C_(S ) be the restriction of F to S. Since p, q? X, equation (3.3) says that cF

cxj( p)lcxcFj(q)l 2λαjj

for all jl 1,…, k. This means that f h(t) has the same value at p and q, where t is a linear variable on S. This is a contradiction, because f should be strictly convex on

S. Hence, for all Š ? Rn, XEHv contains at most one point. This proves the

proposition.

Since T is abelian, the moment map Φ l "#Dα:Fh is T-invariant. By Proposition 3.1, Φ−"(λ) is a real (2nkk)-submanifold of T

Cgiven by

Φ−"(λ) l XiT l oxjNk1[y]:x ? Xq. (3.4)

Let ı be the natural inclusion ofΦ−"(λ) into T

C. The torus T acts onΦ−"(λ), and

we let Bλl Φ−"(λ)\T be the quotient space. Let π be the quotient map from Φ"(λ)

onto Bλ. There exists a symplectic form ωλ on Bλ, satisfying π*ωλl ı*ω. The construction of the symplectic manifold (Bλ,ωλ) is called symplectic reduction.

Proof of Theorem 1.2. When T acts on Φ−"(λ), the weight α has k non-zero

entries. We have arranged the indices so that T acts only along the first k-variables of [ y]. Therefore, equation (3.4) says that Bλis diffeomorphic to the product manifold

XiTn−k, where Tn−kdenotes the subtorus of T spanned by the last (nkk) variables.

To prove the theorem, it remains to check the assertion on symplectic forms. Consider the following diffeomorphismσ followed by two inclusions  and ı,

Bλ,-σ XiTn−k- Φ−"(λ) -ı T

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Letπ be the quotient map from Φ"(λ) to B

λ. By the definition ofωλ, ı*ω l π*ωλ.

Therefore, since π::σ is the identity function on Bλ,

σ*:*:ı*ω l σ*:*:π*ωλl ωλ.

This shows that the diffeomorphismσ identifies ωλwith the pullback ofω to XiTn−k.

Hence Theorem 1.2 holds.

The realization of Bλas XiTn−k has the defect that the real submanifold Xi

Tn−k9 T

Cis generally not complex. This is because the tangent bundle of XiTn−k

may not be preserved by the almost complex structure of TC.

We remark that there is a complex realization of Bλ, in terms of Reinhardt domain, with ωλ being identified with the ‘ linear ’ Ka$ hler structure ωLl Nk1\2 jdzjFdz`j. However, this Ka$ hler realization will not be used below, and we merely describe it in brief here. Recall that U is the image of the gradient function"#Fh. Consider the diffeomorphism

τ:TC,- UiT, τ(xjNk1[y]) l "#Fh(x)jNk1[y].

By the definition (3.3) of X,

τ(XiTn−k)l (VcEU)iTn−k9 VciTn−k, (3.5)

where Vcis the vertical affine space (3.2) corresponding to cl (λ"\α",…,λ

k\αk, 0, … ,

0). Henceτ is a diffeomorphism from XiTn−kto Rl (VcEU)iTn−k. In particular,

Ris a Reinhardt domain in a complex torus VciTn−k. Consider the standard Ka$ hler

structure on TC,ωLl (Nk1\2) jdzjFdz`j. It satisfies τ*ωLl τ*Nk1 2 j dzjFdz`j l τ*  j dxjFdyj l  j d(τ*xj)Fdyj l1 2i,j c#F cxicxjdxiFdyj l ω.

Hence, by (3.5),τ identifies ωλwith the pullback ofωLto R. In other words,τ is a symplectomorphism between the pullback ofω to XiTn−kand the pullback ofω

Lto

R. Unfortunately, the Ka$ hler realization ωL does not reflect the geometry of the original Ka$ hler form ω, and its quantization is not so interesting. Hence we will always stick to the other realization, (XiTn−k,ω

λ). From now on, we think of Bλas

the symplectic submanifold XiTn−k9 T

C, and regard the reduced symplectic form

ωλas the pullback ofω to Bλ.

