Quantization of Kaehler manifolds II

28 Pages·0.305 MB·English

by  Cahen M., Gutt S., Rawnsley J.

QUANTIZATION OF KA¨HLER MANIFOLDS II
Michel Cahen, Simone Gutt* and John Rawnsley
Abstract. WeuseBerezin’sdequantizationproceduretodefineaformal -product
∗
onadensesubalgebraofthealgebraofsmoothfunctionsonacompacthomogeneous
K¨ahler manifold M. We prove that this formal -product is convergent when M is a
∗
hermitian symmetric space.
0. Introduction: In part I of this paper [7] we showed how to quantize cer-
tain compact K¨ahler manifolds (M,ω,J). This means the following: Let (L, ,h)
∇
be a quantization bundle over M (i.e., a holomorphic line bundle L with connec-
tion admitting an invariant hermitian structure h, such that the curvature is
∇
curv( ) = 2iπω). Let be the Hilbert space of holomorphic sections of L. To
∇ − H
ˆ
any linear operator A on is associated a symbol A which is a real analytic func-
H
ˆ
tion on M. Denote by E(L) the space of these symbols. For any positive integer k,
(Lk = kL, (k),h(k)) is a quantization bundle for (M,kω,J). If k is the Hilbert
⊗ ∇ H
space of holomorphic sections of Lk, we denote by Eˆ(Lk) the space of symbols of
linear operators on k. If, for every k, a certain characteristic function (cid:143)(k) (which
H
depends on L and k and which is real analytic on M) is constant, the space Eˆ(Ll)
is contained in the space Eˆ(Lk) for any k l. Furthermore Eˆ(Ll) (denoted
≥ ∪∞l=1
by ) is a dense subspace of the space of continuous functions on M. Any func-
L
C
tion f in belongs to a particular Eˆ(Ll) and is thus the symbol of an operator
L
C
A(k) acting on k for k l. One has thus constructed, for a given f, a family of
f H ≥
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
family of associative products on Eˆ(Ll), with values in , parametrized by an
L
C
integer k; indeed
f g = A(cid:92)(k)A(k) f,g Eˆ(Ll); k l.
∗k f g ∈ ≥
The aim of Part II is twofold. Firstly, we prove that, for any compact generalized
flag manifold, there exists on a formal differential -product with parameter ν
L
C ∗
which coincides with the asymptotic expansion of the previously defined associa-
k
∗
tive products when ν = i/(4πk). Secondly, we prove that, when M is a hermitian
1991 Mathematics Subject Classification. 58F06.
Key words and phrases. Quantization, K¨ahler manifolds.
*Research Associate of the National Fund for Scientific Research (Belgium)
Typeset by -TEX
AMS
1
2 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY
symmetric space, the product of two functions in is a rational function of k
k L
∗ C
having no pole at infinity. 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
§ ∗
product of two elements of Eˆ(Ll). In 3 we turn to flag manifolds and prove that
§
the asymptotic expansion constructed in 2 defines 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 when M is hermitian symmetric.
L
C
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 difficulties which have to be overcome to understand
completely the compact K¨ahler case.
Acknowledgement: We are pleased to thank our friend Joe Wolf who gave
us the benefit of his expertise in the geometry of hermitian symmetric spaces. We
also thank the British Council and the Communaut´e fran¸caise de Belgique for their
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 defined in terms of coherent states. Some of the properties
of this function are described here. Another, somewhat related 2-point function
ψ, defined 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
t(cid:101)hree 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)(cid:101)will be a compact K¨ahler manifold and (L, ,h) a quantiza-
∇
tion bundle over M. Consider a real-analytic, closed, real 2-form Ω on M of type
(1,1). On a contractible open set U M there exists a real 1-form β such that
⊂
Ω = dβ. One may write β = α+α for some (1,0) form α on U. Since Ω is type
(1,1)
Ω = (∂ +∂)(α+α) = ∂α+(∂α+∂α)+∂α
implies ∂α = 0. Thus, by the Dolbeault lemma, there exists a function g defined
on a possibly smaller open set V U, such that
⊂
Ω = ∂∂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
−
∂∂f = 0, ∂f is a holomorphic (1,0)-form, thus there exists locally a holomorphic
1 1
function h such that ∂f = ∂h = dh. The reality of f implies that ∂f = dh and
1 1 1
thus df = d(h+h); i.e., f is the real part of a holomorphic function.
