QUANTIZATION OF KAHLER MANIFOLDS II

Michel Cahen, Simone Gutt* and John Rawnsley

Abstract. We use Berezin™s dequantization procedure to de¬ne a formal —-product

on a dense subalgebra of the algebra of smooth functions on a compact homogeneous

K¨hler manifold M . We prove that this formal —-product is convergent when M is a

a

hermitian symmetric space.

0. Introduction: In part I of this paper [7] we showed how to quantize cer-

tain compact K¨hler manifolds (M, ω, J). This means the following: Let (L, , h)

a

be a quantization bundle over M (i.e., a holomorphic line bundle L with connec-

admitting an invariant hermitian structure h, such that the curvature is

tion

curv( ) = ’2iπω). Let H be the Hilbert space of holomorphic sections of L. To

ˆ

any linear operator A on H is associated a symbol A which is a real analytic func-

ˆ

tion on M . Denote by E(L) the space of these symbols. For any positive integer k,

(Lk = —k L, (k) , h(k) ) is a quantization bundle for (M, kω, J). If Hk is the Hilbert

ˆ

space of holomorphic sections of Lk , we denote by E(Lk ) the space of symbols of

linear operators on Hk . If, for every k, a certain characteristic function (k) (which

ˆ

depends on L and k and which is real analytic on M ) is constant, the space E(Ll )

ˆ ˆ

is contained in the space E(Lk ) for any k ≥ l. Furthermore ∪∞ E(Ll ) (denoted

l=1

by CL ) is a dense subspace of the space of continuous functions on M . Any func-

ˆ

tion f in CL belongs to a particular E(Ll ) and is thus the symbol of an operator

(k)

Af acting on Hk for k ≥ l. One has thus constructed, for a given f , a family of

quantum operators parametrized by an integer k.

From the point of view of deformation theory [1], where quantization is realised

at the level of the algebra of functions, one can say that one has constructed a

ˆ

family of associative products on E(Ll ), with values in CL , parametrized by an

integer k; indeed

(k)

(k) ˆ

f, g ∈ E(Ll ); k ≥ l.

f —k g = Af Ag

The aim of Part II is twofold. Firstly, we prove that, for any compact generalized

¬‚ag manifold, there exists on CL a formal di¬erential —-product with parameter ν

which coincides with the asymptotic expansion of the previously de¬ned —k associa-

tive products when ν = i/(4πk). Secondly, we prove that, when M is a hermitian

1991 Mathematics Subject Classi¬cation. 58F06.

Key words and phrases. Quantization, K¨hler manifolds.

a

*Research Associate of the National Fund for Scienti¬c Research (Belgium)

Typeset by AMS-TEX

1

2 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY

symmetric space, the —k product of two functions in CL is a rational function of k

having no pole at in¬nity. Hence the formal —-product is in fact convergent.

The paper is organised as follows: In §1 we study the geometrical properties of

two 2-point functions on M , one of which is related to Calabi™s diastatic function

[3]. In §2 we prove the existence of an asymptotic expansion in k ’1 of the —k

ˆ

product of two elements of E(Ll ). In §3 we turn to ¬‚ag manifolds and prove that

the asymptotic expansion constructed in §2 de¬nes a formal —-product. The proof

depends crucially both on the homogeneity and on the density property proved in

Part I [7]. Finally §4 is devoted to the proof of the rational dependence in ν of the

associative product of two functions in CL when M is hermitian symmetric.

The relationship between quantization by deformation and a calculus of symbols

has been studied by many authors and in particular by F. Berezin [2] and by

C. Moreno [6]. We hope that results presented here help to elucidate some of the

underlying geometry and the di¬culties which have to be overcome to understand

completely the compact K¨hler case.

a

Acknowledgement: We are pleased to thank our friend Joe Wolf who gave

us the bene¬t of his expertise in the geometry of hermitian symmetric spaces. We

also thank the British Council and the Communaut´ fran¸aise de Belgique for their

e c

support during the preparation of this work.

1. The relationship between Calabi™s diastatic functions and line bun-

dles: The formula for the product of symbols (to be described in §2) introduces

a 2-point function ψ de¬ned in terms of coherent states. Some of the properties

of this function are described here. Another, somewhat related 2-point function

ψ, de¬ned locally in terms of non-zero holomorphic sections, turns out to be ex-

pressible in terms of Calabi™s diastatic function D. In this section we study these

three functions D, ψ, ψ and in particular we give a formula for the Hessian of such

a function at a critical point.

As in [7], (M, ω, J) will be a compact K¨hler manifold and (L, , h) a quantiza-

a

tion bundle over M . Consider a real-analytic, closed, real 2-form „¦ on M of type

(1, 1). On a contractible open set U ‚ M there exists a real 1-form β such that

„¦ = dβ. One may write β = ± + ± for some (1, 0) form ± on U . Since „¦ is type

(1, 1)

„¦ = (‚ + ‚)(± + ±) = ‚± + (‚± + ‚±) + ‚±

implies ‚± = 0. Thus, by the Dolbeault lemma, there exists a function g de¬ned

on a possibly smaller open set V ‚ U , such that

„¦ = ‚‚g + ‚‚g = ‚‚(g ’ g) = (i/4π)‚‚f (1.1)

where f is the real valued function f = 4πi(g ’ g). The function f is not unique: if

‚‚f1 = 0, ‚f1 is a holomorphic (1, 0)-form, thus there exists locally a holomorphic

function h such that ‚f1 = ‚h = dh. The reality of f1 implies that ‚f1 = dh and

thus df1 = d(h + h); i.e., f1 is the real part of a holomorphic function.

A function f satisfying (1.1) is called a potential for „¦. Since „¦ is real-analytic,

f is also real-analytic and may thus be complex-analytically continued to an open

neighbourhood W of the diagonal in V — V . Denote this extension by f (x, y). It is

holomorphic in x, antiholomorphic in y and, with this notation, f (x) = f (x, x).

Consider the function D(„¦) : W ’ C,

D(„¦) (x, y) = f (x, x) + f (y, y) ’ f (x, y) ’ f (y, x). (1.2)

¨

QUANTIZATION OF KAHLER MANIFOLDS II 3

Since f is real-valued on V , one has

f (x, y) = f (y, x)

and thus D(„¦) is real-valued. One checks that D(„¦) does not depend on the choice

of the local potential f and is thus a globally de¬ned function on a neighbourhood

of the diagonal in M — M , depending only on „¦. It is called the Calabi function of

„¦.

