Toeplitz Quantization and Asymptotic Expansions: Geometric Construction

For a real symmetric domain GR/KR, with complexification GC/KC, we introduce the concept of ''star-restriction'' (a real analogue of the ''star-products'' for quantization of Kähler manifolds) and give a geometric construction of the GR-invariant differential...

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Дата:	2009
Автори:	Englis, M., Upmeier, H.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2009
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Toeplitz Quantization and Asymptotic Expansions: Geometric Construction / M. Englis, H. Upmeier // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 37 назв. — англ.

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spelling	irk-123456789-1491822019-02-20T01:27:31Z Toeplitz Quantization and Asymptotic Expansions: Geometric Construction Englis, M. Upmeier, H. For a real symmetric domain GR/KR, with complexification GC/KC, we introduce the concept of ''star-restriction'' (a real analogue of the ''star-products'' for quantization of Kähler manifolds) and give a geometric construction of the GR-invariant differential operators yielding its asymptotic expansion. 2009 Article Toeplitz Quantization and Asymptotic Expansions: Geometric Construction / M. Englis, H. Upmeier // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 37 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 32M15; 46E22; 47B35; 53D55 http://dspace.nbuv.gov.ua/handle/123456789/149182 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	For a real symmetric domain GR/KR, with complexification GC/KC, we introduce the concept of ''star-restriction'' (a real analogue of the ''star-products'' for quantization of Kähler manifolds) and give a geometric construction of the GR-invariant differential operators yielding its asymptotic expansion.
format	Article
author	Englis, M. Upmeier, H.
spellingShingle	Englis, M. Upmeier, H. Toeplitz Quantization and Asymptotic Expansions: Geometric Construction Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Englis, M. Upmeier, H.
author_sort	Englis, M.
title	Toeplitz Quantization and Asymptotic Expansions: Geometric Construction
title_short	Toeplitz Quantization and Asymptotic Expansions: Geometric Construction
title_full	Toeplitz Quantization and Asymptotic Expansions: Geometric Construction
title_fullStr	Toeplitz Quantization and Asymptotic Expansions: Geometric Construction
title_full_unstemmed	Toeplitz Quantization and Asymptotic Expansions: Geometric Construction
title_sort	toeplitz quantization and asymptotic expansions: geometric construction
publisher	Інститут математики НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/149182
citation_txt	Toeplitz Quantization and Asymptotic Expansions: Geometric Construction / M. Englis, H. Upmeier // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 37 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT englism toeplitzquantizationandasymptoticexpansionsgeometricconstruction AT upmeierh toeplitzquantizationandasymptoticexpansionsgeometricconstruction
first_indexed	2025-07-12T21:36:00Z
last_indexed	2025-07-12T21:36:00Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 021, 30 pages Toeplitz Quantization and Asymptotic Expansions: Geometric Construction? Miroslav ENGLIŠ †‡ and Harald UPMEIER § † Mathematics Institute, Silesian University at Opava, Na Rybńıčku 1, 74601 Opava, Czech Republic ‡ Mathematics Institute, Žitná 25, 11567 Prague 1, Czech Republic E-mail: englis@math.cas.cz § Fachbereich Mathematik, Universität Marburg, D-35032 Marburg, Germany E-mail: upmeier@mathematik.uni-marburg.de Received October 01, 2008, in final form February 14, 2009; Published online February 20, 2009 doi:10.3842/SIGMA.2009.021 Abstract. For a real symmetric domain GR/KR, with complexification GC/KC, we intro- duce the concept of “star-restriction” (a real analogue of the “star-products” for quantiza- tion of Kähler manifolds) and give a geometric construction of the GR-invariant differential operators yielding its asymptotic expansion. Key words: bounded symmetric domain; Toeplitz operator; star product; covariant quanti- zation 2000 Mathematics Subject Classification: 32M15; 46E22; 47B35; 53D55 1 Introduction Geometric quantization of (complex) Kähler manifolds is of particular interest for symmetric manifolds B = G/K (of compact or non-compact type). In this case the Hilbert state space H carries an irreducible representation of G, whereas the various star products (Weyl calculus, Toeplitz–Berezin calculus) describe the (associative) product of observables (operators on H) as an asymptotic series of G-invariant bi-differential operators on B. In this paper we introduce and study similar concepts for real symmetric manifolds (of flat or non-compact type), emphasizing the interplay between the real symmetric space and its “hermitification” which is a complex hermitian space (of flat or non-compact type). In general, for a real-analytic manifold BR of dimension n, a complexification BC is a complex manifold of (complex) dimension n, with BR embedded (real-analytically) as a totally real submanifold [1, 16, 29]. If BR = GR/KR is a symmetric space, for a real (reductive) Lie group GR with maximal compact subgroup KR, we write its hermitification as BC = GC/KC, where GC denotes the (real, semi-simple) biholomorphic isometry group and KC is the maximal compact subgroup. Thus, contrary to the usual notational conventions, GC is not the complexification of GR but the real Lie group “in the complex setting”. For example, if GR = SU(1, 1) then GC is given by SU(1, 1)× SU(1, 1) instead of SL(2,C); similarly, for GR = SO(1, 1) we have GC = SU(1, 1). On the level of states, the interplay between a real symmetric space BR = GR/KR and its hermitification BC = GC/KC corresponds to a “real-wave” realization of HC via a Segal– Bargmann transformation [37], which is invariant under the subgroup GR ⊂ GC. On the other hand, the real analogue of the star-product is not so obvious. In this paper (and its companion paper [22]) we introduce such a concept, called “star-restriction” for real symmetric domains ?This paper is a contribution to the Special Issue on Deformation Quantization. The full collection is available at http://www.emis.de/journals/SIGMA/Deformation Quantization.html mailto:englis@math.cas.cz mailto:upmeier@mathematik.uni-marburg.de http://dx.doi.org/10.3842/SIGMA.2009.021 http://www.emis.de/journals/SIGMA/Deformation_Quantization.html 2 M. Englǐs and H. Upmeier of non-compact type and study its asymptotic expansion as a series of GR-invariant differential operators. Whereas the paper [22] establishes existence and uniqueness of the asymptotic expan- sion, closely related to spectral theory and harmonic analysis (spherical functions), the current paper gives a “geometric construction” of the differential operators involved, based on a GR- invariant retraction π : BC → BR. We emphasize that our ∗-restriction operator is a GR-equivariant map C∞(BC) → C∞(BR) instead of a map C∞(BR) ⊗ C∞(BR) → C∞(BR) analogous