4. Quantization commutes with reduction

Recall that T acts on TC by mapping (1.2), preserving ω l Nk1cc`F. We have

assumed the weight of this action to beα l (α",…,αk, 0 … , 0), whereα",…,αk are non-zero. In this way, every integral point in the image of the moment map is of the form λ l (λ",…,λk, 0, … , 0)? Ω. In the previous section, we performed symplectic reduction toλ, and obtained the reduced symplectic manifold (Bλ,ωλ).

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In this section, we apply geometric quantization to Bλ, and prove Theorem 1.3. Using Theorem 1.2, we shall always identify Bλwith the real submanifold XiTn−k9

TC. In this way, the reduced symplectic formωλis just the pullback ofω to Bλ. Since

ω is exact, so is ωλ. We let Lλbe the pre-quantum line bundle over Bλ, with Chern class

[ωλ]l 0. Hence Lλis a trivial bundle, and in fact is the restriction of L to Bλ. As remarked at the end of the previous section, the submanifold Bλ9 TC is not

complex. Therefore, there is no intrinsic polarization on the space of smooth sections

C_(Lλ). To overcome this problem, consider

 l oxjNk1[y] ? TC: Hxintersects Xq, (4.1)

where Hxis the horizontal k-space introduced in (3.1). Clearly, 9 T

C is open. In

fact, since T acts along the first k variables of [ y],  is the smallest complex submanifold of TC which contains Bλ and is preserved by the weighted T-action.

Therefore, we can define H(, L)λ to be the holomorphic sections over  which transform by the weight λ under the T-action. Consequently, among the smooth sections C_(Lλ) over Bλ, we can define

H(Lλ)l os ? C_(Lλ) : s extendable to H(, L)λq. (4.2) In other words, H(Lλ) consists of all smooth sections of Lλ obtained from the restriction of H(, L)λto Bλ.

We restrict the Haar measure dV to Bλ, and use the Hermitian structure of Lλto define an L#-structure on H(Lλ). Let H(ωλ) be the corresponding square-integrable sections :

H(ωλ)l

(

s? H(Lλ) :

&

(s, s) dV _

*

. (4.3)

In this way, H(ωλ) is a complex inner product space. P 4.1. H(ωλ)is a Hilbert space.

Proof. The only thing to check is completeness. We do this by constructing an inner product space isomorphism between H(ω

λ)and a Hilbert space.

From Proposition 3.1, X9 Rn is an (nkk)-submanifold which intersects every

horizontal affine k-space Hv(3.1) at most once. Define

Wl oŠ ? Rn−k9 Rn: Hvintersects Xq. (4.4)

Then we get a diffeomorphism

ψ:X ,- W, x /- (0,…, 0, xk+", … , xn). (4.5) Let dV be the restriction of the Lebesgue measure of Rn−k to its open set Wl

ψ(X), and dVλ the restriction of the Lebesgue measure of Rn to X. Also, let cjl

λjj? Z for j l 1,…, k, and let c:x denote k

"cjxj. Recall that F is the potential

function ofω. Since ψ is a diffeomorphism, it has a Jacobian Jψ? C_(W ) between the volume forms e#c:x−F(x)dV

λon X and dV on W. In other words,

e#c:x−F(x)dV

λl ψ*(JψdV). (4.6)

Let WC9 Tn−k

C be the Reinhardt domain

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The function Jψ extends naturally to WC by Tn−k-invariance. Let B(WC, Jψ) be the

Bergman space of Jψ-weighted L#-holomorphic functions on the Reinhardt domain

WC. In other words, it is the Hilbert space defined by

B(WC, Jψ)l

(

h? C_(WC) : h holomorphic,

&

WC

hh`JψdV _

*

. We now check that the inner product space H(ω

λ) is isomorphic to the weighted

Bergman space B(WC, Jψ), which implies that H(ω

λ) is a Hilbert space.

From the definition (4.2) of H(Lλ), we have the natural restriction map

κ:H(, L)λ,- H(Lλ).

Let s! be the T-invariant holomorphic section of (2.2), restricted to . Pick s?H(ωλ).