Observe that, for any y ∈ M , the set Uy = {x ∈ M | D(„¦) (x, y)is de¬ned} is a

neighbourhood of y. Denote by D(„¦)y the function on Uy de¬ned by D(„¦)y (x) =

D(„¦) (x, y), then

‚‚D(„¦)y (x) = (‚‚f )(x, x)

and thus D(„¦)y is a potential for „¦ on Uy .

De¬nition 1. Calabi™s diastatic function D is the Calabi function of the K¨hler

a

form ω.

n

Example 1. Let M = Cn and let ω = (i/2) dzj §dz j be the canonical K¨hler

a

j=1

form, then a potential f is given by

n

|zj |2

f = 2π

j=1

and the Calabi diastatic function D is

n

|zj |2 + |zj |2 ’ zj z j ’ zj z j

D(z, z ) = 2π

j=1

(1.3)

n

|zj ’ zj |2 .

= 2π

j=1

It is thus, up to a factor, the square of the distance between the points z and z .

Example 2. Let M = CP n and let π : Cn+1 \{0} ’ CP n be the canonical

projection. Denote by , the standard metric on Cn+1 . The Fubini-Study metric

g on CP n is such that

1 2

(π — g)z (X, Y ) = ’ X, Az Y, Az ’ X, Bz Y, Bz

X, Y z

4

πz

where z ∈ Cn+1 and

(z± ‚z± + z ± ‚z± ),

Az =

±¤n

Bz = (’iz± ‚z± + iz ± ‚z± ).

±¤n

4 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY

Considering the chart U = {p|p = π(z)withz n+1 = 0} and the coordinate map

φ : U ’ Cn , φ(p) = u where p = π(z) and uj = z j /z n+1 , j ¤ n, one sees that the

K¨hler form ω on U has the form

a

i

|u» |2 ) du± § du±

(1 +

ω=

|2 )2

2π(1 + » |u» ±

»

’ u± du± § uβ duβ .

±,β

Hence a K¨hler potential f is given by

a

|u» |2 ).

f = 2 log(1 +

»

In these coordinates the diastatic function has the expression

|u» |2 ) + log(1 + |u» |2 )

D(u, u ) = 2 log(1+

» »

u» u» ) .

u» u» ) ’ log(1 +

’ log(1 +

»

»

This can be rewritten in terms of the coordinates in Cn+1 as

z2 z 2

D(π(z), π(z )) = 2 log (1.4)

.

| z, z |2

In particular D > 0 unless π(z) = π(z ), where D = 0.

Observe that Cn+1 \{0} may be identi¬ed with a principal C— -bundle over CP n .

Observe also that the argument of the logarithm in formula (1.4) may be rewritten

for any hermitian line bundle by using local sections. These observations will lead

us to the de¬nition of our second 2-point function.

Let π : L ’ M be a holomorphic line bundle with real analytic hermitian

structure h. Let s : U ’ L be a zero-free holomorphic section of L over the open

set U ‚ M . Then |s|2 (x) = hx (s(x), s(x)) is a real-analytic function on U which

can be analytically continued to a neighbourhood of the diagonal in U — U , to give

a function |s|2 (x, y) holomorphic in x and antiholomorphic in y. This function has

non-zero values for y su¬ciently close to x. Consider then the expression (analogous

to the one in (1.4))

|s|2 (x, x)|s|2 (y, y)

ψ(x, y) = (1.5)

| |s|2 (x, y) |2

wherever this is de¬ned, which will be the case in a neighbourhood of the diagonal.

Remark that if t : U ’ L is another holomorphic section on U without 0, there

exists a holomorphic function f : U ’ C such that t = f.s. Then

|t|2 (x, x) |t|2 (y, y) |f (x)|2 |s|2 (x, x) |f (y)|2 |s|2 (y, y)

=

| |t|2 (x, y) |2 |f (x)f (y)|2 | |s|2 (x, y) |2

= ψ(x, y).

This justi¬es the following de¬nition.

¨

QUANTIZATION OF KAHLER MANIFOLDS II 5

De¬nition 2. If L ’ M is a holomorphic line bundle with real-analytic hermitian

structure h, the 2-point function ψ de¬ned locally by formula (1.5), in a neigh-

bourhoood of the diagonal in M — M , will be called the characteristic function of

the bundle L, and denoted ψL .

Before considering some of the properties of ψ, let us exhibit the relationship

between ψ and Calabi™s function.

Proposition 1. Let π : L ’ M be a holomorphic line bundle with real analytic

hermitian structure h. Let be the unique connection, [5], on L leaving h invariant

and such that X s = 0 for any X of type (0, 1) and for any local holomorphic section

s. Let ω = (i/2π)curv( ). Then

1

ψ = e’ 2 D (1.6)

where D is the Calabi function of ω.

Remark. If (L, , h) is a quantization bundle for M , the K¨hler form satis¬es

a

1

the assumption of Proposition 1 and thus ψ = e’ 2 D where D is Calabi™s diastatic

function.

Proof. Let s : U ’ L be a holomorphic section of L. Then

Xs = ±s (X)s

where ±s is a 1-form of type (1, 0). Also

d|s|2 = (±s + ±s ) |s|2

and thus

±s = ‚ log |s|2 .

The curvature 2-form σ is de¬ned by

σ(X, Y )s = ( ’ ’ [X,Y ] )s

X Y

Y X

and for s holomorphic we get

σ(X, Y )s = (d±s )(X, Y )s.

That is

σ = ‚‚ log |s|2 = ’‚‚ log |s|2 .

Thus for ω = (i/2π)σ, ’2 log |s|2 is a potential and we can compute the Calabi

function D of ω as

|s|2 (x, y)|s|2 (y, x)

D(x, y) = 2 log

|s|2 (x, x)|s|2 (y, y)

= ’2 log ψ(x, y),

hence the conclusion.

The 2-point function has the following two properties:

6 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY

a) Let π : L ’ M , π : L ’ M be two holomorphic hermitian line bundles and

let ψL and ψL be the corresponding characteristic functions. If s : U ’ L and

s : U ’ L are local sections then s — s is a local section of L — L . Hence

ψL—L = ψL .ψL . (1.7)

In particular if one considers the k th power Lk of the bundle L, one has

ψLk = (ψL )k . (1.8)

b) Let f : M ’ N be a holomorphic map and let π : L ’ N be a holomorphic

hermitian line bundle. Then f — L ’ M has natural holomorphic and hermitian

structures such that if s : U ’ L is a local holomorphic section of L over the open

set U ‚ N , then s—¦f , the corresponding section of f — L over f ’1 U , is holomorphic

and |s—¦f |2 = |s|2 —¦f . Formula (1.5) shows that

ψf — L = ψL —¦f . (1.9)

These properties justify the name characteristic function.