1 1
A function f satisfying (1.1) is called a potential for Ω. Since Ω 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(Ω) : W C,
→
D (x,y) = f(x,x)+f(y,y) f(x,y) f(y,x). (1.2)
(Ω)
− −
QUANTIZATION OF KA¨HLER 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
(Ω) (Ω)
of the local potential f and is thus a globally defined function on a neighbourhood
of the diagonal in M M, depending only on Ω. It is called the Calabi function of
×
Ω.
Observe that, for any y M, the set U = x M D (x,y)is defined is a
y (Ω)
∈ { ∈ | }
neighbourhood of y. Denote by D the function on U defined by D (x) =
(Ω)y y (Ω)y
D (x,y), then
(Ω)
∂∂D (x) = (∂∂f)(x,x)
(Ω)y
and thus D is a potential for Ω on U .
(Ω)y y
Definition 1. Calabi’s diastatic function D is the Calabi function of the K¨ahler
form ω.
Example 1. Let M = Cn and let ω = (i/2) jn=1dzj∧dzj be the canonical K¨ahler
form, then a potential f is given by
(cid:80)
n
f = 2π z 2
j
| |
j=1
(cid:88)
and the Calabi diastatic function D is
n
D(z,z ) = 2π z 2 + z 2 z z z z
(cid:48) | j| | j(cid:48)| − j j(cid:48) − j(cid:48) j
(cid:88)j=1 (cid:104) (cid:105) (1.3)
n
= 2π z z 2.
| j − j(cid:48)|
j=1
(cid:88)
It is thus, up to a factor, the square of the distance between the points z and z .
(cid:48)
Example 2. Let M = CPn and let π : Cn+1 0 CPn be the canonical
\{ } →
projection. Denote by , the standard metric on Cn+1. The Fubini-Study metric
(cid:104) (cid:105)
g on CPn is such that
1
(π g) (X,Y) = X,Y z 2 X,A Y,A X,B Y,B
∗ z π z 4 (cid:104) (cid:105)(cid:107) (cid:107) −(cid:104) z(cid:105)(cid:104) z(cid:105)−(cid:104) z(cid:105)(cid:104) z(cid:105)
(cid:34) (cid:35)
(cid:107) (cid:107)
where z Cn+1 and
∈
A = (z ∂ +z ∂ ),
z α zα α zα
α n
(cid:88)≤
B = ( iz ∂ +iz ∂ ).
z − α zα α zα
α n
(cid:88)≤
4 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY
Considering the chart U = p p = π(z)withzn+1 = 0 and the coordinate map
{ | (cid:54) }
φ : U Cn, φ(p) = u where p = π(z) and uj = zj/zn+1, j n, one sees that the
→ ≤
K¨ahler form ω on U has the form
i
ω = (1+ u 2) du du
2π(1+ u 2)2 | λ| α ∧ α
λ| λ| (cid:34) λ α
(cid:88) (cid:88)
(cid:80)
u du u du .
α α β β
− ∧
(cid:35)
α,β
(cid:88)
Hence a K¨ahler potential f is given by
f = 2log(1+ u 2).
λ
| |
λ
(cid:88)
In these coordinates the diastatic function has the expression
D(u,u ) = 2 log(1+ u 2)+log(1+ u 2)
(cid:48) | λ| | (cid:48)λ|
(cid:104) (cid:88)λ (cid:88)λ
log(1+ u u ) log(1+ u u ) .
