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n+ = n’ =
β∈∆+ ’β∈∆+
c
c



Then n+ and n’ are abelian subalgebras of pC and pC = n+ • n’ [8, p 384].
Two roots ±, β ∈ ∆ are said to be strongly orthogonal if neither ± + β nor
± ’ β belongs to ∆. Let a be a maximal abelian subalgebra of p and let r be
¨
QUANTIZATION OF KAHLER MANIFOLDS II 21


its dimension. The number r is called the rank of the symmetric space M . One
proves [8, p 385] that there exists a subset {γ1 , . . . , γr } of ∆+ consisting of strongly
c
C
orthogonal roots. If ± ∈ ∆ let h± ∈ h be the element such that B(h, h± ) = ±(h)
where B is the Killing form of gC . There exist vectors X± ∈ g± such that for all
± ∈ ∆, (X± ’ X’± ) and i(X± + X’± ) belong to g. Furthermore

2
[X± , X’± ] = h± .
±(h± )

Then we can choose the subalgebra a so that it is spanned by the vectors i(Xγj +
X’γj ), (j ¤ r).
Let GC be the simply connected Lie group with algebra gC and let N + , K C , N ’
denote the analytic subgroups of GC with algebras n+ , kC , n’ respectively. The
following facts are standard:
i) The exponential map induces a di¬eomorphism of n+ (respectively n’ ) onto
N (respectively N ’ ), [8, p 388].
+

ii) The map N’ — K C — N+ ’ GC , (n’ , k, n+ ) ’ n’ kn+ is a smooth di¬eo-
morphism onto an open submanifold of GC containing G, [8, p 388].
iii) The map f : G/K ’ GC /K C N+ , gK ’ gK C N+ is a holomorphic di¬eo-
morphism, [8, p 393].
iv) The map ξ : n’ ’ M, Y ’ expY.o is a holomorphic di¬eomorphism of n’
onto an open dense subset of M , [8, p 395].
Let us now go back to the quantization bundle L ’ M ; as a homogeneous
line bundle L = G —χ C where χ is a character of K. Observe that χ extends
holomorphically to K C and trivially to N+ and thus de¬nes a character of the
parabolic K C N+ . One can view L holomorphically as L = GC —χ C.
Denote by g ’ g the complex conjugation of GC with respect to the real form
¯
o
G. If q ∈ L (= L \ zero section) and if eq is the corresponding coherent state one
has
g · eq = egq . (4.1)
¯

Indeed the representation of G on H extends to GC . Formula (4.1) is valid for g in
G, and both sides depend holomorphically on g. Hence (4.1) is valid for GC .
If qo is a non zero element of Lo (= ¬bre above o), the coherent state eqo is a
lowest weight vector for the representation of GC on H. Indeed if n ∈ N’ and
h ∈ K C then
n · eqo = enqo = eqo ,
¯

¯
h · eqo = ehqo = eχ(h)qo = χ(h)’1 eqo = χ(h)eqo .
¯
¯

From its de¬nition the 2-point function ψ(x, y) is invariant under G (i.e. ψ(gx, gy) =
ψ(x, y)). This implies that in the integral given in Lemma 1 one can choose x = o.
Also by the observation (iv) above one can assume that y ∈ ξ(n’ ). We thus com-
pute for any n ∈ N’
| eqo , enqo |2
,
ψ(o, n · o) =
eqo 2 enqo 2

eqo , enqo = eqo , neqo = n’1 eqo , eqo = eqo 2
¯
22 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY


using the same argument as for formula (4.1). Hence

2
eqo
.
ψ(o, n · o) = 2
enqo

Similarly, if z ∈ ξ(n’ ) and if so is a holomorphic section of Lo over ξ(n’ )

| enqo , eso (z) |2
ψ(n · o, z) =
enqo 2 eso (z) 2
| n · eqo , eso (z) |2
¯
.
=
enqo 2 eso (z) 2

Thus

| n · eqo , eso (z) |2q eqo 2p ω n (y)
n ¯
(y)

p
ψ(o, y) ψ(y, z) = .
enqo 2(p+q) eso (z) 2q
n! n!
M ξ(n’ )


Observe that one can choose qo such that eqo 2 = 1. The above integral is the
product of a function of z, independent of p together with

| n · eqo , eso (z) |2q ω n (n · o)
¯
.
n · eqo 2(p+q) n!
¯
ξ(n’ )