The 2-point function ψL may be expressed in terms of coherent states and of the

function which has been introduced in Part I and denoted by θ there. Denote by

HL the Hilbert space of holomorphic sections of L with the scalar product

ω n (x)

s, s = hx (s(x), s (x)) (1.10)

n!

M

where h is the hermitian structure of L, ω the K¨hler form of M and dimM = 2n.

a

Evaluation of a holomorphic section s at a point x of M is a continuous linear

map HL ’ Lx = π ’1 (x). Choose a base-point q ∈ Lx . Then there exists a unique

element eq ∈ HL , called a coherent state, such that

x = π(q).

s(x) = s, eq q,

Since ecq = c’1 eq for c a non zero complex number, the function

2

|q|2 , x = π(q)

(x) = eq

is well-de¬ned.

If s—¦ : U ’ L is a zero-free holomorphic section over U , and s an arbitrary

holomorphic section of L then

s(x) = s, es—¦ (x) s—¦ (x), x ∈ U.

Thus s, es—¦ (x) is a holomorphic function of x and hence es—¦ (x) depends antiholo-

morphically on x. In particular

(x) = es—¦ (x) , es—¦ (x) |s—¦ (x)|2

¨

QUANTIZATION OF KAHLER MANIFOLDS II 7

is real analytic and admits an analytic extension to a neighbourhood of the diagonal

in U — U , holomorphic in the ¬rst variable and antiholomorphic in the second

(x, y) = es—¦ (y) , es—¦ (x) |s—¦ |2 (x, y).

In particular

| es—¦ (y) , es—¦ (x) |2

(x, x) (y, y)

ψ(x, y) =

es—¦ (x) 2 es—¦ (y) | (x, y)|2

2

(1.11)

(x) (y) | eq , eq |2

= q ∈ Lx \ {0}, q ∈ Ly \ {0}.

,

| (x, y)|2 eq 2 eq 2

We have seen in [7], the importance of the condition being constant. This moti-

vates the following de¬nition.

De¬nition 3. A hermitian holomorphic line bundle π : L ’ M will be said to be

regular if the function is constant.

De¬nition 4. Let {eq |q ∈ L—¦ } be the set of coherent states of the hermitian line

bundle π : L ’ M . The 2-point function ψ is de¬ned by

| eq , eq |2

π(q) = x, π(q ) = y.

ψ(x, y) = , (1.12)

eq 2 eq 2

ψ is globally de¬ned on M — M and takes values in [0, 1]. In particular all

points of the diagonal are critical points where ψ takes the value 1. Rewriting the

de¬nition in terms of a local holomorphic section so we have

| eso (x) , eso (y) |2

ψ(x, y) =

eso (x) 2 eso (y) 2

which shows that ψ is real-analytic.

Reformulating the above, we have the next proposition.

Proposition 2. If L is a regular, hermitian, holomorphic line bundle, the char-

acteristic function ψ equals the 2 point function ψ and is thus globally de¬ned on

M — M.

A precise result concerning the zeroes of the diastatic function (or equivalently

the set of points in M — M where ψ = 1) can be obtained for regular bundles which

are su¬ciently positive.

In this situation let H be the Hilbert space of holomorphic sections of L and

let φ : M ’ P (H— ), x = π(q) ’ Clq where q ∈ L—¦ and lq (s) q = s, eq q = s(x)

for any s ∈ H. When the Chern class c1 (L) is su¬ciently positive, this map φ is

an embedding. Using §3 of [7], one sees that the pull-back of H — , the dual of the

tautological bundle over P (H— ), is isomorphic to the given bundle L. Hence ψL =

ψH — —¦φ. From proposition 1 and from the expression of D for complex projective

space given in example 2, one sees that ψ(x, y) = ψL (x, y) = 1 if and only if x = y.

This proves the following proposition.

8 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY

Proposition 3. If the bundle L is regular and su¬ciently positive, the diastatic

function D(x, y) vanishes if and only if x = y.

The function ψ admits the points of the diagonal as critical points. In fact at

these points ψ has its maximum value, namely 1, and if the bundle is su¬ciently

positive, ψ(x, y) = 1 only if x = y. Choosing a point x of M , we plan to compute

the Hessian of the function ψ(x, .) at y = x.

If, as above, H denotes the tautological bundle over P (H— ), then the hermitian

form on H is given by z ’ |z|2 , z ∈ H— \ {0}. Thus on the dual bundle H — it is

given by z ’ 1/|z|2 . Hence we get from (1.4) and Proposition 1

| lq , lq |2 | eq , eq |2

φ(x) = Clq

ψH — (φ(x), φ(y)) = = ,

lq 2 lq 2 e q 2 eq 2

if q (resp. q ) belongs to Lox (resp. Loy ). This means that

φ— ψH — = ψL .

Observe that at a critical point (x, x) of ψL , one has

(Hess ψL )(x,x) (X, Y ) = Hess ψH — (φ— X, φ— Y )(φ(x),φ(y)) . (1.14)

The right hand side can be computed readily :

(Hess2 ψH — ) = ’π g (1.15)

where the su¬x 2 means that one ¬xes the ¬rst variable (and thus ψH — becomes

a function on P (H— )) and where g is the Fubini-Study metric given in example 2.

For L su¬ciently positive we then have the following proposition.

Proposition 4. If π : L ’ M is a su¬ciently positive regular bundle, the Hessian

of the characteristic function ψL (considered as a function of its second argument

only) is given by

Hess2 ψL = ’π φ— g. (1.16)

In particular it is a non singular symmetric bilinear form.

Remark. Using the proposition of §3 of [7] one sees that the Hessian is propor-

tional to the metric of M .

2. The composition of operators and an asymptotic formula: As in

§1, we denote by (L, , h) a quantization bundle of the compact K¨hler manifold

a

(M, ω, J) and by H the Hilbert space of holomorphic sections of L. By compactness

of M , it is ¬nite dimensional. Let A : H ’ H be a linear operator and let

Aeq , eq

ˆ

A(x) = , q ∈ Lox , x∈M

eq , eq

be its symbol.