− λ (cid:48)λ − (cid:48)λ λ
(cid:88)λ (cid:88)λ (cid:105)
This can be rewritten in terms of the coordinates in Cn+1 as
z 2 z 2
(cid:48)
D(π(z),π(z )) = 2log (cid:107) (cid:107) (cid:107) (cid:107) . (1.4)
(cid:48)
z,z 2
(cid:48)
|(cid:104) (cid:105)|
In particular D > 0 unless π(z) = π(z ), where D = 0.
(cid:48)
Observe that Cn+1 0 may be identified with a principal C∗-bundle over CPn.
\{ }
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 definition 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) = h (s(x),s(x)) is a real-analytic function on U which
x
⊂ | |
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-zerovaluesfory sufficientlyclosetox. Considerthentheexpression(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 defined, whi(cid:101)ch 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 justifies the following definition.
(cid:101)
QUANTIZATION OF KA¨HLER MANIFOLDS II 5
Definition 2. If L M is a holomorphic line bundle with real-analytic hermitian
→
structure h, the 2-point function ψ defined 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 ψ . (cid:101)
L
Before considering some of the properties of ψ, let us exhibit the relationship
(cid:101)
between ψ and Calabi’s function.
(cid:101)
Proposition 1. Let π : L M be a holomorphic line bundle with real analytic
(cid:101) →
hermitian structure h. Let be the unique connection, [5], on L leaving h invariant
∇
and such that s = 0 for any X of type (0,1) and for any local holomorphic section
X
∇
s. Let ω = (i/2π)curv( ). Then
∇
ψ = e−21D (1.6)
where D is the Calabi function of ω.(cid:101)
Remark. If (L, ,h) is a quantization bundle for M, the K¨ahler form satisfies
∇
the assumption of Proposition 1 and thus ψ = e 1D where D is Calabi’s diastatic
−2
function.
(cid:101)
Proof. Let s : U L be a holomorphic section of L. Then
→
s = α (X)s
X s
∇
where α is a 1-form of type (1,0). Also
s
d s 2 = (α +α ) s 2
s s
| | | |
and thus
α = ∂log s 2.
s
| |
The curvature 2-form σ is defined by
σ(X,Y)s = ( )s
X Y Y X [X,Y]
∇ ∇ −∇ ∇ −∇
and for s holomorphic we get
σ(X,Y)s = (dα )(X,Y)s.
s
That is
σ = ∂∂log s 2 = ∂∂log s 2.
| | − | |
Thus for ω = (i/2π)σ, 2log 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) = 2log | | | |
s 2(x,x) s 2(y,y)
(cid:104)| | | | (cid:105)
= 2logψ(x,y),
−
hence the conclusion.
(cid:101)
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
(cid:48) (cid:48)
→ →
let ψ and ψ be the corresponding characteristic functions. If s : U L and
L L
(cid:48) →
s : U L are local sections then s s is a local section of L L . Hence
(cid:48) (cid:48) (cid:48) (cid:48)
→ ⊗ ⊗
(cid:101) (cid:101)
ψ = ψ .ψ . (1.7)
L L L L
⊗ (cid:48) (cid:48)
In particular if one considers the(cid:101)kth powe(cid:101)r Lk(cid:101)of the bundle L, one has
ψ = (ψ )k. (1.8)
Lk L
(cid:101) (cid:101)
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 1U, is holomorphic
∗ −
⊂ ◦
and s f 2 = s 2 f. Formula (1.5) shows that
| ◦ | | | ◦
ψ = ψ f . (1.9)
f L L
∗ ◦
(cid:101) (cid:101)
These properties justify the name characteristic function.
The 2-point function ψ may be expressed in terms of coherent states and of the
L
function (cid:143) which has been introduced in Part I and denoted by θ there. Denote by
the Hilbert space of(cid:101)holomorphic sections of L with the scalar product
L
H
ωn(x)
s,s = h (s(x),s (x)) (1.10)
(cid:48) x (cid:48)
(cid:104) (cid:105) n!