By the definition of H(Lλ), s is of the form sl κ(hec:z

s!), where h is a holomorphic

function on  and depends only on the variables zk+", … , zn because the section

hec:z

s!?H(,L)λneeds to transform byλ. Define

L: H(ωλ),- B(WC, Jψ), κ(hec:zs!)/-hQWC. (4.8)

We claim that L is an inner product space isomorphism.

Since h is independent of the first k variables, the function hh` satisfies

(hh`) (q)l (hh`) (ψ(q)) (4.9)

for all q? X. We let R:R" and R:R# denote the norms of H(ωλ)and B(WC, Jψ) respectively.

For sl κ(hec:z s!)?H(ωλ), RsR#"l

&

Xhh`e#c:x−F(x)dV λ by (2.2) l

&

X hh`ψ*(JψdV) by (4.6) l

&

X ψ*(hh`JψdV) by (4.9) l

&

W hh`JψdV by (4.5) l

&

WC hh`JψdV l RL(s)R##. by (4.8) (4.10) Let Tk

C be the complex subtorus spanned by the first k variables. It follows from

the definitions (4.1), (4.4) and (4.7) that l Tk

CiWC. Hence the operation h/- hQWC

in (4.8) is bijective, because a holomorphic function h on independent of z",…,zk is equivalent to a holomorphic function on WC. Therefore, L is a bijection. Then (4.10)

says that L is an isomorphism of inner product spaces from H(ω

λ) to B(WC, Jψ).

Therefore H(ω

λ) is a Hilbert space.

Recall that (Hω)λl H(L)λEHω. Let cjl λjjfor jl 1,…, k. From an argument similar to the one leading to (2.4), we know that, if any cjis not an integer, then (Hω)λ

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vanishes. Assuming cj? Z from now on, our goal is to construct a natural Hilbert space isomorphism (Hω)λ% H(ω

λ), and prove Theorem 1.3. In order to compare these

two Hilbert spaces, the next two propositions provide integrability conditions. Let s! be the holomorphic section of equation (2.2). Given bl(bk+", … , bn)? Zn−k,

we define a one-dimensional subspace

Sbl

(

aexp

0

k "

cjzj

1

exp

0

n

k+"

bjzj

1

s!:a?C

*

9 H(L)λ. (4.11) Proposition 4.2 follows, from [3].

P 4.2 [3]. Let 0  s ? Sb. Then T

C(s, s) dV conŠerges if and only if

(c",…,ck, bk+", … , bn) is in the image of "#Fh.

Recall thatΩ is the image of the moment map. Following Proposition 4.2, it is clear that

(Hω)λ 0  λ ? Ω,λj

αj? Z for jl 1,…, k. (4.12)

Assume thatλ ? Ω, so that we have the reduced space Bλ. Let s? Sb. In Proposition 4.2, we have given a necessary and sufficient condition for (s, s) dV to be integrable over TC. We now restrict it to Bλ9 TC, but for simplicity, we still denote it by (s, s) dV.

The next proposition considers its integrability over Bλ. P 4.3. Let 0  s ? Sb. Then B

λ(s, s) dV conŠerges if and only if

(c",…,ck, bk+", … , bn) is in the image of "#Fh.

Proof. Let s? Sb. Suppose that (c",…,ck, bk+

", … , bn) is in the image of "#Fh.

By Proposition 4.2, T

C(s, s) dV converges. Therefore, when restricted to Bλ9 TC, Bλ(s, s) dV also converges.

