Quantization of K/ihler Manifolds. IV*

Letters in Mathematical Physics 34: 159-168, 1995. 159 1995 Kluwer Academic Publishers. Printed in the Netherlands. Quantization of K/ihler Manifolds. IV* MICHEL CAHEN 1, SIMONE GUTT 1'2,and JOHN RAWNSLEY 3 1Ddpartement de Math~matiques, Universitd Libre de Bruxelles, Campus Plaine CP 218, 15 Bruxelles, Belgique. e-mail Simone Gutt: sgutt@uib.ac.be 2 D@artement de Math~matiques, Universitd de Metz, Ile du Saulcy, F-5745 Metz Cedex 1, France Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom, e-mail: jhr@maths.warwick.ac.uk (Received: 2 August 1994) Abstract. We use Berezin's dequantization procedure to define a formal *-product on the algebra of smooth functions on the bounded symmetric domains. We prove that this formal *-product is convergent on a dense subalgebra of the algebra of smooth functions. Mathematics Subject Classification (1991). 58F6. 1. Introduction The notion of,-product as an operator-free approach to quantization was introduced in [-1, 2]. The method of quantization of K~ihler manifolds due to Berezin [3], as the inverse of taking symbols of operators, has been used systematically to construct such,-products from geometric data in [5,6]. The method involves introducing a suitable parameter into the Berezin composition of symbols, taking the asymptotic expansion in this parameter on a large algebra of functions and then showing that the coefficients of this expansion satisfy the cocycle conditions to define a,-product on the smooth functions. We have only been able to establish these cocycle conditions by showing that the asymptotic expansion actually converges on this subalgebra. This restricts the applicability of our results to the situation where we can find a large enough subalgebra on which the expansion converges. In [5], we showed the existence of the asymptotic series for coadjoint orbits of compact Lie groups and its convergence for compact Hermitian symmetric spaces. In [6], we showed how the symbols of polynomial differential operators gave a dense subalgebra on which the asymptotic expansion converges in the case of the unit disk in C. In this Letter, we turn our attention to general Hermitian symmetric spaces of the noncompact type, and use their realization as bounded domains to define an analogous algebra of symbols of polynomial differential operators. The main result of the Letter is the following theorem. *This work was partially supported by EC contract CHRX-CT92-5.

16 MICHEL CAHEN ET AL. THEOREM. Let ~ be a bounded symmetric domain and ~ the algebra of symbols of polynomial differential operators on a homogeneous holomorphic line bundle L over 9 which gives a realization of a holomorphic discrete series representation of Go; then for f and 9 in g, the Berezin product f*k g has an asymptotic expansion in powers of k-1 which converges to a rational function of k. The coefficients of the asymptotic expansion are bidifferential operators which define an invariant and covariant,-product on C~(~). 2. Existence and Convergence of the,-product for Bounded Domains Let ~ denote a bounded symmetric domain. We shall use the Harish-Chandra embedding to realize ~ as a bounded subset of its Lie algebra of automorphisms. More precisely, if Go is the connected component of the group of holomorphic isometrics then 9 is the homogeneous space Go/Ko, where Go is a noncompact semi-simple Lie group and Ko is a maximal compact subgroup. Let go be the Lie algebra of Go, fo the subalgebra corresponding with Ko, g and ~ the complexifieations and G, K the corresponding complex Lie groups containing Go and K. The complex structure on 9 is determined by Ko-invariant Abelian subalgebras m + and m_ with g = m+ + ~ + m_, [m+,m_] ~ ~, m+ = m_, where - denotes conjugation over the real form go of g. The exponential map sends m e diffeomorphically onto subgroups M_+ of G such that M KM_ is an open set in G containing Go and the multiplication map M x K x M_ ~ M+KM_ is a diffeomorphism. KM_ is a parabolic subgroup of G and the quotient G/KM_ a generalized flag manifold. The Go-orbit of the identity coset can be identified with Go/Ko and lies inside M + KM_/KM_ ~-M+. Composing this identification with the inverse of the exponential map gives the desired Harish-Chandra embedding of as a bounded open subset of m+. We shall assume from now on that @ c m+ via this embedding. In this realization, it is clear that the action of Ko on ~ coincides with the adjoint action of Ko on m+. In [6], we considered the special case of the unit disk in C. In [9, p395], it is shown that every bounded symmetric domain contains a product of one-dimensional disks (which we call a polydisk), ~o, which is totally geodesically embedded in 9. ~o is homogeneous under a subgroup of Go which is locally isomorphic to a product of copies of SL(2, R). The K-orbit of 9o is the whole of ~. It is obvious that, for a disk and, hence, the polydisk ~o, if ~ is in the disk then so is any multiple r~ for < r < 1. This immediately implies: LEMMA 1. If O < a < b, then a~ c bg. ~ c r~, Vr > 1 (here ~ topological closure of ~). means the