(cid:90)M
where h is the hermitian structure of L, ω the K¨ahler form of M and dimM = 2n.
Evaluation of a holomorphic section s at a point x of M is a continuous linear
map L = π 1(x). Choose a base-point q L . Then there exists a unique
L x − x
H → ∈
element e , called a coherent state, such that
q L
∈ H
s(x) = s,e q, x = π(q).
q
(cid:104) (cid:105)
Since e = c 1e for c a non zero complex number, the function
cq − q
(cid:143)(x) = e 2 q 2, x = π(q)
q
(cid:107) (cid:107) | |
is well-defined.
If s : U L is a zero-free holomorphic section over U, and s an arbitrary
◦ →
holomorphic section of L then
s(x) = s,e s (x), x U.
s (x)
(cid:104) ◦ (cid:105) ◦ ∈
Thus s,e is a holomorphic function of x and hence e depends antiholo-
s (x) s (x)
(cid:104) (cid:105)
morphically◦ on x. In particular ◦
(cid:143)(x) = e ,e s (x) 2
s (x) s (x)
(cid:104) ◦ ◦ (cid:105)| ◦ |
QUANTIZATION OF KA¨HLER MANIFOLDS II 7
is real analytic and admits an analytic extension to a neighbourhood of the diagonal
in U U, holomorphic in the first variable and antiholomorphic in the second
×
(cid:143)(x,y) = e ,e s 2(x,y).
s (y) s (x)
(cid:104) ◦ ◦ (cid:105)| ◦|
In particular
(cid:143)(x,x)(cid:143)(y,y) e ,e 2
s (y) s (x)
ψ(x,y) = |(cid:104) (cid:105)|
◦ ◦
e 2 e 2 (cid:143)(x,y) 2
s (x) s (y)
(cid:107) (cid:107) (cid:107) (cid:107) | | (1.11)
◦ ◦
(cid:101) (cid:143)(x)(cid:143)(y) eq ,eq 2
= |(cid:104) (cid:48) (cid:105)| , q L 0 ,q L 0 .
(cid:143)(x,y) 2 e 2 e 2 ∈ x \{ } (cid:48) ∈ y \{ }
q q
| | (cid:107) (cid:48)(cid:107) (cid:107) (cid:107)
We have seen in [7], the importance of the condition (cid:143) being constant. This moti-
vates the following definition.
Definition 3. A hermitian holomorphic line bundle π : L M will be said to be
→
regular if the function (cid:143) is constant.
Definition 4. Let e q L be the set of coherent states of the hermitian line
q
{ | ∈ ◦}
bundle π : L M. The 2-point function ψ is defined by
→
e ,e 2
q q
ψ(x,y) = |(cid:104) (cid:48) (cid:105)| , π(q) = x, π(q ) = y. (1.12)
(cid:48)
e 2 e 2
q q
(cid:107) (cid:48)(cid:107) (cid:107) (cid:107)
ψ is globally defined 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
definition in terms of a local holomorphic section s we have
o
e ,e 2
ψ(x,y) = |(cid:104) so(x) so(y)(cid:105)|
e 2 e 2
(cid:107) so(x)(cid:107) (cid:107) so(y)(cid:107)
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 defined on
M M.
×
(cid:101)
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 sufficiently positive.
In this situation let be the (cid:101)Hilbert space of holomorphic sections of L and
H
let φ : M P( ∗), x = π(q) Clq where q L and lq(s)q = s,eq q = s(x)
→ H (cid:55)→ ∈ ◦ (cid:104) (cid:105)
for any s . When the Chern class c (L) is sufficiently positive, this map φ is
1
∈ H
an embedding. Using 3 of [7], one sees that the pull-back of H , the dual of the
∗
§
tautological bundle over P( ), is isomorphic to the given bundle L. Hence ψ =
∗ L
H
ψ φ. From proposition 1 and from the expression of D for complex projective
H
∗◦
space given in example 2, one sees that ψ(x,y) = ψ (x,y) = 1 if and only if x(cid:101)= y.