Since the group N+ is nilpotent, the numerator of the integrand is a polynomial
function on n’ .
We now wish to write this integral over ξ(n’ ) as an integral over n’ , by making
the change of variables explicit. Let dz = dz1 § . . . § dzn be the n-form on n’
(n = dimC n’ ) which is translation invariant. This induces an invariant form, still

denoted by dz, on N’ . Clearly ξ ’1 (dz § d¯) is a volume form on ξ(n’ ) and so has
z
the form
ω n (z)
’1 —
ξ (dz § d¯) = f
z
n!
for some function f . Since G acts transitively on M , any element g ∈ GC can be
written in the form
g = u(g)k(g)n+ (g)
where u(g) ∈ G, k(g) ∈ K C , n+ (g) ∈ N+ . In particular for n ∈ N’

ω n (u(n) · o)
’1 —
(dz § d¯)|n·o
z = f (n · o)
ξ
n!
n
ω (o)
—¦ u(n)’1
= f (n · o) —
n!

or equivalently

f (n · o) ’1 — —
— —
(ξ ’1 dz)n·o —¦ u(n)— § (ξ ’1 d¯)n·o —¦ u(n)— = dz)o § (ξ ’1 d¯)o .
(ξ z
z
f (o)
¨
QUANTIZATION OF KAHLER MANIFOLDS II 23


But — —
(ξ ’1 dz)n·o —¦ u(n)— = (ξ ’1 dz)n·o —¦ (nn+ (n)’1 k(n)’1 )—

= (ξ ’1 dz)o —¦ (n+ (n)’1 k(n)’1 )—
and since the action of N+ is unipotent, one has

f (n · o)
= |det(Adn’ k(n)’1 )|2 .
f (o)

One can choose normalizations such that f (o) = 1. After this change of variable
the integral becomes

| neqo , eso (z) |2q
¯
|det(Adn’ k(n)’1 )|’2 dz § d¯.
z
2(p+q)
neqo
¯
n’

We still wish to write the denominator and the jacobian more explicitly. Observe
that
n = u(n)k(n)n+ (n),
implies
n = u(n)k(n) n+ (n).
¯
Since conjugation permutes N+ and N’ and since eqo is a lowest weight vector
2 2
= |χ(k(n))|2 eqo 2
= |χ(k(n)’1 )|2 .
= k(n)eqo
n · eq o
¯

Since the centre of K C is one-dimensional, all characters of K C are powers of a
given one µ, which we choose in such a way that

χ(k ’1 ) = µ(k)a

for some positive integer a.
Clearly k ’ det(Adn’ k ’1 ) is also a character of K C and thus there exists an
integer a such that
det(Adn’ k(n)’1 ) = µ(k(n))a .
We conclude this discussion with the following lemma.
Lemma 3. To prove that H(p, q; x, w) is rational in p for an irreducible hermitian
symmetric space M = G/K it is enough to prove that

| neqo , eso (z) |2
¯
dzd¯
z
|µ(k(n))|2p
n’

is rational in p. Here the numerator is a polynomial on n’ and µ is the fundamental
character of K.
Proof. Combining the preceeding arguments, we see that the rationality of H(p, q; x, w)
is equivalent to the rationality of

| neqo , eso (z) |2
¯
dzd¯
z
|µ(k(n))|2(ap+aq+a )
n’
24 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY


with respect to p. Replacing ap + aq + a by p will not change the question.
Since n’ is isomorphic (as a vector space) to p, this integral may be viewed as
an integral on p. Our next aim is to decompose this integral into “radial” and
“angular” integrals, but ¬rst we introduce some more notation. Let m be the
centralizer of a in k, and let M be the corresponding connected subgroup of K. Let
t be a maximal abelian subalgebra of g containing a and let tC be the corresponding
Cartan subalgebra of gC . Denote by ∆ the set of roots of gC relative to tC , and by
∆1 the set of roots in ∆ which do not vanish identically on a. As above the roots
take real values on it— . Choose compatible Weyl chambers in it— and in ia— .
Consider the map β: K/M — a ’ p, (kM, a) ’ Ad k(a). It is surjective. Now
K/M , p, a have invariant metrics induced by the Killing form of g. Denote by
dkM and da the corresponding Riemannian measures on K/M and a, respectively.
The measure on p ∼ n’ was already denoted dz § d¯. Then we have the identity
z
=
[9, p 382]
β — (dz § d¯)(kM,a) = »(a) dkM da
z +
»∈∆1

and thus the following lemma holds.
Lemma 4. Let F be a function on p which is integrable. Then

F (z)dz § d¯ =
z dkM »(a) F (Ad k(a))da (4.2)
»∈∆+
p K/M a 1


and in (4.2) one can replace the integral over K/M by an integral over K, intro-
ducing some normalization constant C.
Consider, as above, the vectors Xj = i(Xγj + X’γj ), (j ¤ r), which form a
basis of a and also the vectors Yj = Xγj ’ X’γj and Hj = 2ihj /B(γj , γj ). They
determine r mutually commuting su(2)-subalgebras of g

[Xj , Yj ] = ’2Hj , [Hj , Xj ] = ’2Yj , [Hj , Yj ] = 2Xj .