to the usual ∗-products. Thus we do not propose a quantization method for general real symmetric domains (which may not be symplectic nor even dimensional) but instead consider invariant operators which somewhat resemble boundary restriction operators such as Szegö or Poisson kernel integrals. In case BR is the underlying real manifold of a complex hermitian domain B, then both concepts coincide and indeed yield the well-known covariant quantization methods applied to the Kähler manifold B. In order to illustrate the two concepts, consider the simplest non-flat case of the open unit disk B ⊂ C and its real form BR = (−1, 1) ⊂ R. The complexification BC R coincides with B, and we have a restriction operator ρ, mapping a smooth function f on B = BC R to its restriction ρf on BR. A star-restriction is a deformation of the operator ρ, obtained by adding smooth, but non-holomorphic, differential operators on B as higher order terms. In the context of symmetric domains, these differential operators should be invariant under the subgroup GR of the holomorphic automorphism group G of B which leaves BR invariant. Now consider instead the (usual) complex situation. Here B is regarded as a real (symplectic) manifold, denoted by BR, whose complexification BR C is the product of B and its complex conjugate B, with BR embedded as the diagonal. Then a star-product, regarded as a bilinear operator acting on f ⊗ g (with f , g smooth functions on B), is precisely a deformation of the usual product f · g by (G-invariant) bi-differential operators on B or, equivalently, differential operators on BR C = B × B. Since f · g is nothing but the restriction of f ⊗ g to the diagonal BR ⊂ BR C , we see that the concept of star-restriction yields in fact the star-product for the special case where the complexified domain is of product type. The higher-dimensional case is analogous. In order to state our main result concerning the asymptotic expansion (in the deformation parameter ν) of a ∗-restriction operator as above, we first note that for the basic Toeplitz– Berezin calculus (the only case considered in detail here) the ∗-restriction operator is trivial for anti-holomorphic functions so that we may concentrate on the holomorphic part, which is a GR-covariant map ρν : O(BC) → C∞(BR). Using deep facts from representation theory (of the compact Lie groups KR and KC), we con- struct a family of differential operators ρm : O(BC) → C∞(BR) indexed by integer partitions m1 ≥ · · · ≥ mr ≥ 0 (cf. Definition 3.1), and (in Theorem 3.1) express ρν as an asymptotic series ρν ∼ ∑ m 1 [ν]m ρm, (1.1) where the constants [ν]m are generalized Pochhammer symbols. Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 3 Using the Fourier–Helgason transform on BR, it is conceivable (see [22] for the details) that ρν can also be expressed as an oscillatory integral ρνF (x) ∼ ∫ BC F (z) aν(z, x) eνS(z,x) dz, (1.2) where aν is a suitable power series in 1 ν , and the “phase” S is a function on BC ×BR invariant under the diagonal action of GR. This is reminiscent of the WKB-quantization programme of Karasev, Weinstein and Zakrzewski [35], studied extensively in the context of symplectic (i.e. not necessarily Hermitian, or even Riemannian) symmetric spaces by Bieliavsky, Pevzner, Gutt, and other authors, see e.g. [11, 10, 12, 13]. As has already been pointed out above, real symmetric domains need not be symplectic (in fact, they can even be of odd dimension), so neither of the two approaches contains the other, and the situations where they both apply include the original Kähler case of an Hermitian symmetric space. A thorough comparison of both methods is, however, beyond the scope of this paper. For the flat cases of BR = Rd and BR = Cd, the expansions (1.2) were obtained quite explicitly, and for a whole one-parameter class of calculi which includes the Toeplitz calculus, by Arazy and the second author [5]. In Section 4 the asymptotic series (1.1) are computed for the simplest cases of (real or complex) dimension 1. In general, finding explicit formulas may be quite difficult, but there is some hope that at least all symmetric domains of rank 1 (i.e., hyperbolic spaces in Rn, Cn, Hn and the Cayley plane) can be treated in a unified and explicit way. 2 Preliminaries One of the most inspiring examples of deformation quantization is the Berezin quantization [7, 8] using the Berezin transform and Toeplitz operators (originally called co- and contra-variant symbols, respectively). Although it has subsequently been generalized and extended to various classes of compact and noncompact Kähler manifolds [15, 19, 28, 32], the theory is still richest in its original setting of complex symmetric spaces, or bounded symmetric domains, in Cd [9], due to the powerful machinery available from Lie groups and their representation theory on the one hand [26, 34], and from the theory of Jordan triple systems on the other [30]. More specifically, let B = G/K be an irreducible bounded symmetric domain in Cd in the Harish-Chandra realization, with G the identity connected component of the group of all biholomorphic self-maps of B and K the stabilizer of the origin. For ν > p − 1, p being the genus of B, let H2 ν (B) denote the standard weighted Bergman space on B, i.e. the subspace of all holomorphic functions in L2(B, dµν), with dµν(z) = cν K(z, z)1−ν/p dz, where dz stands for the Lebesgue measure, K(z, w) is the ordinary (unweighted) Bergman kernel of B, and cν is a normalizing constant to make dµν a probability measure. The space H2 ν (B) carries the unitary representation U (ν) of G given by U (ν) g f(z) = f(g−1(z)) · Jg−1(z)ν/p, g ∈ G, f ∈ H2 ν (B), where Jg denotes the complex Jacobian of the mapping g. (In general, if ν/p is not an integer, then U (ν) is only a projective representation due to the ambiguity in the choice of the power Jg−1(z)ν/p.) By a covariant operator calculus, or covariant quantization, on B one understands a mapping A : f 7→ Af from functions on B into operators on H2 ν (B) which is G-covariant in the sense that Af◦g = U (ν) g ∗AfU (ν) g , ∀ g ∈ G. 4 M. Englǐs and H. Upmeier In most cases, such calculi can be built by the recipe Af = ∫ B f(ζ)Aζ dµ0(ζ) where dµ0 is a G-invariant measure on B, and Aζ is a family of operators on H2 ν (B) labelled by ζ ∈ B such that Ag(ζ) = U (ν) g AζU (ν) g ∗, ∀ g ∈ G. (One calls such a family a covariant operator field on B. One also usually normalizes the measure dµ0 so that A1 is the identity operator.) Note that in view of the transitivity of the action of G on B, any covariant operator field is uniquely determined by its value A0 at the origin ζ = 0. The best known examples of such calculi are the Toeplitz calculus T and the Weyl calcu- lus W, corresponding to T0 = (·\|1)1 (the projection onto the constants) and W0f(z) = f(−z) (the reflection operator), respectively. In addition to bounded symmetric domains, we will also consider the complex flat case of a Hermitian vector space B = Z ≈ Cd, with B = G/K for G the group of all orientation- preserving rigid motions of