¨

QUANTIZATION OF KAHLER MANIFOLDS II 9

The composition of operators on H gives rise to a product for the corresponding

symbols, which is associative and which we shall denote by — following Berezin. For

the basic facts about — quantization see [1]. We have

—

ˆ — B)(x) = AB(x) = ABeq , eq = Beq , A eq

ˆ

(A

eq 2 eq 2

1 ω n (y)

—

hy ((Beq )(y), (A eq )(y))

=

eq 2 n!

M

|q |2 ω n (y)

(2.1)

= Aeq , eq

Beq , eq

eq 2 n!

M

n

2

ˆ y)B(y, x) | eq , eq | (y) ω (y)

ˆ

A(x,

=

eq 2 eq 2 n!

M

n

ˆ y)B(y, x)ψ(x, y) (y) ω (y) .

ˆ

= A(x,

n!

M

Let k be a positive integer. The bundle (Lk = —k L, k , hk ) is a quantization

bundle for (M, kω, J) and we denote by Hk the corresponding space of holomorphic

ˆ

sections and by E(Lk ) the space of symbols of linear operators on Hk . We have

proven in [7] the following facts

i) When (k) is constant for all k (i.e., when all bundles Lk are regular) one has

ˆ ˆ

the nesting property E(Lk ) ‚ E(Lk+1 ).

ˆ

ii) With the same assumption ∪k E(Lk ) is dense in C —¦ (M ).

From §1, we recall that formula (1.8) and proposition 2 prove that

ψ (k) (x, y) = (ψ(x, y))k . (2.2)

From formula (2.1), the nesting property and formula (2.2) one sees that if A, B

ˆ

belong to E(Ll ) and if k ≥ l one may de¬ne

n

ωn

(k) k

ˆ ˆ ˆ ˆ

A(x, y)B(y, x)ψ k (x, y)

(A —k B)(x) = . (2.3)

n!

M

Remark 1. Let G be a Lie group of isometries of the K¨hler manifold (M, ω, J)

a

which lifts to a group of automorphisms of the quantization bundle (L, , h). This

automorphism group acts naturally on the bundles (Lk , (k) , hk ). We have proven

(l) (l) (l)

in [7] that if g ∈ G and if eq is a coherent state of Ll , then g.eq = egq . From this

ˆ

ˆˆ

one deduces that for any A, B in E(Ll ) and any k ≥ l:

ˆ ˆ ˆ ˆ

g — (A —k B)(x) = (A —k B)(gx)

= ((g ’1 Ag —k g ’1 Bg)(x)

ˆ

ˆ

= (g — A — g — B)(x)

which means that the product —k is G-invariant in the geometrical sense [1].

It is also a consequence of [7], that if g is the Lie algebra of G and X ∈ g, the

symbol of the quantum operator associated to the function »X , where i(X — )ω =

d»X and X — is the fundamental vector ¬eld on M corresponding to X, is precisely

that function »X . Hence the product —k is covariant, [1].

We plan to analyze the k-dependence of the formula (2.3). The ¬rst step in this

direction is a the rationality of (k) .

10 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY

Proposition 1. Let (M, ω, J) be a ¬‚ag manifold, let (L, , h) be a quantization

bundle for (M, ω, J) and let Lk = —k L and let (k) be the -function of the bundle

Lk . Then (k) is a rational function of k, with no pole at in¬nity.

Proof. Recall that by the theorem of §3 of [7]

1

(k)

dimH(k)

= n

(volM )k

1

where n = 2 dimM and H(k) is the space of holomorphic sections of Lk . Now

H(k) carries an irreducible representation of G (M = G/K where G is a compact

connected, simply connected Lie group and K is the centralizer of a torus T1 of G).

Denote by » the highest weight of this representation. If µ is the highest weight of

the irreducible representation of G on H, one has » = kµ.

Let g be the Lie algebra of G and t be the algebra of a maximal torus T ⊇ T1 .

Let ∆ be the set of roots of gC relative to tC . There exists an “admissible” Weyl

chamber C of it— such that if ∆+ is the set of positive roots relative to C and if ¦

is the set of positive simple roots, then ¦ = ¦1 ∪ ¦2 where ¦1 = {± ∈ ¦ | ±|t1 = 0}.

Furthermore the algebra k of the isotropy group K is such that

k C = tC • (g± + g’± )

+

±∈∆1

where the elements of ∆1 are sums of elements of ¦1 . Denote by ∆+ = ∆+ \∆1 .

+ +

2

One has then

dimR G/K = dimG ’ dimt ’ 2#{± ∈ ∆+ } 1

1

n = (dimG ’ dimt) ’ #{± ∈ ∆+ }. 1

2

Recall that Weyl™s dimension formula tells us that the dimension d» of H is given

by

+ » + δ, ±

d» = ±∈∆

±∈∆+ δ, ±

where δ denotes half the sum of the positive roots and , is the scalar product

on it— induced by the Killing form of g.

Thus d» is a polynomial in k with degree q equal to the number of positive roots

which are not orthogonal to ». Clearly

q = #{± ∈ ∆+ }.

2

Indeed the stabilizer of the highest weight is equal to K as the geometrical quanti-

zation conditions are satis¬ed. Hence the conclusion since q = n.

Remark 2. This proposition generalizes to the compact regular case. Indeed, in

this situation L ’ M is a holomorphic, hermitian line bundle with connection

and the curvature of is 2πiω. Thus ω is an integral form representing the

¬rst Chern class c1 (L). As ω is a positive (1, 1)-form, c1 (L) > 0 and thus L is an

ample bundle. So passing to a su¬ciently high power k of L, Kodaira™s map φ is

an embedding of M in projective space. In particular by Chow™s theorem, M is a

projective algebraic variety.

¨

QUANTIZATION OF KAHLER MANIFOLDS II 11

Now by the Riemann-Roch-Hirzebruch formula the Euler-Poincar characteristic

χ of the ‚-complex is given by an integral of a polynomial in c1 (L) of degree equal

to the dimension of M . Kodaira™s vanishing theorem replaces the Borel-Weil-Bott

theorem to tell us that, if c1 (L) is su¬ciently positive (i.e. for k su¬ciently large),

all cohomology spaces vanish in positive dimension. Hence χ reduces to dimH(Lk )

and is thus a polynomial of degree dimM in k.