L
T(cid:101)his proves the following proposition.
(cid:101)
8 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY
Proposition 3. If the bundle L is regular and sufficiently 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 sufficiently
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( ), then the hermitian
∗
H
form on H is given by z z 2, z 0 . Thus on the dual bundle H it is
∗ ∗
(cid:55)→ | | ∈ H \{ }
given by z 1/ z 2. Hence we get from (1.4) and Proposition 1
(cid:55)→ | |
l ,l 2 e ,e 2
q q q q
ψH∗(φ(x),φ(y)) = |l(cid:104) 2 l(cid:48)(cid:105)| 2 = |e(cid:104) (cid:48)2 e(cid:105)| 2, φ(x) = Clq
q q q q
(cid:107) (cid:107) (cid:107) (cid:48)(cid:107) (cid:107) (cid:107) (cid:107) (cid:48)(cid:107)
(cid:101)
if q (resp. q ) belongs to L (resp. L ). This means that
(cid:48) ox oy
φ ψ = ψ .
∗ H L
∗
Observe that at a critical point (x,x)(cid:101)of ψ , one has
L
(Hessψ ) (X,Y) = Hessψ (φ X,φ Y) . (1.14)
L (x,x) H (φ(x),φ(y))
∗ ∗ ∗
The right hand side can be computed readil(cid:101)y :
(Hess ψ ) = πg (1.15)
2 H
∗ −
where the suffix 2 means that one fixe(cid:101)s the first variable (and thus ψ becomes
H
∗
a function on P( )) and where g is the Fubini-Study metric given in example 2.
∗
H
For L sufficiently positive we then have the following proposition. (cid:101)
Proposition 4. If π : L M is a sufficiently positive regular bundle, the Hessian
→
of the characteristic function ψ (considered as a function of its second argument
L
only) is given by
Hess ψ = π φ g. (1.16)
2 L ∗
−
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¨ahler manifold
§ ∇
(M,ω,J) and by the Hilbert space of holomorphic sections of L. By compactness
H
of M, it is finite dimensional. Let A : be a linear operator and let
H → H
Ae ,e
ˆ q q
A(x) = (cid:104) (cid:105), q L , x M
o
e ,e ∈ x ∈
q q
(cid:104) (cid:105)
be its symbol.
QUANTIZATION OF KA¨HLER MANIFOLDS II 9
The composition of operators on gives rise to a product for the corresponding
H
symbols, which is associative and which we shall denote by following Berezin. For
∗
the basic facts about quantization see [1]. We have
∗
ABe ,e Be ,A e
(Aˆ Bˆ)(x) = AB(x) = (cid:104) q q(cid:105) = (cid:104) q ∗ q(cid:105)
∗ e 2 e 2
q q
(cid:107) (cid:107) (cid:107) (cid:107)
1 ωn(y)
(cid:100)
= h ((Be )(y),(A e )(y))
y q ∗ q e 2 n!
(cid:90)M (cid:107) q(cid:107)
q 2 ωn(y)
(cid:48)
= Be ,e Ae ,e | | (2.1)
(cid:104) q q(cid:48)(cid:105)(cid:104) q(cid:48) q(cid:105) e 2 n!
(cid:90)M (cid:107) q(cid:107)
e ,e 2 ωn(y)
= Aˆ(x,y)Bˆ(y,x) |(cid:104) q(cid:48) q(cid:105)| (cid:143)(y)
e 2 e 2 n!
(cid:90)M (cid:107) q(cid:48)(cid:107) (cid:107) q(cid:107)
ωn(y)
ˆ ˆ
= A(x,y)B(y,x)ψ(x,y)(cid:143)(y) .
n!