Z, and K = U(Z) ≈ Ud(C) the stabilizer of the origin in G; the spaces H2 ν (Z) will then be the Segal–Bargmann spaces of all entire functions which are square-integrable with respect to the Gaussian measure dµν(z) = (ν π )d e−ν‖z‖2 dz, and U (ν) will be the usual Schrödinger representation. In this setting, the Weyl calculus W above is just the well-known Weyl calculus from the theory of pseudodifferential operators [25]. Given a covariant operator calculus A, the associated star product ∗ on functions on B is defined by Af∗g = AfAg. (2.1) It follows from the construction that the star-product is G-invariant in the sense that (f ◦ φ) ∗ (g ◦ φ) = (f ∗ g) ◦ φ ∀φ ∈ G. (2.2) While f ∗ g is a well-defined object for some calculi (e.g. for A = W, at least on Cd and rank one symmetric domains, see [5]), in most cases (e.g. for A = T , the Toeplitz calculus), it makes sense only for very special functions f , g and (2.1) is then usually understood as an equality of asymptotic expansions as the Wallach parameter ν tends to infinity. For instance, for A = T , it was shown in [14] that for any f, g ∈ C∞(B) with compact support, ‖TfTg − T∑N j=0 ν−jCj(f,g)‖ = O ( ν−N−1 ) as ν →∞, for some bilinear differential operators Cj (not depending on f , g and ν). (The as- sumption of compact support can be relaxed [18].) We can thus define f ∗g as the formal power series f ∗ g := ∞∑ j=0 ν−j Cj(f, g). Interpreting ν as the reciprocal of the Planck constant, we recover the Berezin–Toeplitz star product [33], which is the dual to Berezin’s original star-product mentioned above [19]. (This ap- proach to the Berezin and Berezin–Toeplitz star-products, i.e. using covariant operator calculi Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 5 and the definition (2.1), is not the traditional way of constructing the G-invariant Berezin quan- tization, however, for the case of bounded symmetric domains these two are equivalent [20].) Viewing the Planck parameter ν as fixed for the moment, the formula (2.2) means that one can view ∗ as a mapping from the tensor product ∗ : C∞(B ×B) ∼= C∞(B)⊗ C∞(B) → C∞(B), f ⊗ g → f ∗ g, which intertwines the G-action f 7→ f ◦ φ, φ ∈ G, on C∞(B) with the diagonal G-action f⊗g 7→ (f ◦φ)⊗ (g ◦φ) of G on C∞(B×B). This observation can be used as a starting point for extending the whole quantization procedure also to real bounded symmetric domains BR ⊂ Rd, as follows. Suppose ZC is an irreducible hermitian Jordan triple [30, 34] endowed with a (conjugate- linear) involution z 7→ z# which preserves the Jordan triple product and therefore the unit ball BC of ZC, i.e. (BC)# = BC. Define the real forms ZR := {z ∈ ZC : z# = z}, BR := {z ∈ BC : z# = z} = Z ∩BC. For the groups GC := Aut(BC), KC := Aut(ZC) we have the subgroups GR := {g ∈ GC : g(z#) = g(z)#}, KR := {k ∈ KC : kz# = (kz)#} = GR ∩KC acting on BR and ZR, respectively. In this situation ZR is an irreducible real Jordan triple, GR is a reductive Lie group (it may have a nontrivial center), and BR = GR/KR is an irreducible real bounded symmetric domain. Up to a few exceptions, all non-hermitian Riemannian symmetric spaces of non-compact type arise in this way [30, Chapter 11]. A covariant quantization (or covariant extension) on the real bounded symmetric domain BR is a map f 7→ Af from C∞(BR) into H2 ν (BC) such that Af◦g = U (ν) g ∗Af for all g ∈ GR. The counterpart of the star product, associated to a covariant quantization A on BR and a covariant quantization AC on BC, is the star restriction ρ = ρν : C∞(BC) → C∞(BR) defined by AρF = AC F I, (2.3) where I(z) = Iν(z) = K(ν)(z, z#)1/2 is the unique GR-invariant holomorphic function on BC satisfying I(0) = 1. In addition, we will again consider the above construction also in the case of the Segal–Bargmann spaces for an 6 M. Englǐs and H. Upmeier involutive Hermitian vector space ZC ≈ Cd, with the ordinary complex conjugation as the involution z 7→ z#; thus B = ZR ≈ Rd. In most cases, covariant extensions can again be constructed by the recipe Af = ∫ BR f(ζ)Aζ dµ0(ζ), where dµ0 is the GR-invariant measure in BR, and Aζ is a family of holomorphic functions (not necessarily belonging to H2 ν (BC)) labelled by ζ ∈ BR which is covariant in the sense that Ag(ζ) = U (ν) g Aζ , ∀ g ∈ GR, ζ ∈ BR. As before, one usually normalizes dµ0 so that A1 = I. The prime example is now the real Toeplitz calculus A = T corresponding to A0 = 1 (the function constant one) [36, 31, 17, 6, 3]; there is also a notion of real Weyl calculus, but it is more complicated [4]. Here is how the complex hermitian case of a bounded symmetric domain B ⊂ Cd from the beginning of this section can be recovered within the more general real framework. Identify B with the “diagonal” domain BR := {(z, z) : z ∈ B} ⊂ ZR := {(z, z) : z ∈ Z}, where the bar indicates that we consider the “conjugate” complex structure for the second component. The complexifications BR C = {(z, w) : z, w ∈ B} = B ×B, ZR C = {(z, w) : z, w ∈ Z} = Z × Z are endowed with the flip involution (z, w)# := (w, z) having fixed points BR and ZR, respectively. Similarly we can identify G, K with the groups GR := {(g, g) : g ∈ G}, KR := {(k, k) : k ∈ K} which act “diagonally” on BR and ZR, respectively, and whose complexifications GC := {(g1, g2) : g1, g2 ∈ G} = G×G, KC := {(k1, k2) : k1, k2 ∈ K} = K ×K act on BC and ZC, with BC = GC/KC. Since H2 ν (B) is a reproducing kernel space (with reproducing kernel K(ν)(x, y) = h(x, y)−ν , where h(x, y) = [K(x, y)/cp]−1/p is the Jordan determinant polynomial), any bounded linear operator on H2 ν (B) is automatically an integral operator: namely, Tf(z) = ∫ B f(w)T̃ (z, w) dµν(w), with T̃ (z, w) = (T ∗K(ν)(·, z))(w) = (TK(ν)(·, w)\|K(ν)(·, z)). This follows from the identity Tf(z) = (Tf \|K(ν)(·, z)) = (f \|T ∗K(ν)(·, z)). In this way, we may identify operators on H2 ν (B) with (some) functions on B × B, holomorphic in the first and Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 7 anti-holomorphic in the second variable; that is, with holomorphic functions on BC. Upon this identification, the covariant quantization rule f 7→ Af becomes simply a (densely defined) operator f 7→ Ãf from C∞(BR) into the Hilbert space H2 ν (BC) ≈ H2 ν (B)⊗H2 ν (B) corresponding to the Hilbert–Schmidt operators, and the covariance condition means that Ã is equivariant under GR ≈ G, i.e. intertwines the G-action on the former with the diagonal G-action on the latter: Ãf◦g = ( U (ν) g ∗ ⊗ U (ν) g ∗ ) Ãf . Similarly, upon taking AC = A⊗A, and identifying pairs f , g of functions on B with the function F (x, y) = f(x)g(y) on BC, (2.3) reduces just to (2.1). Note, however, that the complexified domain BC is now no longer irreducible, but of “product type”. We will henceforth refer to this situation, i.e. of BR = B, BC = B × B with a bounded symmetric domain B ⊂ Cd, as the “complex” case. To each covariant extension (or quantization) A we can consider its adjoint A∗ from H2 ν (BC) into functions on BR, defined with respect to the inner products in H2 ν (BC) and L2(BR, dµ0). That is, (A∗f \|φ)L2 = (f \|Aφ)ν , ∀φ ∈ L2(BR, dµ0), ∀ f ∈ H2 ν (BC). (2.4) One sometimes calls A∗ a covariant restriction; this should not be confused with the star- restriction ρ, which is a map from C∞(BC) into functions on BR. One can also consider the associated link transform, which is the composition A∗A, a GR- invariant operator on functions on BR. In particular, for the Toeplitz calculus A = T , the link transform T ∗T =: Bν is the Berezin transform, introduced for the ‘complex‘” case in Berezin’s original papers (cf. Sec- tion 4 below). A crucial role in the analysis on complex bounded symmetric domains is played by the Peter– Weyl decomposition of holomorphic functions on B under the composition action f 7→ f ◦ k of the (compact) group K. Namely, the vector space P of all holomorphic polynomials on Cd decomposes under this action into non-equivalent irreducible components P = ∑ m Pm labelled by partitions (or signatures) m ∈ Nr +, that is, by r-tuples of integers m1 ≥ m2 ≥ · · · ≥ mr ≥ 0, where r is the rank of B. With respect to the Fock inner product (p\|q)F := ∫ Cd p(z)q(z)e−\|z\| 2 dz = p(∂)q∗(0), q∗(z) := q(z#), each Peter–Weyl space Pm possesses a reproducing kernel Km(z, w), z, w ∈ Z. It was shown by Arazy and Ørsted [2] that the Berezin transform Bν admits the asymptotic expansion Bν = ∑ m Em (ν)m as ν → +∞, 8 M. Englǐs and H. Upmeier where Em is the G-invariant differential operator on B determined (uniquely) by the requirement that Emf(0) = Km(∂, ∂)f(0), ∀ f ∈ C∞(B); while (ν)m is the multi-Pochhammer symbol (ν)m = r∏ j=1 Γ(ν − a 2 (j − 1) +mj) Γ(ν − a 2 (j − 1)) , a being the so-called characteristic multiplicity of B. Analogously, it was shown in [18] that the star-product (2.1) arising from the Toeplitz calculus A = T admits an expansion f ∗ g = ∑ m Am(f, g) (ν)m , (2.5) where Am are certain (rather complicated) G-invariant (cf. (2.2)) bi-differential operators. The main purpose of the present paper is an extension of the last formula to real symmetric domains. That is, to obtain a decomposition of the star restriction operator ρν = ∑ m ρm [ν]m (2.6) with some GR-invariant differential operators ρm : C∞(BC) → C∞(BR) (independent of ν) and generalized “Pochhammer symbols” [ν]m. A prominent role in our analysis is played by holomorphic polynomials on ZC which are invariant under the group KR. In the Peter–Weyl decomposition under KC mentioned above, partitions n for which Pn contains a nonzero KR-invariant vector are called “even”, and are in one-to-one correspondence with partitions m of length rR = rankBR; furthermore, for each “even” Peter–Weyl space the subspace of KR-invariant vectors is one dimensional, consisting only of multiples of a certain polynomial which (under an appropriate normalization) we denote by Em. For more details, including the description of Em, bibliographic references, etc., as well as for the various preliminaries and notation not introduced here, we refer to [36, 22]. The construction of the decomposition (2.6) for general real symmetric domains is carried out in Section 3. In Section 4 it is shown that the decomposition obtained indeed reduces to (2.5) for the “complex” case. The final Section 5 contains a few examples with more or less explicit formulas for ρm and [ν]m. For the reader’s convenience, we are also attaching a table of all real bounded symmetric domains and their various parameters. In some sense, our results can be perceived as a step towards building a version of Berezin’s quantization for real (as opposed to Kähler) manifolds as phase spaces. 3 Invariant retractions and Moyal restrictions As a first step towards a geometric construction of asymptotic expansions for the Moyal type restriction, we obtain an integral representation for the Moyal restriction operator, defined in terms of GR-invariant retractions π : BC → BR. Here BR is an irreducible real symmetric domain of rank r, in its bounded realization (real Cartan domain) and BC is the open unit ball of the complexification ZC, which is a complex hermitian bounded symmetric domain, not necessarily irreducible [30, 24, 34, 27]. Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 9 We will assume that the preimage π−1(0) of the origin 0 ∈ BR has the form π−1(0) = BC ∩ Y = ΛBR (3.1) for some real vector subspace Y ⊂ BC and real-linear KR-invariant map Λ : ZR → ZC. Our con- struction in fact works even without these assumptions (cf. Remark 3.1 below), but all situations studied in this paper will be of the above form. Let hC : ZC × ZC → C denote the Jordan triple determinant (cf. [30]) of ZC and define the Berezin kernel Bν : BC → C by Bν(z) := hC(z, z)ν/\|hC(z, z])\|ν , (3.2) where z 7→ z] is the involution with real form BR. Note that hC(z, w) 6= 0 for all z, w ∈ BC. Proposition 3.1. The Berezin kernel Bν is GR-invariant, i.e., Bν(gz) = Bν(z) for all g ∈ GR and z ∈ BC. Proof. Since GR ⊂ GC we have hC(gz, gw)ν = jν(g, z) hC(z, w)ν jν(g, w) for all z, w ∈ BC, where jν(g, z) = [det g′(z)]ν/p and p is the (complex) genus of BC. For g ∈ GR we have g(z)] = g(z]) and jν(g, z) = jν(g, z]) since these relations are anti-holomorphic in z ∈ BC and hold for z = z]. It follows that hC(gz, gz)ν \|hC(gz, (gz)])\|ν = hC(gz, gz)ν hC(gz, g(z]))ν/2 hC(g(z]), gz)ν/2 = jν(g, z) hC(z, z)ν jν(g, z) jν/2(g, z) hC(z, z])ν/2 jν/2(g, z]) jν/2(g, z]) hC(z], z)ν/2 jν/2(g, z) = hC(z, z)ν \|hC(z, z])\|ν jν/2(g, z) jν/2(g, z) jν/2(g, z]) jν/2(g, z]) = hC(z, z)ν \|hC(z, z])\|ν . Another proof can be given by observing that, using the familiar transformation rule for hC, hC(gz, gz) \|hC(gz, gz#)\| = hC(z, z)hC(a, a) \|hC(z, a)\|2∣∣∣hC(z, z#)hC(a, a) hC(z, a)hC(a, z#) ∣∣∣ = hC(z, z) \|hC(z, z#)\| ∣∣∣ hC(a, z) hC(a, z#) , where g ∈ GC and a = g−1(0). If g ∈ GR, then gz# = (gz)#, while hC(a, z#) = hC(a#, z) = hC(a, z) (as a# = a) whence \|hC(a, z#)\| = \|hC(a, z)\|. Thus Bν(gz) = Bν(z). � 10 M. Englǐs and H. Upmeier The relationship between the Moyal restriction operator ρν : C∞(BC) → C∞(BR) and the Berezin kernel Bν is given by the following result. Proposition 3.2. For G ∈ O(BC) and F ∈ C∞(BC) we have, if ν is large enough,∫ BR dxhC(x, x) ν−p 2 G(x)(ρν F )(x) = ∫ BC dz hC(z, z)−pBν(z)(G/Iν)(z)F (z), where Iν(z) = hC(z, z])−ν/2. Proof. The Toeplitz restriction map T ∗ R satisfies (T ∗ R G)(x) = hC(x, x)ν/2G(x) = (G/Iν)(x) for all x ∈ BR [36, 6]. Using the duality relation (2.4) and the definition (2.3) of ρν we obtain∫ BR dxhC(x, x) ν−p 2 G(x) (ρν F )(x) = ∫ BR dxhC(x, x)−p/2(G/Iν)(x)(ρνF )(x) = ∫ BR dxhC(x, x)−p/2(T ∗ RG)(x)(ρνF )(x) = (T ∗ RG\|ρνF )BR = (G\|TR ρνF )ν = (G\|TC(F ) Iν)ν = (G\|F · Iν)ν = ∫ BC dz hC(z, z)ν−pG(z)F (z)Iν(z) = ∫ BC dz hC(z, z)ν−p(G/Iν)(z)F (z)\|Iν(z)\|2. Since hC(z, z)ν \|Iν(z)\|2 = Bν(z) the assertion follows. � Corollary 3.1. For G ∈ O(BC) and F ∈ C∞(BC), we have ρν(GF ) = G ρνF . It follows from (3.1) that Y ⊂ ZC is a KR-invariant subspace such that ZC = ZR ⊕ Y (direct sum of real vector spaces). For x ∈ BR, let γx ∈ GR be the “transvection” sending 0 to x, explicitly given by γx(y) = x+B(x, x)1/2(y−x), where B is the Bergman operator and yx = B(y, x)−1(y −Qyx) is the so-called quasi-inverse [30]. Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 11 Lemma 3.1. The mapping Φ : BR × (Y ∩BC) → BC defined by Φ(x, y) = γx(y) (x ∈ BR, y ∈ Y ∩BC) is a real-analytic isomorphism, whose derivative at (0, y) is given by Φ′(0, y)(ξ, η) = ξ + η − {yξy} for all ξ ∈ ZR = Tx(BR), η ∈ Y = Ty(Y ∩BC). Proof. For z ∈ BC, set x := πz and y = γ−xz (= γ−1 x z). Then x ∈ BR while, by the GR- invariance of π, πy = γ−xπz = γ−xx = 0, so y ∈ Y ∩ BC. This proves that Φ is surjective. Similarly, if Φ(x, y) = Φ(x′, y′) for some x, x′ ∈ BR and y, y′ ∈ Y ∩ BC, then x = γx0 = γxπy = πΦ(x, y) = πΦ(x′, y′) = x′ and y = γ−xΦ(x, y) = γ−x′Φ(x′, y′) = y′, showing that Φ is injective. It remains to prove the formula for the derivative. For this, we will use some of the formulas collected in [30, Appendix A1–A3]. For the quasi-inverse Ψ(x, y) = xy we obtain, by definition, Ψ(ξ, y) = B(ξ, y)−1(ξ −Qξ y) and hence (∂1 Ψ)(0, y) ξ = ξ. Using the symmetry formula [30, A3] we obtain Ψ(x, η) = xη = x+Qx(ηx) = x+Qx B(η, x)−1(η −Qη x) and hence (∂2Ψ)(x, 0)η = Qxη. Now the addition formulas [30, A3] yield (x+ ξ)y = xy +B(x, y)−1(ξ(y x)) and hence (∂1Ψ)(x, y)ξ = B(x, y)−1(∂1Ψ)(0, yx)ξ = B(x, y)−1ξ. Similarly, we have x(y+η) = (xy)η and hence, with (JP28) from [30, A2], (∂2Ψ)(x, y)η = (∂2Ψ)(xy, 0)η = Qxyη = B(x, y)−1Qxη. It follows that Ψ′(x, y)(ξ, η) = B(x, y)−1(ξ +Qxη). 12 M. Englǐs and H. Upmeier Since B(x, x)1/2 is an even function of x, its derivative at x = 0 vanishes and we obtain for Φ(x, y) = γx(y) = x+B(x, x)1/2y−x = x+B(x, x)1/2Ψ(y,−x) the derivatives ∂1Φ(0, y)ξ = ξ − ∂2Ψ(y, 0)ξ = ξ −Qyξ and ∂2Φ(0, y)η = ∂1Ψ(y, 0)η = B(y, 0)−1η = η. Therefore Φ′(0, y)(ξ, η) = (∂1Φ)(0, y)ξ + (∂2Φ)(0, y)η = ξ + η −Qyξ. � Corollary 3.2. For all y ∈ Y ∩BC we have det Φ′(0, y) = detZR(I −Qy). Define Pν : C∞(BC) → C∞(BR) by (PνF )(x) := hC(x, x)p/2 ∫ Y ∩BC dy F (γx y)\|det Φ′(x, y)\| · hC(γxy, γxy)−pBν(y) = hC(x, x)p/2 ∫ Y ∩BC dy F (γx y)\|det Φ′(x, y)\|hC(γxy, γxy)ν−p\|hC(γxy, (γxy)])\|−ν for all F ∈ C∞(BC) and x ∈ BR. Here Φ′(x, y) is the derivative of Φ at (x, y) ∈ BR × (Y ∩BC). If f ∈ C∞(BR), then f ◦ π ∈ C∞(BC) and (f ◦ π)(γxy) = f(γxπ(y)) = f(γx0) = f(x). It follows that Pν((f ◦ π)F ) = f · (PνF ), (3.3) i.e. Pν behaves like a “conditional” expectation. Proposition 3.3. For F ∈ C∞(BC) we have∫ BR dxhC(x, x)−p/2(PνF )(x) = ∫ BC dz hC(z, z)−pBν(z)F (z). Proof. The change of variables z = γx(y) = Φ(x, y) yields in view of the invariance of Bν∫ BC dz hC(z, z)−pBν(z)F (z) = ∫ BR dx ∫ Y ∩BC dy \|det Φ′(x, y)\|hC(γx y, γxy)−pBν(y)F (γxy) = ∫ BR dxhC(x, x)−p/2(PνF )(x). � Corollary 3.3. The operator Pν is GR-invariant, i.e., we have Pν(F ◦ g) = (PνF ) ◦ g for all F ∈ C∞(BC) and g ∈ GR. Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 13 Proof. Let f ∈ C∞(BR) be arbitrary. Using (3.3) and the GR-invariance of Bν and π, we obtain∫ BR dxhC(x, x)−p/2f(gx)(PνF )(gx) = ∫ BR dxhC(x, x)−p/2Pν((f ◦ π)F )(gx) = ∫ BR dxhC(x, x)−p/2Pν((f ◦ π)F )(x) = ∫ BC dz hC(z, z)−pBν(z)(f ◦ π)(z)F (z) = ∫ BC dz hC(z, z)−pBν(gz)(f ◦ π)(gz)F (gz) = ∫ BC dz hC(z, z)−pBν(z)(f ◦ g)(π(z))(F ◦ g)(z) = ∫ BR dxhC(x, x)−p/2Pν(((f ◦ g) ◦ π)(F ◦ g))(x) = ∫ BR dxhC(x, x)−p/2(f ◦ g)(x)Pν(F ◦ g)(x). � For x = 0 ∈ BR we have in particular (PνF )(0) = ∫ Y ∩BC dy \|det Φ′(0, y)\|Bν(y)F (y)hC(y, y)−p. (3.4) Our next goal is to obtain an asymptotic expansion of (3.4), as ν → ∞, using the method of stationary phase but also the more refined “KR-invariant” Taylor expansion of F at 0 ∈ Y . As a first step we recall that Y = ΛZR = {Λx : x ∈ ZR}, for an R-linear (but not necessarily C-linear) isomorphism Λ : ZC → ZC which commutes with KR. For f ∈ C∞(BR) we have f ◦ Λ−1 ∈ C∞(Y ∩BC). Consider the distribution f 7→ Pν ( f ◦ Λ−1 ) (0) (3.5) on BR, which by construction is KR-invariant. For any partition m ∈ Nr + let Em R be the KR- invariant constant coefficient differential operator on ZR corresponding to the polynomial Em introduced in Section 2. Using multi-indices κ ∈ Nd we may write Em(x) = ∑ κ cmκ x κ, Em R = ∑ κ cmκ ∂ κ R , where ∂κ R is the “real” partial derivative operator on ZR associated with κ and \|κ\| ≤ \|m\|. Ex- pressing ∂κ R in terms of Wirtinger type derivatives ∂σ C, ∂τ C on ZC, for multi-indices σ, τ ∈ Nd, such that \|σ\| ≤ \|m\| ≥ \|τ \|, Em R determines a complexified constant coefficient differential operator Em C = ∑ σ,τ cmσ,τ∂ σ C∂ τ C for suitable constants cmσ,τ ∈ C. Pulling back by the (real-linear) map Λ we get ∂σ C∂ τ C(F ◦ Λ) = ∑ α,β Λσ,τ α,β(∂α C∂ β CF ) ◦ Λ for suitable constants Λσ,τ α,β ∈ C, and hence Em C (F ◦ Λ) = ∑ σ,τ cmσ,τ∂ σ C∂ τ C(F ◦ Λ) = ∑ σ,τ cmσ,τ ∑ α,β Λσ,τ α,β(∂α C∂ β CF ) ◦ Λ = ∑ α,β Pm α,β(∂α C∂ β CF ) ◦ Λ, 14 M. Englǐs and H. Upmeier where Pm α,β = ∑ σ,τ cmσ,τ Λσ,τ α,β. (3.6) Returning to the distribution (3.5) on BR, one has Proposition 3.4. There exist unique constants [ν]m, for m ∈ Nr +, such that for all F ∈ C∞(BC) (Pν F )(0) ∼ ∑ m 1 [ν]m Em C (F ◦ Λ)(0) (3.7) as an asymptotic expansion. Proof. By the definition of Y , the real-linear operator y 7→ y# from Y into ZC is injective, and thus bounded below. It follows that also the (GC-invariant) pseudohyperbolic distance ρ(y, y#) := ‖γy(y#)‖, y ∈ BC, is bounded below by a multiple of ‖y− y#‖ if y ∈ Y . Since, by the familiar transformation rule for the Jordan determinant hC, Bν(z)2 = h(z, z)νh(z#, z#)ν \|h(z, z#)\|2ν = h(γzz #, γzz #)ν and h(w,w) ≤ 1 on the closure of BC, with equality if and only if w = 0, it follows that Bν has a global maximum on Y at y = 0, which also dominates the boundary values of Bν in the sense that Bν(yk) → 1, yk ∈ Y , implies that yk → 0. We may therefore apply the method of stationary phase exactly as in Section 3 of [22] to conclude that for any F ∈ C∞(BC), for which the right-hand side exists for some ν > p− 1, the integral PνF (0) = ∫ Y F (y)\|det Φ′(0, y)\|Bν(y)hC(y, y)−p/2 dy = \|det Λ\| ∫ BR F (Λx)\|det Φ′(0,Λx)\|Bν(Λx)hC(Λx,Λx)−p/2 dx has an asymptotic expansion as ν → +∞ PνF (0) ∼ ν−d/2 ∑ k≥0 Sk(∂R)(F ◦ Λ)(0)ν−k for some constant coefficient differential operators Sk(∂R), with Sk polynomials on ZR. Since Pν is KR-invariant, so must be the Sk; thus they admit a decomposition Sk = ∑ \|m\|≤k qkmE m, qkm ∈ C, into the “even” Peter–Weyl components Em. Interchanging the two summations and setting 1 [ν]m := ν−d/2 ∑ k qkmν −k, the claim follows. � Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 15 Using the transvections γx ∈ GR, for x ∈ BR, we define a GR-invariant differential operator Pm : C∞(BC) → C∞(BR) by putting Pm(F )(x) := Em C (F ◦ γx ◦ Λ)(0) = ∑ α,β Pm α,β∂ α C∂ β C(F ◦ γx)(0). (3.8) Since γx : BC → BC is holomorphic, there exist smooth functions γα ι : BR → C, with \|ι\| ≤ \|α\|, such that ∂α C(H ◦ γx)(0) = ∑ ι γα ι (x)(∂ ι CH)(x) for all H ∈ O(BC) and x ∈ BR. Since Pν is GR-invariant, Proposition 3.4 implies (Pν F )(x) ∼ ∑ m 1 [ν]m (Pm F )(x) (3.9) for all F ∈ C∞(BC) and x ∈ BR. Now let m ∈ Nr + and κ ∈ Nd be fixed, with \|κ\| ≤ \|m\|. Define a (non-invariant) “holomor- phic” differential operator Pm κ : O(BC) → C∞(BR) by the formula (Pm κ H)(x) = ∑ α,β Pm α,β∂ α C(H ◦ γx)(0)γβ κ(x) = ∑ α,β,ι Pm α,βγ α ι (x)γβ κ(x) (∂ ι CH)(x) (3.10) for all x ∈ BR and H ∈ O(BC), where the constants Pm α,β are defined by (3.6). Lemma 3.2. Let G,H ∈ O(BC). Then Pm(GH)(x) = ∑ κ (Pm κ H)(x)∂κ CG(x). Proof. Since γx preserves holomorphy, (3.10) implies Pm(GH)(x) = ∑ α,β Pm α,β∂ α C∂ β C(G ◦ γx(H ◦ γx))(0) = ∑ α,β Pm α,β∂ α C(H ◦ γx)(0)∂β C(G ◦ γx)(0) = ∑ α,β,ι,κ Pm α,βγ α ι (x)(∂ ι CH)(x)γβ κ(x)(∂κ CG)(x) = ∑ α,β,ι,κ Pm α,βγ α ι (x)γβ κ(x)(∂ ι CH)(x)(∂κ CG)(x) = ∑ κ (Pm κ H)(x)∂κ CG(x). � Definition 3.1. For m ∈ Nr +, the m-th Moyal component is the differential operator ρm : O(BC) → C∞(BR) defined by the formula (ρmH)(x) = hC(x, x)p/2 ∑ κ (−1)κ∂κ R (h−p/2 C Pm κ H)(x) (3.11) for all x ∈ BR and H ∈ O(BC). Here ∂κ R is the “real” partial derivative operator on BR ⊂ ZR, and (−1)κ ∂κ R is its (Euclidean) adjoint. We also write just hC for hC(x, x). 