QUANTIZATION OF KXHLER MANIFOLDS. IV 161 Proof. If ~ e D, then Ad k~ e Do for some k E Ko. Hence, (a/b) Ad k e 9o so a~ e bg. If 4,, is a sequence in D converging to ~ e ~, then there is a sequence k,, e Ko such that Ad k,,~m ~ 9o for all m. Since Ko is compact, by passing to a subsequence if necessary, we can assume that k,, converges to some k e Ko. Thus, Ad k,,,, converges to Ad k~ ~ Do. In the case of the unit disk, it is obvious that its closure is contained in any disk of larger radius. Hence, the lemma. LEMMA 2. 9 = kdo<r<l rg. Proof. By Lemma 1, the union is contained in D. Conversely, suppose { e 9. The map 2 ~ 2~ is a continuous map of R into m + and its value is in the open subset 9 when )v = 1. Hence, there is a neighbourhood of 1 where the values are in 9 and, thus, there is 2 > 1 with 2~ e 9 and so ~ e (1/2)9. COROLLARY. If X is a compact subset of g, there is an r with < r < 1 such that Xcrg. Proof. The {rdi < r < 1} form an open covering of X. Hence, a finite number cover X and, since they are nested, X is contained in one of them. Following Satake [11], we define maps k:9 x 9--+ K, m+:d x 9-+m+ by exp-z'expz = expm+(z,z')k(z,z')-lexpm_(z,z'), Z,Z' ed. They satisfy k(z,z') = k(zt, Z) -1, m+(g,z') ~- -m_(z',z). In what follows, the derivatives of k will be important, so we pause to prove the following lemma. LEMMA 3. For ~ em+, we have (i) d,=o k(z + t, Z') k(z, Z') -1 = -[m_(z, Z'), ~], (ii) d,=o m_ (Z + t, Z') = ½Era (Z, Z'), Ira_ (Z, Z'), ~]], (iii) d ~=om_(z,z'+t~) = -Ad k(z, Z ')~. - Proof. k(z + t~, Z') = exp m_ (Z + t{, Z') exp - t{(exp - Z' exp Z)- 1 exp m+(z + t~,z') = exp m_ (Z + t~, Z') exp - t~ exp - m_ (Z, Z') x x k(z,z')exp - m+(z,z')expm+(z + t{,z')

162 MICHEL CAHEN ET AL. so d k(z + t~, Z') k(z, Z') -~ dt t=o = -(Ad exp m_(z, Z')~) ~ = -[m_(z,z'), ] using the fact that m_ is Abelian. Similarly, exp m_ (Z + t~, Z') = k(z + t~,z')exp- m+(z + td, Z')exp- Z'expZexpt~ = k(z + t~,z')exp - m+(z + t~,z')expm+(z,z') x x k(z, Z')- 1 exp m_ (Z, Z') exp t~ SO d t=om_(z + t~,z') d exp m_ (Z + t~, Z') exp - m_ (Z, Z') dt t=o = (Ad exp m_(z, Z') ~)m = ½ [m_ (z, z'), [m_ (z, z'), ~]]. Finally, exp m_ (Z, Z' + t~) = k(z, Z' + t~) exp - m+ (Z, Z' + t~) exp - t~-exp - Z~exp Z = k(z,z' + t~)exp - m+(z,z' + t~)exp- t~expm+(z,z') x x k(z,z') -1 expm_(z,z') so dr=ore (Z,Z'+t~) d expm_(z,z' + t~) exp m_(z,z') dt t=o = - (Ad k(z, Z') Ad exp - m+ (Z, Z') ~)~ = - Ad k(z, Z')~

