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¨
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

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

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

k
f (x, y)(1 ’ χx (y))ψ (x, y)k
n!
U1,x
n

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

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

k
χ(x, y)f (x, y)ψ (x, y)k
n!
U1,x

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

χ(x, y)f (x, y)ψ(x, y)k
n!
U1,x

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)

¨
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 .

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