Remark 3. The integral in (2.3) is an absolutely convergent integral which makes

sense for any real number k ≥ l. Indeed it may be rewritten as :

(l) (l) (l) (l) 2

(l) (l) 2 l

eq

Aeq , eq Beq , eq eq |q l |2 |q |2 k ωn

n

(k) k

ψ (x, y)

2 |q l |2 |q l |2

(l) (l) (l) (l) (l) 2 (l) n!

eq

eq , eq eq , eq eq

M

(k) n

ωn

l 2k

(l) (l)

(l) k’l

Aeq , e(l) l2

ψ

Beq , eq (x, y)

= |q | |q |

q (l)2 n!

M

(k) nn

l 2k ω

k’l

(l) (l) — (l)

hy (Beq , A eq )ψ (x, y) (l)2 |q |

=

n!

M

and thus

n

ωn

(k) k

ˆ ˆ

A(x, y)B(y, x)ψ k’l (x, y)

n!

M

kn ωn

ˆ ˆ k (k)

A(x, y)B(y, x)ψ (x, y)e

¤

n!

M

n

(k)

ω

|h(l) (ABe(l) , e(l) )|

l2 n

¤ |q | k y q q

(l) )2 n!

( M

which is clearly bounded.

The second step consists in localizing the integral (2.3) in a neighbourhood V of

x; or, more precisely, to de¬ne a neighbourhood U of the diagonal in M — M such

that, for any x in M , V = {y ∈ M |(x, y) ∈ U }. This is done by means of a version

of the Morse lemma which we take from Combet [4].

Proposition 2. Let (M, ω, J) be a compact K¨hler manifold and let g denote its

a

metric. Let V be an open neighbourhood of the zero section of the tangent bundle

p : T M ’ M , such that the map ± : V ’ M — M , X ’ (p(X), expp(X) X) is

well de¬ned. Let (L, , h) be a regular quantization bundle over M and let ψ be the

corresponding 2-point function on M —M . Then there exists an open neighbourhood

W of the zero section in T M , and a smooth embedding ν : W ’ T M such that

π

gp(X) (X, X), X ∈ W. (2.4)

(’ log ψ—¦±—¦ν)(X) =

2

Proof. By compactness of M , there exists an open neighbourhood V1 of the zero

section of T M and an open neighbourhood U1 of the diagonal ∆ in M — M such

that

i) V1 ‚ V ;

ii) ±|V1 : V1 ’ U1 is a smooth di¬eomorphism;

12 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY

iii) U 1 © ψ ’1 (0) = ….

Denote by f : V1 ’ R the smooth function f = ’ log ψ—¦±. Observe that f (0x ) =

’ log ψ(x, x) = 0. If we denote by a subscript 2 di¬erentiation in the vertical

directions in T M one has

1

(D2 f )ox = (D2 ψ)(x,x) = 0

ψ(x, x)

since all points of the diagonal are critical points of ψ. Finally, using Proposition 4

of §1 we get

(Hess2 f )ox = πgx .

For v ∈ V1 , de¬ne the function gv : [0, 1] ’ R : t ’ f (tv). Clearly

˜

gv (0) = 0, gv (0) = (D2 f )o v = 0

˜

˜

and

gv (o) = (Hess2 f )o (v, v) = πgx (v, v) > 0

˜

whatever v we choose. Taylor™s formula with remainder gives us

1

f (v) = (1 ’ s)˜v (s)ds

g

0

and one sees that

gv (s) = (D2 f )sv v,

˜ gv (s) = (Hess2 f )sv (v, v).

˜

We can thus introduce on each tangent space Mx a family of symmetric bilinear

forms, indexed by an element v ∈ V1 © Mx

1

Bv (u, u ) = (1 ’ s)(Hess2 f )sv (u, u )ds.

0

Clearly Bo (u, u ) = 2 (Hess2 f )o (u, u ) = π gx (u, u ) is positive de¬nite.

1

2

Using compactness again, we can ¬nd a neighbourhood V2 of the zero section in

T M such that i) V2 ‚ V1 ; ii) Bv is positive de¬nite for any v ∈ V2 . Recall that

f (v) = Bv (v, v). There exists a unique non-singular element Cv of GL(Mx ) which

is symmetric relative to Bo such that

Bv (u, u ) = Bo (Cv u, u )

and all eigenvalues of Cv are strictly positive. Furthermore the map V2 ‚ T M ’

End(T M ), v ’ Cv is smooth. Finally the endormorphism Cv admits a unique

1/2

symmetric, positive de¬nite square root Cv and

1/2 1/2

Bv (u, u ) = Bo (Cv u, Cv u ).

1/2 1/2

Also Co = I and the map v ’ Cv is smooth.

1/2

De¬ne the map β : V2 ’ T M , v ’ Cv v. This maps the zero-section onto

the zero-section and one can ¬nd a neighbourhood W of the zero-section such that

’1

β|W : W ’ T M is an embedding. Clearly one may choose W = β(W ), ν = β|W

and the proposition is proven.

¨

QUANTIZATION OF KAHLER MANIFOLDS II 13

Proposition 3. Let (M, ω, J) be a compact K¨hler manifold, (L, , h) be a quan-

a

tization bundle for M and ψ be the corresponding 2-point function. Then for any

ˆ

f belonging to E(Ll ) the integral

n

nω

k

for k ≥ l

, (2.5)

Fk (x) = f (x, y)ψ (x, y)k

n!

M

admits an asymptotic expansion

k ’r Cr (f )(x)

Fk (x) ∼ (2.6)

r≥0

where Cr is a smooth di¬erential operator depending only on the geometry of M .

Proof. Use Proposition 2 to construct a neighbourhood U1 of the diagonal ∆ in

M — M and a neighbourhood V1 of the zero section in T M such that the following

hold.

i) ± : V1 ’ U1 , X ’ (x, expx X) is a smooth di¬eomorphism;

ii) ∃ ν ’1 : V1 ’ ν ’1 (V1 ) ‚ T M a smooth embedding such that ’ log ψ—¦±—¦ν =

π ’1

2 g on ν (V1 );

iii) U 1 © ψ ’1 (0) = ….

Going back to the proof of Proposition 2, one observes that ±—¦ν : ν ’1 (V1 ) © Mx ’

M = {x}—M is an embedding and hence one may de¬ne a non zero smooth function

θ by

n

—ω

((±—¦ν) )(x, v) = θ(x, v)dv

n!

where dv denotes the linear Lebesgue measure on Mx . Shrinking V1 , if necessary,

one may assume that θ is de¬ned on V 1 and hence is bounded as well as all its

derivatives.