QUANTIZATION OF K)I.HLER MANIFOLDS. IV 163 3. The Holomorphic Quantizafion For any unitary character g of Ko there is a Hermitian holomorphic line bundle L over 9 whose curvature is the K/ihler form of an invariant Hermitian metric on 9. If X also denotes the holomorphic extension to K, then the Hermitian metric has K/ihler potential log g(k(z, Z)) and L has a zero-free holomorphic section So with Iso(Z) l 2 = z(k(z, Z)) -1 For Z sufficiently positive, So is square-integrable and the representation U of Go on the space ~ of square-integrable sections of L is one of Harish-Chandra's holomorphic discrete series [7, 8]. So is a highest weight vector for the extremal K-type so is a smooth vector for the representation. We form the coherent states eq and see that eso(o) transforms the same way as So and so they must be equal up to a multiple. This means that the coherent states are also smooth vectors of the representation. Further, since the quantization is homogeneous, ~ will be constant. Thus ( e~o~z,), eso~z ) ) = ~Z(k(Z, Z')). LEMMA 4. Up to a constant (determined by the normalization of Haar measure on Go) e is the formal degree dv of the discrete series representation U. Proof. dv is the positive constant such that fg!(ugv, w)12dg=dvlllvll211wl]2 Vv, we~. If we take v to be the coherent state based at the identity coset eko, then multiplying the left-hand side of this equation by Iso(eKo)l 2, it becomes foo '(U ~ w)(ek)12 dg = fgo ]w(gk)[2 dg = cllwl[2' where c depends on the normalization of Haar measure (relative to the Riemannian volume on 9). However, the right-hand side becomes du l ellwll 2 and a comparison of the two formulas gives dv - ce. COROLLARY. ~ is a polynomial function of the differential dz of the character )~. Proof. This is a consequence of Harish-Chandra's formula for the formal degree dv. See, for example, [12, 1.2.4]. The two-point function ~ is given by ~,(z, z') = [z(k(z,z'))i 2 z(k(z, Z))z(k(Z', z')) LEMMA 5. The quantization is regular in the sense of [6].

164 MICHEL CAHEN ET AL. Proof. Since we already know ~ is constant, showing regularity amounts to establishing the property that ~b takes the value 1 only on the diagonal. We recall that ~b is Go-invariant under simultaneous transformation of both variables, and everywhere bounded by 1. Suppose we have a pair of points where ~b has the value 1, then we can assume that one of these points is the origin in m + so that it is enough to consider solutions of ~b(z, ) = 1. The stabilizer Ko of is still available to move Z around and so can move it into the polydisk 9. Since ~b is natural with respect to holomorphic isometrics, then its restriction to ~o will be the product of the corresponding 2-point functions for one-dimensional disks where we know the result to be true from [6]. 4. Polynomial Differential Operators and Symbols We let d denote the algebra of holomorphic differential operators on functions on 9 with polynomial coefficients. We filter d by both the orders of the differentiation and the degrees of the coefficients: dr, q denotes the subspaee of operators of order at most p with coefficients of degree at most q. Obviously, the composition of operators gives a map The global trivialization by So of the holomorphic line bundle L corresponding with the character Z allows us to transport the above operators to act on sections of L by sending D ~ d to D x, where DZ(f so) = (D f)so. Let d(x) denote the resulting algebra of operators on sections of L and ~/v,q(z) the corresponding subspaces. In this noncompact situation, elements of d(z) do not define bounded operators on the Hilbert space ~, but the fact that the coherent states are smooth vectors of the holomorphic discrete series representation and that polynomials are bounded on 9, means that each operator in d(z) maps the coherent states into Jf so that it makes sense to speak of the symbols of these operators. A The analytically continued symbol Dz(Z,Z ') of an operator D z in LEMMA 6. dv,q( Z) is a polynomial in Z and m_ (Z, Z') of bidegree p, q. Proof. This follows from Lemma 1 and the following observation: Dz(Z, Z')so(Z) = (DZeso~Z'), esocz)) so(z) @so(z'), e,o(z) ) (D~e,o(z,))(Z) (%(z'), e~o(z) ) '