16 M. Englǐs and H. Upmeier Proposition 3.5. Let G,H ∈ O(BC). Then∫ BR dxhC(x, x) ν−p 2 G(x)(ρmH)(x) = ∫ BR dxhC(x, x)−p/2Pm(G/IνH)(x). (3.12) Proof. Since G is holomorphic, we have ∂κ C G(x) = ∂κ R G(x) for all κ ∈ Nd and x ∈ BR. Applying Lemma 3.2 to G/Iν we obtain∫ BR dxhC(x, x) ν−p 2 G(x)(ρmH)(x) = ∫ BR dxhC(x, x)−p/2(G/Iν)(x)(ρmH)(x) = ∑ κ ∫ BR dx (G/Iν)(x)(−1)κ ∂κ R (h−p/2 C Pm κ H)(x) = ∑ κ ∫ BR dx ∂κ R (G/Iν)(x)hC(x, x)−p/2(Pm κ H)(x) = ∑ κ ∫ BR dx ∂κ C (G/Iν)(x)hC(x, x)−p/2(Pm κ H)(x) = ∫ BR dxhC(x, x)−p/2Pm(G/IνH)(x). � As a consequence of Proposition 3.5 we obtain Corollary 3.4. The differential operators ρm are GR-invariant, i.e., ρm(H ◦ g)(x) = (ρmH)(g(x)) for all H ∈ O(BC), g ∈ GR and x ∈ BR. Proof. Replacing G/Iν = Φ, (3.12) can be written as∫ BR dxhC(x, x)−p/2Φ(x)(ρmH)(x) = ∫ BR dxhC(x, x)−p/2Pm(ΦH)(x) for Φ,H∈O(BC). Since Pm is GR-invariant by construction (cf. (3.8)), the assertion follows. � The main result of this section yields the desired asymptotic expansion of the Moyal type restriction operator ρν in terms of the invariant differential operators ρm: Theorem 3.1. For H ∈ O(BC) we have an asymptotic expansion ρνH ∼ ∑ m 1 [ν]m ρmH, where ρm : O(BC) → C∞(DR) are GR-invariant holomorphic differential operators independent of ν and the constants [ν]m are determined by (3.7). Proof. Let G,H ∈ O(BC). Applying Proposition 3.2 and 3.3, we obtain with (3.9) and (3.12)∫ BR dxhC(x, x) ν−p 2 G(x)(ρνH)(x) = ∫ BC dz hC(z, z)−pBν(z)(G/Iν)(z)H(z) = ∫ BR dxhC(x, x)−p/2Pν(G/IνH)(x) = ∑ m 1 [ν]m ∫ BR dxhC(x, x)−p/2Pm (G/IνH)(x) = ∑ m 1 [ν]m ∫ BR dxhC(x, x) ν−p 2 G(x)(ρmH)(x). Since G ∈ O(BC) is arbitrary, the assertion follows. � Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 17 Remark 3.1. Most – probably all – of the above extends also to the case of general GR-invariant smooth retractions π : BC → BR, i.e. when π−1(0) is not necessarily an intersection of BC with some real subspace Y , or that the parameterization Λ : BR → π−1(0) is not necessarily linear but only smooth. In fact, the application of the stationary phase method in the proof of Proposition 3.4 involves only the germs of F and π−1(0) (or, equivalently, Λ) at the origin. Thus we may replace the variety π−1(0) by its tangent space at 0 ∈ π−1(0), and Λ : BR → π−1(0) by its differential at the origin. We omit the details. 4 Asymptotic expansion: the complex case In the complex case, where BC = B ×B = {(z, w) : z, w ∈ B}, BR = {(z, z) : z ∈ B} and B is an irreducible complex Hermitian bounded symmetric domain (of rank r), the Moyal type restriction operator ρν : C∞(BC) = C∞(B)⊗C∞(B) → C∞(BR) can be identified with the Moyal type (star-) product ]ν via the formula ρν(f ⊗ g) = f ]ν g for all f, g ∈ C∞(B). In this case an asymptotic expansion has been constructed in [18], and here we show that the general construction described in Section 3 yields precisely the expansion of [18]. This is not completely obvious, since the construction in [18] is based on the complex structure of B whereas the general construction of Section 3 uses the “real” structure of BR. The first step is to identify the Berezin kernel Bν on BC, defined in (3.2), for the complex case. We have hC((z, w), (ζ, ω)) = h(z, ζ)h(ω,w) for z, w, ζ, ω ∈ B and the involution is given by (z, w)] := (w, z). Therefore Bν(z, w) = hC((z, w), (z, w))ν \|hC((z, w), (w, z))\|ν = h(z, z)νh(w,w)ν h(z, w)ν h(w, z)ν coincides with the integral kernel for the G-invariant Berezin transform Bν : C∞(B) → C∞(B) on B. This is clearly invariant under GR = {(g, g) : g ∈ G}, where G = Aut (B). The construction in [18] starts with the asymptotic expansion (Bνf)(0) = ∫ B dz h(z, z)ν−pf(z) = ∑ m 1 (ν)m (Em R f)(0) 18 M. Englǐs and H. Upmeier of the ν-Berezin transform Bν associated with the usual Toeplitz–Berezin quantization of B. Here, for any partition m ∈ Nr +, the Pochhammer symbol (ν)m := ΓΩ(ν + m) ΓΩ(ν) is defined via the Koecher–Gindikin Γ-function, and the “sesqui-holomorphic” constant coeffi- cient differential operators Em R are defined via the Fock space expansion e(z\|w) = ∑ m Em(z, w) for all z, w ∈ Z. In multi-index notation, Em(z, w) = ∑ α,β cmα,βz αwβ, Em R = ∑ α,β cmα β∂ α ∂β (4.1) for suitable constants cmα β and multi-indices α, β ∈ Nd, such that \|α\| ≤ \|m\| ≥ \|β\|. Since Em(z, w) = Em(w, z) it follows that cmα β = cmβ α. (4.2) Passing to the complexification ZC = Z × Z, with variables (z, w) for z, w ∈ Z, we use pairs of multi-indices and write ∂αβ C and ∂γδ C for the associated Wirtinger derivatives. Thus, for functions on BC of the form (f ⊗ g)(z, w) = f(z)g(w), (4.3) we have ∂αβ C ∂ γδ C (f ⊗ g) = (∂α∂γf)⊗ (∂β∂δg). (4.4) Note that the first and second variable are treated differently, since holomorphic functions on BC correspond to the case where f is holomorphic and g is anti-holomorphic. Let Λ : BR → BC denote the R-linear mapping Λ(z, z) = (z, 0) which is clearly KR-invariant. Consider the GR-invariant retraction π : BC → BR defined by π(z, w) := (w,w). Then π−1(0) = {(z, 0) : z ∈ B} = ΛBR. Lemma 4.1. For F ∈ C∞(BC) we have Em C (F ◦ Λ)(0) = ∑ α,β cmα β∂ α0 C ∂β0 C F (0). Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 19 Proof. We may assume that F (z, w) = f(z)g(w) is of the form (4.3) . Since ((f ⊗ g) ◦ Λ)(z, z) = (f ⊗ g)(z, 0) = f(z)g(0) it follows from (4.4) and (4.1) that Em C ((f ⊗ g) ◦ Λ)(0) = (Emf)(0)g(0) = ∑ α,β cmα β(∂α∂βf)(0)g(0) = ∑ α,β cmα β(∂α 0 C ∂β 0 C (f ⊗ g))(0). � Comparing with the coefficients Pm α,β introduced by (3.6) in the general case, it follows that Pm α0,β0 = cmα β (4.5) for α, β ∈ Nd, whereas all other such coefficients vanish. This reflects the fact that Λ is trivial on the second component. For z ∈ B, let as before γz ∈ G be the transvection mapping 0 to z. Then we have for α ∈ Nd and f ∈ O(B) ∂α(f ◦ γz)(0) = ∑ ι≤α γα ι (z)(∂ ιf)(z), where γα ι are smooth functions on B. As in [18] define a G-invariant operator Em : C∞(B) → C∞(B) by putting (Em f)(z) := Em R (f ◦ γz)(0). Then we have for f, g ∈ O(B) (Em(fg))(z) = Em R ((fg) ◦ γz)(0) = Em R ((f ◦ γz)g ◦ γz)(0) = ∑ α,β cmα β∂ α∂β((f ◦ γz)g ◦ γz)(0) = ∑ α,β cmα β∂ α(f ◦ γz)(0)∂β(g ◦ γz)(0) = ∑ α,β ∑ κ,ι cmα βγ α κ(z)(∂κf)(z)γβ ι (z)(∂ ιg)(z). (4.6) Following [18, Section 4] one defines (non-invariant) differential operators Rm κ , for any partition m ∈ Nr + and any multi-index κ ∈ Nd with \|κ\| ≤ \|m\|, via the expansion (Em(fg))(z) = ∑ κ (∂κf)(z)(Rm κ g)(z), where f ∈ O(B), g ∈ C∞(B). Comparing with (4.6) it follows that (Rm κ g)(z) = ∑ α,β ∑ ι cmα βγ α κ(z)γβ ι (z)(∂ ιg)(z) whenever g is holomorphic. On the other hand, putting γz,z := (γz, γz) ∈ GR ⊂ G×G 20 M. Englǐs and H. Upmeier we have for the “holomorphic” Wirtinger derivatives ∂α β C [(f ⊗ g) ◦ γz,z](0, 0) = ∂α β C [(f ◦ γz)⊗ g ◦ γz](0, 0) = ∂α(f ◦ γz)(0)∂β(g ◦ γz)(0) = ∑ κ,ι γα κ(z)(∂κf)(z)γβ ι (z) ∂ ιg(z) = ∑ κ,ι γα κ(z)γβ ι (z)∂κ ι C (f ⊗ g)(z, z). (4.7) We will now compute the (non-invariant) operators Pm κ , introduced in (3.10), for the complex case. Combining (4.5) and (4.7) it follows that the non-zero operators correspond to multi-index pairs (κ, 0) for κ ∈ Nd and, in view of (4.2), Pm κ0(f ⊗ g)(z, z) = ∑ α,β,ι Pm α0, β0γ α ι (z)γβ κ(z)∂ι0 C (f ⊗ g)(z, z) = ∑ α,β,ι cmα βγ α ι (z)γβ κ(z)(∂ ιf)(z) g(z) = (Rm κ f)(z)g(z). This passing to the complex conjugate (also in the proof of the following Proposition) could be avoided by working with the “anti-holomorphic” second variable instead. The G-invariant bi-differential operators Am on B, introduced in [18, Section 4], satisfy Am(f, g)(z) = ∑ κ h(z, z)p(−∂)κ(h−p f(Rm κ g))(z) for all f, g ∈ O(B), and are uniquely determined by this property since Am involves only holomorphic derivatives in the first variable and anti-holomorphic derivatives in the second variable. By [18, Proposition 6], Am(f, g) = Am(g, f) for all f, g ∈ O(B). Proposition 4.1. Let f, g ∈ O(B). Then ρm(f ⊗ g)(z, z) = Am(f, g)(z) for all z ∈ B. Proof. Since the operators ρm are defined by taking suitable adjoints on BR, which requires another identification, we instead verify that both operators satisfy the same integral duality formula. Thus let f, g, φ, ψ ∈ O(B). Then, by (3.11),∫ B dz h(z, z)−pφ(z) ψ(z)ρm(f ⊗ g)(z, z) = ∫ BR d(z, z)hC((z, z), (z, z))−p/2(φ⊗ ψ)(z, z)ρm(f ⊗ g)(z, z) = ∑ κ ∫ BR d(z, z)hC((z, z), (z, z))−p/2∂κ0 C (φ⊗ ψ)(z, z)Pm κ0(f ⊗ g)(z, z) = ∑ κ ∫ B dz h(z, z)−p(∂κφ)(z) ψ(z)g(z)(Rm κ f)(z) = ∑ κ ∫ B dz h(z, z)−p(∂κφ)(z)ψ(z)g(z)(Rm κ f)(z) = ∑ κ ∫ B dz φ(z)(−∂)κ(h−pψg(Rm κ f))(z) = ∑ κ ∫ B dz φ(z)ψ(z)(−∂)κ(h−pg(Rm κ f))(z) Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 21 = ∫ B dz h(z, z)−pφ(z)ψ(z)Am(g, f)(z) = ∫ B dz h(z, z)−pφ(z)ψ(z)Am(f, g)(z). Since φ, ψ ∈ O(B) are arbitrary, the assertion follows. Note that the formula (3.11) defining ρm uses the real derivatives ∂R, whereas in this section we are using rather the Wirtinger deriva- tives ∂ and ∂ on B (corresponding to viewing BR = B as a domain in Cd rather than R2d); this is reflected by the appearance of the Hermitian adjoint −∂κ0 (rather than −∂κ0) of ∂κ0 on the third line in the computation above. � 5 Examples We begin with the case of the Euclidean space where everything can be computed explicitly. Example 5.1. Let BR = R, so that BC = C, and Λx := εx for some ε ∈ C \ R, \|ε\| = 1. The corresponding retraction π is just the oblique R-linear projection associated to the direct sum decomposition C = R⊕ εR; the mapping φ is just Φ(x, y) = x + y, and detΦ′ = 1. The role of the Jordan determinant polynomial hC(x, y) is played by the function e−xy, x, y ∈ C, (5.1) and the “genus” p = 0 while “rank” r = 1. The partitions are just nonnegative integers m = (m), and the polynomials Em are simply Em(x) = x2m (2m)! . Thus Em C = (∂ + ∂)2m/(2m)!, and Pm α,β =  εα−β α!β! if α+ β = 2m, 0 otherwise. (5.2) The “transvections” γx are just the ordinary translations γxy = x + y, which implies that the functions γα ι equal constant one if α = ι, and vanish otherwise. Feeding all this information into (3.10) and (3.11), we get Pm κ = ε2m−2κ (2m− κ)!κ! ∂2m−κ and, for H ∈ O(C), ρmH = 1 (2m)! 2m∑ κ=0 ( 2m κ ) ε2m−2κ(−1)κ∂κ R (∂2m−κH) = (ε− ε)2m (2m)! ∂2mH. We next compute the “Pochhammer” symbols [ν]m, using the formula (3.4). By (5.1), PνF (0) = ∫ εR F (y) e−ν\|y\|2 \|e−y2 \|ν dy = ∫ R F (εy)e−νy2(1−Re ε2) dy. 22 M. Englǐs and H. Upmeier Denoting for brevity 1 − Re ε2 = −1 2(ε − ε)2 =: c > 0 and making the change of variable y = x/ √ cν yields PνF (0) = 1√ cν ∫ R F ( ε√ cν x ) e−x2 dx. We may assume that F is holomorphic; replacing then F by its Taylor expansion, integrating term by term (which is easily justified), and using the fact that ∫ R x 2je−x2 dx = Γ(j + 1 2), we finally arrive at PνF (0) = 1√ cν ∞∑ j=0 ( ε√ cν )2j F (2j)(0) (2j)! Γ(j + 1 2). As Em C (F ◦ Λ)(0) = ε2j (2j)!F (2j)(0), we thus get 1 [ν]m = Γ(m+ 1 2) (cν)m+ 1 2 = (2m)!Γ(1 2) m!4m(cν)m+ 1 2 (5.3) where the last equality used the doubling formula for the Gamma function. This corresponds to the unnormalized Lebesgue measure on C; it is usual to make a normal- ization so that ρν1 = 1, i.e. [ν](0) = 1. If this is done then (5.3) gets divided by the same thing with m = 0, that is, it becomes, 1 [ν]m = Γ(m+ 1 2) Γ(1 2)(cν)m = (2m)! (ε− ε)2mνmm!(−2)m . Note that even though both ρm and [ν]m depend on ε, the sum ρν = ∑ m ρm [ν]m = ∞∑ m=0 ∂2m m!(−2)mνm = e−∂2/2ν is independent of it, as it should. A similar analysis can be done for BR = Rd, d > 1; cf. the next example. Example 5.2. BR = Cd ∼= R2d, so that BC = Cd × Cd, where as always we identify BR with {(z, z) : z ∈ Cd} ⊂ BC. For Λ, we let Λz = (z, az) (5.4) with some fixed a ∈ C, a 6= 1. The retraction π is the oblique real-linear projection associated to the direct sum decomposition Cd × Cd = ΛCd ⊕ Λ1Cd, where Λ1 is as in (5.4) but with a = 1. Using again the Wirtinger derivatives ∂, ∂ rather than ∂R on Cd ∼= R2d, we have for any partition m = (m) Em R = ∑ \|β\|=m ∂β∂β β! with the usual multi-index notation. Recalling that the numbers Pm ρ,σ are, quite generally, defined by∑ ρ,σ Pm ρ,σ∂ ρ C∂ σ CF (0) = Em R (F ◦ Λ)(0), (5.5) Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 23 it follows that Pm αβ,γδ =  ρ! α!β!γ!δ! a\|δ\|a\|β\| if α+ δ = β + γ = ρ, \|ρ\| = m, 0 otherwise. (5.6) (Here we are again using the “double” Wirtinger derivatives ∂αβ C etc. as in Section 4.) As in the preceding example, the role of the “Jordan determinant” hC is played by the function hC((z, w), (z1, w1)) = e−(z\|z1)−(w1\|w) (5.7) and p = 0. Taking H ∈ O(BC) of the form H(z, w) = f(z)g(w) with f, g ∈ O(Cd), we get as in the preceding example ρmH = ∑ κ,λ (−∂)κ(−∂)λ ∑ α,β,γ,δ ∑ ι,η Pm αβ,γδ γαβ ιη γ γδ κλ ∂ιηH = ∑ κ,λ (−∂)κ(−∂)λ ∑ α,β Pm αβ,κλ ∂αβH = ∑ α,β,κ,λ (−1)\|κ+λ\|Pm αβ,κλ ∂α+λf∂β+κg = ∑ \|ρ\|=m ∑ β,λ≤ρ (−1)\|ρ−β+λ\| ( ρ β )( ρ λ ) a\|λ\|a\|β\| ρ! ∂ρf∂ρg = ∑ \|ρ\|=m (−1)\|ρ\| ρ! \|1− a\|2\|ρ\|∂ρf∂ρg = (−1)m m! \|1− a\|2m ( d∑ j=1 ∂zj∂wj )m H. (Here the appearance of (−∂)κ(−∂)λ, rather than (−∂)κ(−∂)λ, is for the same reason as indi- cated at the end of the proof of Proposition 4.1.) Thus, symbolically, ρm = (−1)m m! \|1− a\|2m(∂ ⊗ ∂)m. To compute [ν]m, we again start from (3.4). Observe that for the function F ∈ O(BC) given by F (z, w) = zαzβwγwδ, where α+ γ = β + δ = ρ, we have by (5.5) Em R (F ◦ Λ)(0) = α!β!γ!