(cid:90)M
Let k be a positive integer. The bundle (Lk = kL, k,hk) is a quantization
⊗ ∇
bundle for (M,kω,J) and we denote by k the corresponding space of holomorphic
H
sections and by Eˆ(Lk) the space of symbols of linear operators on k. We have
H
proven in [7] the following facts
i) When (cid:143)(k) is constant for all k (i.e., when all bundles Lk are regular) one has
the nesting property Eˆ(Lk) Eˆ(Lk+1).
⊂
ii) With the same assumption Eˆ(Lk) is dense in (M).
k ◦
∪ C
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
belong to Eˆ(Ll) and if k l one may define
≥
knωn
(Aˆ Bˆ)(x) = Aˆ(x,y)Bˆ(y,x)ψk(x,y)(cid:143)(k) . (2.3)
k
∗ n!
(cid:90)M
Remark 1. Let G be a Lie group of isometries of the K¨ahler manifold (M,ω,J)
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
∇
in [7] that if g G and if e(l) is a coherent state of Ll, then g.e(l) = e(l). From this
q q gq
∈
one deduces that for any Aˆ,Bˆ in Eˆ(Ll) and any k l:
≥
ˆ ˆ ˆ ˆ
g (A B)(x) = (A B)(gx)
∗ k k
∗ ∗
= ((g(cid:92)1Ag g(cid:92)1Bg)(x)
− k −
∗
= (g Aˆ g Bˆ)(x)
∗ ∗
∗
which means that the product is G-invariant in the geometrical sense [1].
k
∗
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 λ , where i(X )ω =
X ∗
dλ and X is the fundamental vector field on M corresponding to X, is precisely
X ∗
that function λ . Hence the product is covariant, [1].
X k
∗
We plan to analyze the k-dependence of the formula (2.3). The first step in this
direction is a the rationality of (cid:143)(k).
10 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY
Proposition 1. Let (M,ω,J) be a flag manifold, let (L, ,h) be a quantization
∇
bundle for (M,ω,J) and let Lk = kL and let (cid:143)(k) be the (cid:143)-function of the bundle
⊗
Lk . Then (cid:143)(k) is a rational function of k, with no pole at infinity.
Proof. Recall that by the theorem of 3 of [7]
§
1
(cid:143)(k) = dim (k)
(volM)kn H
where n = 1 dimM and (k) is the space of holomorphic sections of Lk. Now
2 H
(k) carries an irreducible representation of G (M = G/K where G is a compact
H
connected, simply connected Lie group and K is the centralizer of a torus T of G).
1
Denote by λ the highest weight of this representation. If µ is the highest weight of
the irreducible representation of G on , one has λ = kµ.
H
Let g be the Lie algebra of G and t be the algebra of a maximal torus T T .
1
⊇
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 Φ
∗ +
is the set of positive simple roots, then Φ = Φ Φ where Φ = α Φ α = 0 .
1∪ 2 1 { ∈ | |t1 }
Furthermore the algebra k of the isotropy group K is such that
kC = tC (gα +g α)
−
⊕
α(cid:88)∈∆1+
where the elements of ∆+ are sums of elements of Φ . Denote by ∆+ = ∆+ ∆+.
1 1 2 \ 1
One has then
dim G/K = dimG dimt 2# α ∆+
R − − { ∈ 1 }
1
n = (dimG dimt) # α ∆+ .
2 − − { ∈ 1 }
Recall that Weyl’s dimension formula tells us that the dimension d of is given
λ
H
by
λ+δ,α
d = α ∆+(cid:104) (cid:105)
λ ∈
δ,α
(cid:81) α ∆+(cid:104) (cid:105)
∈
where δ denotes half the sum of the p(cid:81)ositive roots and , is the scalar product
(cid:104) (cid:105)
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
λ
which are not orthogonal to λ. Clearly
q = # α ∆+ .
{ ∈ 2 }
Indeed the stabilizer of the highest weight is equal to K as the geometrical quanti-
zation conditions are satisfied. 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
∇ ∇
first Chern class c (L). As ω is a positive (1,1)-form, c (L) > 0 and thus L is an
1 1
ample bundle. So passing to a sufficiently 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.