Choose an open neighbourhood U2 of ∆ in M — M , with U 2 ‚ U1 and de¬ne

V2 = ±’1 (U2 ). Let χ : M — M ’ [0, 1] be a smooth function such that χ|U2 = 1

and supp χ ‚ U1 . Set · = maxx,y∈U2 ψ(x, y). Clearly · < 1 and ψ(x, y) ¤ · on

/

M \ U2 . Let U1,x = {y ∈ M | (x, y) ∈ U1 }, U2,x = {y ∈ M | (x, y) ∈ U2 } and

χx (y) = χ(x, y). The function χx is equal to 1 on U2,x and has support in U1,x .

The function f appearing in the statement of the proposition is a smooth function

on (M — M ) \ ψ ’1 (0) (it may have singularities where ψ vanishes). In particular it

is well-behaved in a neighbourhood of the diagonal. One has

n n

nω nω

k k

f (x, y)ψ (x, y)k

Fk (x) = + f (x, y)ψ (x, y)k .

n! n!

U1,x M \U1,x

Going back to the Remark 1 of the previous section one sees that there exists a

positive constant C1 such that |f ψ l | ¤ C1 on M — M. Thus

n

nω

k

¤ C1 · k’l k n volM .

f (x, y)ψ (x, y)k

n!

M \U1,x

14 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY

Also

n

nω

k

f (x, y)(1 ’ χx (y))ψ (x, y)k

n!

U1,x

n

nω

k

= f (x, y)(1 ’ χx (y))ψ (x, y)k

n!

U1,x \U2,x

¤ C2 · k k n volM

if |f | ¤ C2 on U 1 . Grouping the terms we get

n

nω

k

¤ C· k k n ,

Fk (x) ’ χ(x, y)f (x, y)ψ (x, y)k ∀k ≥ l

n!

U1,x

where we have de¬ned C = vol(M )(C1 · ’l + C2 ). Thus this di¬erence is expo-

nentially uniformly small. The integral may be computed in the tangent space Mx

as

n

nω

k

χ(x, y)f (x, y)ψ (x, y)k

n!

U1,x

kπ

k n θ(x, v)dv

2 g(v,v)

χ(±—¦ν)(x, v)f (±—¦ν)(x, v)e’

=

V1,x

where V1,x = (±—¦ν)’1 U1,x .

Denote by G(x, v) the function on T M de¬ned by

χ((±—¦ν)(x, v))f ((±—¦ν)(x, v))θ(x, v), if (x, v) ∈ V1 ;

G(x, v) = (2.7)

0, if (x, v) ∈ V1 .

/

It is smooth and compactly supported and

n

nω

χ(x, y)f (x, y)ψ(x, y)k

n!

U1,x

kπ

k n dv

2 g(v,v)

G(x, v)e’

=

Mx

∞

kπ 2

G(x, rv)e’ 2r r2n’1 k n dv

dr

=

Sx M

0

where r(v) = g(v, v)1/2 and Sx M is the unit sphere in Mx . Now use Taylor™s

formula with integral remainder for G(x, rv)

2N

rp p

G(x, rv) = (D G)(x, 0)

p! v

p=0

1

(1 ’ s)2N 2N +1

2N +1

+r (Dv G)(x, rsv)ds.

(2N )!

0

¨

QUANTIZATION OF KAHLER MANIFOLDS II 15

The integral of the remainder term is easily bounded since G is compactly sup-

ported.

1

∞

(1 ’ s)2N 2N +1 kπ 2

n 2n+2N +2

G)(x, rsv)e’ 2 r

ds

dv k r

dr (Dv

(2N )!

0

Sx M

0

1

∞ 1

(2t)n+N ’ 2 (1 ’ s)2N

’N

=k dv

dt ds 1

(2N )!

πN + 2

0 Sx M 0

e’t

2t

(Dv +1 G)(x,

2N

sv) √

—

kπ k

C3

¤ k ’N √ .

k

Observe ¬nally that if p is odd

p

(Dv G)(x, 0)dv = 0

Sx M

since this is the integral of the restriction to the sphere of a homogeneous polynomial

of odd degree. Putting these facts together we get

N p+n k ’p

(p + n ’ 1)! 2

N 2p

k Fk (x) ’ (Dv G)(x, 0)dv

(2p)! π 2 Sx M

p=0

C3

¤ C· k k n+N + √ . (2.8)

k

This proves the proposition since the derivatives of the function G in the vertical

direction for v = 0 do not depend on the choice of the cut-o¬ function χ, but depend

only on f and θ (which is related to the geometry alone).

ˆ

Remark 4. If A (resp. B) is an element of E(Ll ) which corresponds to a rank one

operator As = s, u v; u, v, s ∈ Hl (resp. Bs = s, u v ; u, v , s ∈ Hl ) the formula

˜˜ ˜˜

(2.3) for their — product takes a very special form. Indeed

u, eq

v, eq u, eq v , eq

˜ ˜

A(x, y)B(y, x) =

| eq , eq |2

hx (v(x), u(x))hy (˜(y), u(y))

˜ v 1

= .

ψ l (x, y)

( (l) )2

Hence

n

hx (v(x), u(x))

˜ hy (˜(y), u(y)) (k) n ω

v k’l

(A—k B)(x) = (x, y)e

ψ k .

(l) (l) n!

M

ˆ ˆ

Thus it is the product of a symbol C ∈ E(Ll ), (Cs = s, u v; u, v, s ∈ Hl ) by an

˜ ˜

integral of the form (2.5). Observe also that k has been shifted by l.

16 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY

Remark 5. Using Remark 4, we compute the ¬rst 2 terms of the asymptotic

ˆ

expansion of A —k B for A and B in E(Ll ) and k ≥ l.

n (k)

2

ˆ ˆ

(A —k B)(x) ∼ (n ’ 1)! G(x, 0)dv

π 2 Sx M

n+1 (k)

n! 2 1 2

+ Dv G(x, 0)dv + · · ·

2π 2k Sx M

(k)

where we still need to expand and where

ˆ ˆ

G(x, 0) = A(x)B(x)θ(x, 0),

ˆ ˆ

G(x, v) = A(x, exp h(v))B(exp h(v), x)θ(x, v).

ˆ ˆ

Hence the ¬rst term is proportional to the product A(x)B(x), the coe¬cient being

2 n1

θ(x, 0) volS 2n’1 0 (2.9)

± = (n ’ 1)!