This implies that
π
π
Ad exp Yj (Xj ) = Hj , Ad exp Yj (Hj ) = ’Xj .
4 4
In particular the automorphism „ of g de¬ned by
π π
„ = Ad exp Y1 . . . Ad exp Yr
4 4
sends the algebra a onto the subalgebra l of h spanned by the Hj ™s, (j ¤ r) and
sends l onto a. The system of roots of gC relative to tC corresponds to the system
of roots of gC relative to hC . In particular the roots in ∆1 correspond to the set ¦
of roots in ∆ which do not vanish identically on l. The roots in ¦ come in pairs
±, ’± and we shall denote by ¦+ any subset of ¦ obtained by choosing arbitrarily
one root in each pair. Then

»(a) = ρ(„ (a)) .
ρ∈¦+
+
»∈∆1
¨
QUANTIZATION OF KAHLER MANIFOLDS II 25


Lemma 5. i) The function k ’ µ(k(exp Ad k (A))) is constant on K, where A
denotes the n’ component of an element A of a.
ii) Any K-invariant polynomial on p is necessarily of even degree when restricted
to a.
Proof. i)
’1
exp Ad k (A) = k exp A k
’1
= k u(exp A)k(exp A)n+ (exp A)k
and thus
’1
k(exp Ad k (A)) = k k(exp A)k
and i) follows, since µ is a character of K.
ii) If P is the invariant polynomial and A an element of a, write A = aj Xj ;
j
then P (A) is a polynomial in the aj ™s. Observe that
Ad exp πHj (Xk ) = Xk if k = j,
Ad exp πHj (Xj ) = ’Xj
and thus P (a1 , . . . , ar ) is even in each variable aj .
aj X’γj )) is a product of r functions each
Lemma 6. The function µ(k(exp j¤r
depending on only one of the aj .
Proof. The subgroup exp tX’γj is contained in the SL(2, C)-subgroup of GC whose
Lie algebra is spanned by Xj , Yj , Hj . Since these r subgroups are commuting
aj X’γj = exp aj X’γj .
exp
j¤r
j¤r

Write each of the terms as a product
exp aj X’γj = uj kj n+
j

with uj ∈ G, kj ∈ K C , n+ ∈ N+ and observe that each of these elements belongs
j
to the corresponding commuting SL(2, C)-subgroups. Hence
uj kj n+ = n+
uj kj
j j
j¤r j¤r j¤r j¤r

and thus
µ(k(exp aj X’γj )) = µ(kj )
j¤r j¤r
where each of the kj ™s depends on aj alone. This proves the lemma.
If we set
Q(z) = | neqo , eso (z) |2
¯
then Q(z) is a polynomial on n’ and we may summarize the above analysis by
rewriting the integral appearing in Lemma 3 as
»(a)|Q(Ad k (a))
|
Q(z)dz § d¯z +
»∈∆1
= da
dkM
|µ(k(exp Ad k (a)))|2p
|µ(k(n))|2p a
K/M
n’
| ρ(„ (a))|
ρ∈¦+
=C da Q(Ad k (a))dk
2p
| j¤r µ(kj )|
a K

where dk is the Haar measure on K. The integral of Q over K gives a polynomial
Q(a1 , . . . , ar ) invariant under K. Thus it is of even degree in each aj .
26 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY


Lemma 7. The polynomial

R(a1 , . . . , ar ) = ρ(„ ( aj Hj ))
ρ∈¦+
j¤r


is of odd degree in each argument aj .
Proof. To see this we ¬x some j and partition ¦+