QUANTIZATION OF KNHLER MANIFOLDS. IV 165 SO if we set fz" = (eso(z,), eso(z))/e, then eso(z,) = fz' eso and, hence, DAx(Z, Z') - (Dfz,) (Z) fz,(z) Thus, to calculate the symbol of D x, we apply D to the Z argument of fz,(z) = 7~(k(Z,Z')). Lemma 3 (i) tells us that applying one derivative gives us components of m_ (Z, Z'), and Lemma 3 (ii) that each successive derivative of this raises the degree in m_(z, Z') by one. COROLLARY. The space of symbols of the operators in dk, z(z) is the space of polynomials in Z and m_ (Z, Z') of bidegree k, l so, in particular, is independent ofz. Proof. We know that the operation of taking symbols is an injective map, and the dimension of sjp,q(z) is the same as that of @,q which is also the same as the dimension of the space of polynomials, since ~ and m_ have the same dimension. Denote by gp, q the space of polynomials in Z and m_ (Z, Z') of bidegree p, q and by g the union of these spaces, o ~ is an algebra under pointwise multiplication - the algebra of symbols - which we shall utilize in the rest of the Letter. Remark. We have shown more in the proof of Lemma 6 than we have so far claimed, since the process outlined in the proof shows that the symbol of an operator in ~ p,q(z) is a polynomial of degree q in the differential dz of Z. So if we take a symbol in gp,q, then it is the symbol of an operator in dp,q(z). Taking two such operators and composing them corresponds with the composition of two polynomial operators in @,q and so can be expressed in terms of a basis for ~42p,2q as a rational function of dz. In other words, the Berezin product f g of two symbols f, g in gp,q is a symbol in g2p. 2q depending rationally on dx. 5. The,-Product Let us now try to follow the method of [6] to construct a formal deformation of the algebra C ~(9). We do so by first constructing it on the subalgebra & We consider the powers L k of the line bundle L which correspond with the powers k of Z. These powers have differentials k d)~, so the Berezin product f *k g of two symbols f, 9 in Cp,q is a rational function of k by the results of the previous section. In [6], we introduced the algebra ~1 c C~(M) of functions f which have an analytic continuation off the diagonal in M x/~r so that f(x, y)o(x, y)" is globally defined, smooth and bounded on K x M and on M x K for each compact subset K of M for some positive power r and ~ denotes those for which the power r suffices. If we assume for the moment that the symbols in ~p,q are in one of the spaces -~r then, as proved in [6], we will also have the existence of the asymptotic expansion (and, hence, also its convergence). Since the Berezin product is associative for each k, the same argument as in [5] shows that its asymptotic expansion in k-1 is an

166 MICHEL CAHEN ET AL. associative formal deformation on ~ with bidifferential operators as coefficients. To see that it extends to all of C ~ (~), we need to see that ~ contains enough functions to determine these operators. Suppose D is a differential operator of order r such that Df = for all f ~ g. Then we can analytically continue such equations off the diagonal in @ x ~ and suppose that we have a holomorphic differential operator D such that the same equation holds, where now f is an analytically continued symbol. D will have order r in each of Z and Z' and annihilates all polynomials in Z and m_ (Z, Z'). We now change variables from Z, Z' to Z, W, where W= m_ (Z, Z'). We claim this is a diffeomorphism, and so will transform D into another differential operator of degree 2r in Z and W vanishing on all polynomials of Z and W. So it clearly must vanish. To see that the change of variables is a diffeomorphism, we check its differential near the diagonal Z = Z'. So let (z, z') = (z, m_ (z, z')) then, in an obvious notation, q) has differential m_ (z, z') " ~Z' But Lemma 3 (iii) shows that m_ has invertible derivative in its second variable giving the following result. LEMMA 7. If the symbol spaces @,q are contained in some ~r (where r may depend on p, q) then the resulting asymptotic expansion of f *k g has bidifferential operator coefficients which satisfy the cocycle conditions to define a formal product on C ~(~) which is associative. It remains to verify the boundedness condition on the symbols. In fact, we show that they are in ~. It is enough to show that ad m_(z, Z') is bounded on X x and ~ x X for any compact subset X of 9, since the adjoint representation is injective. We follow the method of [1, lemmas 2.11, 2.12]. If we choose a basis t/i of teo which is orthonormal for - B, the Killing form of 9o, and a basis ~i for m+ such that B(41, 4j ) = 6ij then ~1,..., 4,, th,... t/,,, 41...,4, is a basis for g and for Z E m + where adz~i =, ad Zt/i = ~Zji~j, adz~ = ~ Wjirtj, J J m Zj~ = B(~j, adz~h), Wj~ = -B(~lj, adz~).