Therefore, it only remains to prove the converse. Suppose that B

λ(s, s) dV

converges for all s? Sb. Define G? C_(Rn) by

G(x)l F(x)k2 k "

cixik2 n

k+"

bjxj. (4.13)

Since F and G have the same Hessian, G is strictly convex. Since (s!,s!)le−F,

equations (4.11) and (4.13) imply that up to a positive constant, e−Gl (s, s). Hence

&

X

e−GdV λl

&

(s, s) dV _, (4.14)

where dVλis the restriction of the Lebesgue measure to X9 Rn. By Proposition 3.1,

Xis unbounded and has no boundary. Thus (4.14) implies that e−Gapproaches 0 along

every direction of X, in the sense that, for anyε  0, there exists a compact subset of

Xsuch that e−G(x) ε for x outside this compact set. This means that e−Gacquires a

maximum point in X. Equivalently, G has a minimum point p in X :

G( p) G(x), x ? X. (4.15)

Recall the notions of horizontal and vertical affine spaces, defined in equations (3.1) and (3.2). Let GQVpdenote the restricted function on the vertical space Vp. We

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want to show that p is the global minimum of GQVp. However, since GQVpis strictly

convex, it suffices to show that p is a local minimum of GQVp. By Proposition 3.1, X

intersects Hpexactly once, at p. Hence, for eachŠ ? Vpthat is sufficiently near p,

XEHvl oq

vq (4.16)

for some qv. By the definition of X (3.3) and the definition of G in equation (4.13), cG

cxj(qv)l 0, j l 1,…, k. (4.17)

Since the restriction of G to Hvis strictly convex, equation (4.17) says that q vis the

global minimum of the restriction of G to Hv. In particular, sinceŠ ? Hv, it gives

G(qv) G(Š). (4.18)

Since qv? X, equations (4.15) and (4.18) imply that G(p)  G(Š) whenever Š ? Vp is

sufficiently near p. This proves that p is a local minimum of GQVp. However, GQVp

is strictly convex, so p is a global minimum of it. Therefore, cG

cxj( p)l 0, j l kj1,…, n. (4.19) SetŠ l p in (4.16), so that qvl p. Then (4.17) becomes

cG

cxj( p)l 0, j l 1,…, k. (4.20)

Using equations (4.19) and (4.20), we conclude that p is the global minimum of G, and so 0 is in the image of Gh. Then (4.13) implies that (c",…,ck, bk+

", … , bn) is in the

image of "#Fh. Hence the proposition holds.

If s? H(L)λ, we let ρ(s) be its restriction to Bλ. By the definition (4.2) of H(Lλ),

ρ(s) ? H(Lλ). Therefore, we have the restriction map

ρ:H(L)λ,- H(Lλ).

We applyρ to the one-dimensional spaces Sbof (4.11). Recall that cjl λjj? Z, for

jl 1,…, k. Let Il o(bk+ ", … , bn)? Zn−k: (c",…,ck, bk+", … , bn)? Image "#Fhq. (4.21) By Propositions 4.2 and 4.3, Sb9 (Hω)λ ρ(Sb)9 H(ω λ) b ? I. (4.22)

It follows from definition (4.11) that, if 0 s ? Sb, then s is non-vanishing, so in particular its restrictionρ(s) is non-zero. Thus ρ is injective on each Sb. Therefore, since each Sbis one-dimensional, we obtain a constant mb 0 for each b ? I by

RsR l mbRρ(s)R, s ? Sb. (4.23) HereR:R denotes the norms of both (Hω)λand H(ω

λ). For b? I, define ρg on Sbby

ρg(s) l mbρ(s), s ? Sb. (4.24)

Proof of Theorem1.3. Ifλ is not in the image Ω of the moment map, then there is no reduced space Bλor H(ω

λ). Also, (Hω)λl 0 by (4.12), and there is nothing to

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Proposition 4.1 says that H(ω

λ)is a Hilbert space. To complete the proof, we show

thatρg gives the desired Hilbert space isomorphism. It follows from (4.22), (4.23) and (4.24) that

Rρg(s)R l RsR _ for s ? Sb, b? I. (4.25) Consider the standard subgroup action of Tn−k on T

C, which lifts to a Tn−k

-representation on H(L). This restricts to a Tn−k-representation on H(L)

λ, because the

standard Tn−k-action commutes with the weighted T-action defined in (1.2). Since

Tn−kpreserves the L#-structure (1.3) on H(L)

λ, we get a unitary representation

π":Tn−k,- Aut(H ω)λ.