π2

where 0 denotes the constant term in the asymptotic expansion of the rational

function (k) . Observe that one has the identity

1 —k 1 = 1

which tells us that ± = 1. For the second term, we shall only compute its antisym-

metric part. We have

ˆ ˆ ˆ ˆ

(A —k B ’ B —k A)(x) ∼

n+1

n! 2 0 2 2

Dv G(A,B) (x, 0) ’ (Dv G)(B,A) (x, 0) dv + · · ·

2π 2k Sx M

Observe that one has the identities

1 —k B = B —k 1 = B

which imply that the above integrand reduces to

n+1

ˆ —k B ’ B —k A)(x) ∼ n! 2 0 ˆ ˆ

ˆ ˆ ˆ Dv,2 A(x, x)Dv,1 B(x, x)

(A

2π k Sx M

ˆ ˆ

’ Dv,2 B(x, x)Dv,1 A(x, x) θ(x, 0)dv + · · · (2.10)

where the indices 1, 2 refer to the ¬rst (second) variable in a function of the form

ˆ

A(x, y). The integrand is a homogeneous polynomial of degree 2 which one in-

2n’1

tegrates over the sphere Sx . Hence if one decomposes this polynomial into a

multiple of |v|2 and a harmonic polynomial, only the multiple of |v|2 plays a rle in

ˆ

the integration. Since A(x, y) is holomorphic in x and antiholomorphic in y one

gets

β ˆˆ

ˆ ˆ

ˆ ˆ

(A —k B ’ B —k A)(x) ∼ {A, B}(x) + · · · (2.11)

k

where { , } is the Poisson bracket of functions on M associated to ω. From (2.10)

and (2.11) one gets

i

β= . (2.12)

2π

¨

QUANTIZATION OF KAHLER MANIFOLDS II 17

Remark 6. Going back to formula (2.9) applied to the product —k of elements

ˆˆ ˆ

A, B of E(Ll ) one sees that expanding the derivatives (compare 2.6) will give rise

to bidi¬erential operators which are invariant under all the automorphisms of the

quantization. Summarizing the above analysis we have

Theorem 1. Let (M, ω, J) be a compact K¨hler manifold and (L, , h) be a quan-

a

tization bundle over M . Assume this quantization is regular (i.e. ∀k ≥ 1, the

ˆˆ

function (k) corresponding to Lk = —k L is a constant). Let A, B be symbols of

linear operators on Hl (= space of holomorphic sections of Ll ). Then the product

—k

n

ˆ y)B(y, x)ψ k (x, y) (k) k n ω (y)

ˆ —k B)(x) =

ˆ ˆ

(A A(x,

n!

M

de¬ned for any k ≥ l admits an asymptotic expansion for k tending to in¬nity

ˆ ˆ ˆˆ

k ’r Cr (A, B)(x)

(A —k B)(x) ∼ (2.13)

r≥0

where the cochains Cr are smooth bidi¬erential operators, invariant under the auto-

morphisms of the quantization and determined by the geometry alone. Furthermore

ˆˆ ˆ ˆ

C0 (A, B)(x) = A(x)B(x), (2.14)

i ˆˆ

1 ˆˆ ˆˆ

(C1 (A, B) ’ C1 (B, A))(x) = {A, B}(x). (2.15)

2 4π

3. A — product for ¬‚ag manifolds: We would like to show that the

asymptotic expansion obtained above de¬nes an associative formal — product. For

this we have, so far, only a proof when (M, ω, J) is a ¬‚ag manifold.

Lemma 1. Let (M, ω, J) be a ¬‚ag manifold with M = G/K where G is a compact

simply-connected Lie group and K the centralizer of a torus. Assume the geometric

quantization conditions are satis¬ed and let (L, , h) be a quantization bundle over

ˆ

M . Let CL = ∪k E(Lk ) be the union of the symbol spaces. Then CL coincides with

the space E of vectors in C ∞ (M ) whose G-orbit is contained in a ¬nite dimensional

subspace.

ˆˆ

Proof. Any symbol A ∈ E(Ll ) for some l. Its G-orbit is clearly contained in a ¬nite

ˆ

dimensional subspace, namely E(Ll ) itself. Hence CL ‚ E ‚ C ∞ (M ).

We have proven in [7] that CL is dense in C 0 (M ) for the topology of uniform

convergence, hence is dense in L2 (M ) for the convergence in norm. Suppose V is a

¬nite dimensional invariant subspace of L2 (M ) then it is a direct sum of irreducible

subspaces each of which is then closed in L2 (M ). Hence each irreducible subspace

of V must intersect CL . By invariance and irreducibility it is contained in CL . Thus

V and hence E ‚ CL .

ˆˆ ˆ

Corollary 1. If A, B belong to E(Ll ), there exists an integer a(l) such that i)

ˆ ˆ ˆˆ ˆ

ˆ

A —k B belongs to E(La(l) ); ii) for every integer r, Cr (A, B) belongs to E(La(l) ).

ˆ

Proof. It was proved in Remark 2 of the previous section that the map E(Ll ) —

ˆ ˆ ˆ

ˆ ˆ

E(Ll ) ’ C ∞ (M ) given by A — B ’ A —k B intertwines the action of G, hence

18 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY

i). Similarly it was observed in Remark 6 that the bidi¬erential operators Cr were

invariant under G, hence ii).

Consider now the asymptotic development given by Theorem 1:

N

ˆ

ˆ ˆˆ ˆˆ

k ’r Cr (A, B) + RN (A, B, k)

A —k B =

r=0

where

ˆˆ

lim k N RN (A, B, k) = 0

k’∞

ˆ

ˆˆ

uniformly in x. Corollary 1 tells us that RN (A, B, k) belongs to E(La(l) ) where

a(l) is independent of k. Then

N

ˆ ˆˆ ˆˆ ˆ

ˆ ˆ ˆ

k ’r Cr (A, B) —k C + RN (A, B) —k C

(A —k B) —k C =

r=0

N

ˆˆ ˆ

k ’r’s Cs (Cr (A, B), C)

=

r,s=0

N

ˆˆ ˆ

ˆˆ ˆ

k ’r RN (Cr (A, B), C, k) + RN (A, B, k) —k C.

+

r=0

The second term multiplied by k N clearly tends to zero when k tends to in¬nity.