¦+ = {γj } ∪ {ρ ∈ ¦+ |ρ ⊥ γj } ∪ ¦+
j

where the last set consists of all elements not in the ¬rst two sets. If ρ is orthogonal
to γj then ρ( j aj Hj ) is independent of aj . Also γj ( j aj Hj ) = aj γj (Hj ), so is
of degree one. Finally, if ρ ∈ ¦+ and sj denotes the element of the Weyl group
j
associated to γj , then sj ρ di¬ers from both ±ρ and one of ±sj ρ is in ¦+ . Since sj
¬xes all the γk for k = j,

ak Hk = ±ρ ’aj Hj + ak Hk
±(sj ρ)
k k=j


and thus the root which belongs to ¦+ actually is in ¦+ . Hence ¦+ has an even
j
j
number of terms which pair o¬ to contribute factors of the form

ρ( ρ( ak Hk ) + ρ(aj Hj )
ak Hk ) ’ ρ(aj Hj )
k=j
k=j

ak Hk )2 ’ ρ(aj Hj )2 .
= ρ(
k=j


The lemma now follows.
Lemma 8. The function |µ(kj )|2 of aj is of the form (1 + a2 )± for some positive
j
power ±.
Proof. The calculation can be done in a ¬xed SL(2, C). We have

01 0 0
∈ n+ , X’ =
X+ = ∈ n’
00 1 0

and we decompose
10
= ukn+
exp aX’ =
a1
with u ∈ SU (2), k diagonal and n+ upper triangular with 1s on the diagonal. If

1z
0
c
, n+ =
k= ’1
01
0 c

with c ∈ C— , z ∈ C then multiplying out and solving gives

|c|2 = 1 + a2 .
¨
QUANTIZATION OF KAHLER MANIFOLDS II 27


The lemma follows since every character of the diagonal subgroup has the form

c 0
’ c± .
0 c’1


If we expand the polynomial RQ in monomials, only odd powers will occur as a
consequence of Lemma 7. Using Lemma 8, it follows that

Q(a1 , . . . , ar )
da
ρ(„ (a))
µ(kj )|2p
|
ρ∈¦+
a j¤r


is a sum of terms each of which is a product of integrals of the form

|x2s+1 |
dx
(1 + x2 )±p
’∞

where s is in a ¬xed range independent of p. A simple calculation yields the next
lemma.
Lemma 9. ∞
|x2s+1 | s!
dx = .
(1 + x2 )±p (±p ’ 1) · · · (±p ’ s ’ 1)
’∞

In particular the integral is a rational function of p without pole at in¬nity.
This can be summarized in the following theorem.
Theorem. Let M be a compact hermitian symmetric space and let (L, , h) be a
quantization bundle over M. Let Lk = —k L and let Hk be the space of holomorphic
ˆˆ
ˆ
sections of Lk . Let E(Lk ) be the space of symbols of operators on Hk . If A, B
ˆ ˆ
ˆ
belong to E(Ll ) and k ≥ l, the product A —k B depends rationally on k and has no
pole at in¬nity.
ˆ ˆ
Corollary. The asymptotic expansion of A —k B is convergent.
Remark. A calculation along similar lines for the non-hermitian symmetric ¬‚ag
manifold U (3)/U (1)—U (1)—U (1) leads to a similar result. It thus seems reasonable
to suggest, in conclusion, the following conjecture.
Conjecture. For any generalized ¬‚ag manifold the —k product of two symbols is a
rational function of k without pole at in¬nity.

References
1. F. Bayen et al., Deformation theory and quantisation, Ann. of Phys. 111 (1978), 1“151.
F. A. Berezin, Quantisation of K¨hler manifold, Commun. Math. Phys. 40 (1975), 153.
2. a
3. E. Calabi, Isometric imbeddings of comlex manifolds, Ann. Math. 58 (1953), 1“23.
4. E. Combet, Int´grales exponentielles., Lecture Notes in Mathematics 937, Springer-Verlag,
e
Berlin-Heidelberg-New York, 1982.
5. P. Gri¬ths and J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978.
6. C. Moreno, — products on some K¨hler manifolds, Lett. Math. Phys. 11 (1986), 361“372.
a
7. M. Cahen, S. Gutt and J. Rawnsley, Quantisation of K¨hler manifolds I: Geometric inter-
a
pretation of Berezin™s quantisation, (To appear), Journal of Geometry and Physics.
28 MICHEL CAHEN, SIMONE GUTT* AND JOHN RAWNSLEY


8. S. Helgason, Di¬erential geometry, Lie groups and symmetric spaces, Academic Press, New
York, 1978.
9. S. Helgason, Di¬erential geometry and symmetric spaces, Academic Press, New York, 1962.

Departement de Mathematique, ULB Campus Plaine CP 218, 1050 Brussels, Bel-
gium

Departement de Mathematique, ULB Campus Plaine CP 218, 1050 Brussels, Bel-
gium

Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

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