QUANTIZATION OF K,~HLER MANIFOLDS. IV 167 Since B is invariant, it follows that Wji = Zij and, hence, the matrix of ad Z in this basis has the form Exponentiating we have Similarly Ad exp Z = 1. (, i) Ad exp - ~r = - ~ 1, SO Ad exp - Z' exp Z If Z, Z' e 9, then 1 --~V! t W 1 - WT~W _*WT + ½*W,W,,W ½ tww w-iw~ww ]. 1 - 'W ~ w+ ¼'W ~ W ~ 'ww/ ad m+ (Z, Z') = ad rn_ (Z, Z') =, 'F Ad k(z, Z') = then Ad exp m+ (Z, Z')k(Z, Z') exp m_ (Z, Z') = BV+ ½ UCtVF B + UCtF~ UC ½ C 'F? C 'F" C

168 MICHEL CAHEN ET AL. and, thus, c= 1 -'w'w+ ¼'w'w'~ww, c' = -tw~ + ½~W~W~w. It follows that the boundedness of V (and, hence, m_) is determined by that of C - 1. Observe that C is unchanged if we replace Z by rz and Z' by (1/r)Z'. If X c is compact we know from the corollary to Lemma 2 that X c r~ for some < r < 1 and, hence, that C -1 is defined on X (1/r)~ which contains the compact set X ~. Hence, C- 1 is bounded on X 9. C is symmetrical in Z and Z' so we have the boundedness we need to prove the following theorem: THEOREM. Let ~ be a bounded symmetric domain and ~ the functions on ~ which are polynomials in Z and m_ (Z, Z), then for f and g in ~ the Berezin product f *k g has an asymptotic expansion in powers of k- 1 which converges to a rational function of k. The coefficients of the asymptotic expansion are bidifferential operators which define an invariant and covariant,-product on C ~(~). Acknowledgements We thank Joe Wolf for his help with the proof of boundedness of symbols on general domains. The third author thanks the University of Metz for its hospitality during part of this work. References 1. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., Deformation theory and quantization, Lett. Math. Phys. 1, 521-53 (1977). 2. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., Deformation theory and quantization, Ann. Phys. 111, 61-11 (1978). 3. Berezin, F. A., Quantisation of K~hler manifold, Comm. Math. Phys. 4, 153 (1975). 4. Cahen, M., Gutt, S., and Rawnsley, J., Quantization of K~ihler manifolds I: Geometric interpretation of Berezin's quantisation, J. Geom. Phys. 7, 45-62 (199). 5. Cahen, M., Gutt, S., and Rawnsley, J., Quantization of Kfihler manifolds. II, Trans. Amer. Math. Soc. 337, 73-98 (1993). 6. Cahen, M., Gutt, S., and Rawnsley, J., Quantization of K~ihler manifolds. III, Lett. Math. Phys. 3, 291-35 (1994). 7. Harish-Chandra, Representations of Lie groups, IV, Amer. J. Math. 77, 743-777 (1955). 8. Harish-Chandra, Representations of Lie groups, V, Amer. J. Math. 78, 1-41 (1956). 9. Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces, 2nd edn, Academic Press, New York, 1978. 1. Herb, R. A. and Wolf, J. A., Wave packets for the relative discrete series I. The holomorphic case, J. Funct Anal. 73, 1-37 (1987). 11. Satake, I., Factors of automorphy and Fock representations, Adv. in Math. 7, 83-11 (1971). 12. Warner, G., Harmonic Analysis on Semi-simple Lie Groups II, Springer-Verlag, Berlin, Heidelberg, New York, 1972.