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 2point 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)
qω
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 nform 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 onedimensional, 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 Kinvariant 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
c2 = 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 nonhermitian 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, SpringerVerlag,
e
BerlinHeidelbergNew 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