Since Tn−kalso acts on B

λby acting on its toral component, we similarly get a

Tn−k-representation on H(L

λ). It preserves the L#-structure (4.3) on H(Lλ), so we get

a unitary representation

π#:Tn−k,- Aut H (ωλ).

By the definition (4.11) and property (4.22) of Sb, the irreducible sub-representations of π",π# are given by oSbqb?I and oρ(Sb)qb?I respectively. Apply the Peter–Weyl theorem [2, Chapter III] to these subrepresentations. It says thatoSbqb?I and oρ(Sb)qb?I are collections of mutually orthogonal subspaces in (Hω)λ and H(ωλ) respectively, and their linear spans are dense in these Hilbert spaces. Let

S9 (Hω)λ, R9 H(ωλ)

be the dense subsets given by their linear spans. Since bothoSbqb?Iandoρ(Sb)qb?Iare collections of mutually orthogonal subspaces, definition (4.25) says that ρg is an isometry from S to R.

If I is finite, then the Hilbert spaces are finite-dimensional, and Sl (Hω)λ, Rl

H(ωλ). Thusρg is the required isomorphism. Suppose that I is infinite. Since ρg is an isometry between the dense subsets S and R, it extends continuously to a Hilbert space isomorphismρg:(Hω)λ,- H(ωλ). This proves Theorem 1.3.

5. Open cones

In this section, we give some simple examples to show that the Hilbert spaces of Theorem 1.3 can have any dimension. It suffices to consider a torus of dimension 2. From the previous section, we see that the dimension of (Hω)λis the cardinality of the index set I of (4.21). Considerω l Nk1cc`F, invariant under the T-action (1.2) with weightα l (1, 0). Let λ l (0, 0) ? R#. Then the set I of (4.21) becomes

Il ob ? Z:(0, b) ? Image "#Fhq. (5.1) We now show thatQIQ l dim(Hω)λcan be any of 0, 1, 2, … ,_.

LetŠ",Š#?R# be a basis. Define G?C_(R#) by

G(x)l exp(Š":x)jexp(Š#:x),

whereŠi:x is the usual dot product. Then G is strictly convex. The image of "#Gh is the open cone consisting of all positive linear combinations ofŠ",Š#. By adjusting Š" and Š#, we get all the open cones of R# that emit from the origin.In fact, if G is any strictly convex function and w? R#, then F(x) l G(x)jw:x is also strictly convex. Further, the images of"#Fh and "#Gh differ by an affine translation of "#w. From this observation, we consider the strictly convex function

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By choosing different parametersŠ",Š#,w?R# for F, the image of "#Fh can be any given open cone C9 R#. For every s l 0, 1, 2,…, _, we can always find an open cone C whose intersection with the y-axis contains s integral points, so that I in (5.1) has

selements.

Acknowledgements. The author would like to thank Victor Guillemin, for providing many helpful ideas in geometric quantization.

References

1. R. A and J. M, Foundations of mechanics, 2nd edn (Addison–Wesley, 1985).

2. T. B$ and T.  D, Representations of compact Lie groups (Springer, 1985).

3. M. K. C, ‘Ka$hler structures on complex torus’, J. Geom. Anal. 10 (2000) 253–263.

4. I. G and A. Z, ‘Models of representations of classical groups and their hidden symmetries ’, Funct. Anal. Appl. 18 (1984) 183–198.

5. V. G and S. S, ‘Geometric quantization and multiplicities of group represen-tations ’, InŠent. Math. 67 (1982) 515–538.

6. B. K, Quantization and unitary representations, Lecture Notes in Mathematics 170 (Springer, 1970) 87–208.

7. J. M and A. W, ‘Reduction of symplectic manifolds with symmetry’, Rep. Math. Phys. 5 (1974) 121–130.

8. R. S, ‘Symplectic reduction and Riemann–Roch formulas for multiplicities’, Bull. Amer. Math. Soc. 33 (1996) 327–338.

Department of Applied Mathematics National Chiao Tung UniŠersity 1001 Ta Hsueh Road

Hsinchu Taiwan

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