For the third observe that we can write

ˆˆ

RN (A, B, k) = u± (k)ˆ±

e

±

ˆ

where e± is a basis of E(La(l) ) and where for each ±, limk’∞ k N u± (k) = 0. Now

ˆ

ˆ

e± —k C is well de¬ned and the limit for k large is the usual pointwise prod-

ˆ

ˆˆ ˆ

uct. Hence limk’∞ k N RN (A, B, k) —k C = 0. This implies that the formal series

ˆ ˆ ’r de¬nes an associative deformation on CL . Hence we have proved

r≥0 Cr (A, B)k

the following theorem.

Theorem 2. The asymptotic expansion r≥0 k ’r Cr (u, v) de¬nes a formal asso-

ciative deformation of the usual product of functions in CL .

Remark. This is called a formal — product since the antisymmetrization of the

second term is the Poisson bracket of the functions u and v. The product extends

to all of C ∞ (M ), using uniform convergence.

4. Convergence of the — product for hermitian symmetric spaces: We

ˆ

ˆ

prove that A —k B is a rational function of k with no pole at in¬nity, when the

¬‚ag manifold is a hermitian symmetric space. This implies that the asymptotic

ˆ ˆ

expansion converges for k su¬ciently large. If we know already that A —k B is a

rational function of k, then the existence of the asymptotic expansion says it is

smooth at in¬nity and so has no pole there. Hence it is enough to establish the

rationality. As the proof is relatively long we shall split it into a series of lemmas.

¨

QUANTIZATION OF KAHLER MANIFOLDS II 19

ˆˆ

Lemma 1. To prove that A—k B is a rational function of k, for any pair of symbols

ˆˆ ˆ

A, B ∈ E(Ll ), (l < k) it is enough to prove that

ω n (y) q

p

ψ(w, y) ψ(y, z)

n!

M

depends rationally on p.

Proof. Formula (2.3) tells us that

n

(y)

(k) n ω

ˆ ˆ ˆ ˆ

A(x, y)B(y, x)ψ k (x, y)

(A —k B)(x) = k

n!

M

ˆˆ ˆ

for A, B ∈ E(Ll ), (l < k). Since Hl is ¬nite dimensional it is enough to prove

ˆ ˆ

rationality in the particular case where A and B are rank one operators

Bs = s, u v .

˜˜

As = s, u v,

Then

n

eq , u v, eq eq , u v , e q

˜ ˜ (k) n ω (y)

ˆ ˆ ψ k’l (x, y)

(A —k B)(x) = k

eq 2 2

eq n!

M

where π(q) = x, π(q ) = y. The term outside the integral is the symbol of an

ˆ

operator and belongs to E(Ll ); it is independent of k.

There exists a basis of Hl composed of coherent states and so we may write

u= ui eqi , vi eqi

v=

˜

i¤N i¤N

and thus the integral reduces to

n

e q , e qi e qj , e q (k) n ω (y)

ψ k’l (x, y) k .

ui vj

¯

eq 2 n!

M

i,j¤N

Assume we have proven that

n

eq , eq

eq , eq (k) n ω (y)

k’l

G(k, x, w) = (x, y)

ψ k

2

eq n!

M

(where π(q ) = w) depends rationally on k. Since G depends in a real analytic way

on w, and admits a unique analytic extension to M — M , this analytic extension

still depends in a rational way on k. Thus rationality of G(k, x, w) is su¬cient.

Notice that

n

(k) n ω (y)

eq 2 .

l k’l

G(k, x, w) = ψ (y, w)ψ (x, y) k

n!

M

Since eq 2 is independent of k and (k) is a rational function of k, the rationality

of G is equivalent to the rationality of

ω n (y)

l k’l

ψ (y, w)ψ .

(x, y)

n!

M

Set p = k ’ l, q = l and

ω n (y) q

p

ψ (x, y)ψ (y, w)

H(p, q; x, w) = = H(q, p; w, x).

n!

M

It follows that if we prove the rationality of H(p, q; x, w) with respect to p then we

have proved the rationality with respect to k of the product —k .

20 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY

Lemma 2. To prove that H(p, q; x, w) is rational in p for a ¬‚ag manifold M =

G/K (G a compact Lie group, K the centralizer of a torus in G) it is enough to

consider G simple.

Proof. Flag manifolds are simply connected and one may also assume that the

compact group G is simply connected. Hence G = G1 —· · ·—Gr , K = K1 —· · ·—Kr

and M = G1 /K1 — · · · — Gr /Kr = M1 — · · · — Mr . We shall denote by pj : M ’ Mj

the canonical projection. Let χ be the character of K de¬ning the line bundle

L = G —χ C and let χj be the restriction of χ to Kj . Denote by Lj = Gj —χj C

the corresponding line bundle on Mj and consider pj — Lj , the pull-back bundle. L

is isomorphic to —j pj — Lj , hence

ψL = ψpj — Lj = (ψLj —¦ pj ).

j

j

ωn

Since the volume form is a product of volume forms ωj on each of the factors

n!

Mj one sees that

q

p

H(p, q; x, w) = ψLj (pj (x), pj (y))ψLj (pj (y), pj (w))ωj

j

Mj

and the lemma is proven.

Let us recall some basic facts about hermitian symmetric spaces. Let M = G/K

be a compact irreducible hermitian symmetric space. Then M is simply connected

[8, p 376], and we may take G to be connected, compact, simple with trivial centre

and K to be a maximal connected proper subgroup of G [8, p 382]. Furthermore

the centre Z(K) of K is isomorphic to U(1) [8, p 382]. Let π : G ’ M be the

canonical projection and let o = π(e) denote the identity coset. The symmetry so

belongs to Z(K) [8, p 375]. Let g denote the Lie algebra of G and k be the Lie

algebra of K. Let σ denote the involutive automorphism of g de¬ned by Ad so and

let g = k • p be the decomposition of g into the +1 and ’1 eigenspaces of σ. Let

h be a maximal abelian subalgebra of k. h contains z, the Lie algebra of Z(K) and

is a maximal abelian subalgebra of g. Hence if gC = kC • pC is the complexi¬ed

algebra of g, the subalgebra hC is a Cartan subalgebra of gC . Let ∆ be the set of

roots of g with respect to hC and let ∆c be the set of roots whose retrictions to zC

do not vanish identically. If ∆1 = ∆ \ ∆c then

kC = hC • pC =

gβ , gβ .

β∈∆1 β∈∆c

The roots take real values on ih. Choose compatible Weyl chambers in (ih)— and

(iz)— and denote by ∆+ the corresponding positive roots. Let

gβ , gβ .