δ!Pm αβ,γδ . Hence PνF (0) = ∑ m 1 [ν]m Em C (F ◦ Λ)(0) = ρ! [ν]\|ρ\| a\|γ\|a\|δ\|. On the other hand, from (3.4) and (5.7), PνF (0) = ∫ Cd F (z, az)e−ν\|z\|2\|1−a\|2 dz = a\|γ\|a\|δ\| ∫ Cd zρzρe−ν\|1−a\|2\|z\|2 dz = ρ! (\|1− a\|2ν)\|ρ\| a\|γ\|a\|δ\|, provided dz is normalized so that [ν]0 = 1. Thus (under this normalization) [ν]m = νm\|1− a\|2m. 24 M. Englǐs and H. Upmeier Note that, again, ρν = ∑ m ρm [ν]m = ∞∑ m=0 (−1)m m!νm (∂ ⊗ ∂)m = e−∂⊗∂/ν does not depend on a, even though ρm and [ν]m both do. Example 5.3. As a first “non-flat” situation, consider the unit interval BR = (−1,+1) with complexification BC = D, the unit disc in C; and we take the same Λ as in Example 5.1, i.e. Λx = εx, ε ∈ T \ R. The constants Pm α,β are thus still given by (5.2), and hC(x, y) = 1− xy while p = 2. Thus for H ∈ O(D), ρmH(x) = ( 1− x2 ) ∑ κ (−1)κ ( d dx )κ ( 1 1− x2 ∑ α,β,ι α+β=2m εα−β α!β! γα ι (x)γβ κ(x)∂ιH(x) ) . This time explicit formulas are hard to come by, since the expressions γα ι (x) are quite compli- cated. One has, of course, ρ(0)H = H, while ρ(1)H(x) = (ε− ε)2 [( 1− x2 )2 H ′′(x)− 2x ( 1− x2 ) H ′(x) ] = (ε− ε)2(H ◦ γx)′′(0) is the GR-invariant operator uniquely determined by ρ(1)H(0) = (ε−ε)2H ′′(0). Computer-aided calculation similarly gives ρ(2)H(0) = 24(ε− ε)2H ′′(0) + (ε− ε)4H(4)(0), ρ(3)H(0) = 1080(ε− ε)2H ′′(0) + 120(ε− ε)4H(4)(0) + (ε− ε)6H(6)(0). The leading coefficient in ρ(m)H(0) is always m2(2m− 1)!. To compute [ν]m, noting that detΦ′(0, y) = 1 − y2 by Corollary 3.2, we get from (3.4) and (3.7),∫ 1 −1 F (εx)\|1− ε2x2\| ( 1− x2 \|1− ε2x2\| )ν dx (1− x2)2 ∼ ∑ m (ε∂ + ε∂)2mF (0) (2m)![ν]m . Denoting F (εx) =: f(x) yields∫ 1 −1 f(x) ( 1− x2 \|1− ε2x2\| )ν−1 dx 1− x2 ∼ ∑ m f (2m)(0) (2m)![ν]m . Taking in particular f(x) = x2m we obtain 1 [ν]m = ∫ 1 −1 x2m (1− x2)ν−2 \|1− ε2x2\|ν−1 dx = ∫ 1 0 tm− 1 2 (1− t)ν−2 \|1− ε2t\|ν−1 dt. (5.8) Writing 1 \|1− ε2t\|ν−1 = (1− ε2t)−(ν−1)/2(1− ε2t)−(ν−1)/2 = ∞∑ j,k=0 (ν−1 2 )j(ν−1 2 )k j!k! ε2(j−k)tj+k we arrive at the double series 1 [ν]m = Γ(m+ 1 2)Γ(ν − 1) Γ(m+ ν − 1 2) ∑ j,k≥0 (ν−1 2 )j(ν−1 2 )k j!k! (m+ 1 2)j+k (m+ ν − 1 2)j+k ε2(j−k). Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 25 The double sum on the right-hand side is the value at x = ε, y = ε of the Horn hypergeometric function of two variables [23, § 5.7.1] F1 ( m+ 1 2 , ν−1 2 , ν−1 2 ,m+ ν − 1 2 , x, y ) and in general cannot be evaluated in closed form. For particular values of ε, there may be some simplifications; for instance, for ε = i the integral (5.8) becomes 1 [ν]m = ∫ 1 0 tm− 1 2 (1− t)ν−2(1 + t)1−ν dt = Γ(m+ 1 2)Γ(ν − 1) Γ(m+ ν − 1 2) 2F1 ( m+ 1 2 , ν − 1;m+ ν − 1 2 ;−1 ) , where 2F1 is the ordinary Gauss hypergeometric function. We remark that expressions involving values of 2F1 at −1 occur as eigenvalues of the Berezin (or “link”) transform corresponding to the Weyl calculus on rank 1 real symmetric spaces, cf. [5, Theorem 4.1]. (Also, Horn’s hypergeometric functions of another kind – namely, Φ2 in the notation of [23] – appear in the formula for the harmonic Segal–Bargmann kernel on Cd, see [21]; it is however unclear if there is any deeper relationship.) Example 5.4. In this final example we consider BR = D, embedded in BC = D×D in the usual way as {(z, z) : z ∈ D}. For Λ we take the same map Λz = (z, az) as in Example 5.2, with some fixed a ∈ C, a 6= 1. The corresponding retraction π : BC → BR assigns to (z, w) ∈ D × D the (unique) point x ∈ D such that γxw = aγxz. (The existence of such x follows by the following argument. For any z, w, u, v ∈ D, the existence of g ∈ G such that gz = u, gw = v is equivalent to the equality ρ(z, w) = ρ(u, v) (5.9) of the pseudohyperbolic distances ρ(u, v) := \| u−v 1−uv \|. On the other hand, if u runs through the interval [0,min{1, 1 \|a\|}) and v = au, then ρ(u, v) runs from 0 to 1; thus (5.9) holds for some u. With g as above, take x = −g−1(0).) The constants Pm αβ,γδ are then still given by the formula (5.6) from Example 5.2, while the corresponding functions γαβ ιη are easily seen to be given by γαβ ιη (z, w) = γα ι (z)γβ η (w), where γα ι are the one-variable functions for the disc from the preceding example. By (3.11) we thus get for H(z, w) = f(z)g(w), f, g ∈ O(D), and m = (m), ρmH(z, z) = (1− zw)2 ∑ κ,λ (−∂w)κ(−∂z)λ [ (1− zw)−2 ∑ α,β,γ,δ,ι,η Pm αβ,γδ γα ι (z)γβ η (w)γγ κ(w)γδ λ(z)∂ιf(z)∂ηg(w) ]∣∣∣ w=z . Here again (−∂w)κ(−∂z)λ occurs rather than (−∂w)λ(−∂z)κ, and likewise γγ κ(w)γδ λ(z) rather than γγ κ(z)γδ λ(w), for the same reasons as in Example 5.2 and in the proof of Proposition 4.1. For low values of m, one computes that ρ(0)(fg) = fg (of course), while ρ(1)(fg)(z) = −\|1− a\|2 ( 1− \|z\|2 )2 f ′(z)g′(z) is the G-invariant operator from O(D× D) into C∞(D) uniquely determined by ρ(1)(fg)(0) = −\|1− a\|2f ′(0)g′(0). 26 M. Englǐs and H. Upmeier Computer-aided calculations give ρ(2)(fg)(0) = −\|1− a\|2 2 [ 4(1 + \|a\|2)f ′(0)g′(0)− \|1− a\|2f ′′(0)g′′(0) ] and ρ(3)(fg)(0) = −\|1− a\|2 6 [ 36(1 + \|a\|2 + \|a\|4)f ′(0)g′(0)− 18\|1− a\|2(1 + \|a\|2)f ′′(0)g′′(0) + \|1− a\|4f ′′′(0)g′′′(0) ] . To compute [ν]m, we note as in Example 5.2 that for the function F ∈ C∞(D× D) given by F (z, w) = zαzβwγwδ, where α+ γ = β + δ = ρ, one has by (3.7) PνF (0) = aγaδ ρ! [ν]ρ . On the other hand, since now det Φ′(0,Λy) = \|1 − a\|2(1 − \|a\|2\|y\|4) by Lemma 3.1, we have from (3.4) PνF (0) = aγaδ\|1− a\|2 ∫ D zρzρ ( 1− \|a\|2\|z\|4 )(1− \|z\|2)ν(1− \|az\|2)ν \|1− a\|z\|2\|2ν dz (1− \|z\|2)2(1− \|az\|2)2 . Passing to polar coordinates, we thus obtain (writing m instead of ρ), m! [ν]m = \|1− a\|2 ∫ 1 0 tm (1− \|a\|2t2) (1− t)2(1− \|a\|2t)2 (1− t)ν(1− \|a\|2t)ν \|1− at\|2ν dt. (5.10) Using series expansions, the integral can again be expressed in terms of Horn-type two-variable hypergeometric functions, and simplifies for some special values of a. In particular, for a = 0 the right-hand side of (5.10) is just∫ 1 0 tm(1− t)ν−2 dt = m!Γ(ν + 1) Γ(ν +m) , so that 1 [ν]m = Γ(ν − 1) Γ(ν +m) , or, upon renormalizing so that [ν]0 = 1, [ν]m = Γ(ν +m) Γ(ν) = (ν)m, in agreement with the result ρν(fg) = ∑ m Am(f, g) (ν)m from [18] reviewed in Section 4. Similarly, for a = −1, (5.10) becomes 1 [ν]m = 4Γ(2ν − 2) Γ(m+ 2ν − 1) 2F1(2ν − 1,m+ 1;m+ 2ν − 1;−1), Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 27 or, upon renormalizing dz so that [ν]0 = 1, 1 [ν]m = 2 (2ν − 1)m 2F1(2ν − 1,m+ 1;m+ 2ν − 1;−1). Crude estimates also show that [ν]m ∼ \|1− a\|2mνm [ 1− 2(1 + \|a\|2) \|1− a\|2ν +O ( 1 ν2 )] as ν → +∞, which can be used to check at least for the first few terms that, again, ρν = ∑ m ρm [ν]m is indeed independent of a, although both ρm and [ν]m are not. Note that the retraction π in this case (a = −1) is simply π(z, w) = mz,w, the geodesic mid-point between z and w. A Table of parameters of real bounded symmetric domains The table on the next page lists the groups GR, KR, the root type Σ, the rank rR, characteristic multiplicities aR, bR, cR and the dimension d of real bounded symmetric domains BR, as well as the analogous parameters rC, aC, bC of the complex domains BC and the labellings of BC and BR following the notation in [30, Chapter 11]. The table was mostly compiled using [37, 26, 24] and [30]. The low-dimensional isomorphisms between the various types, and the resulting restrictions on subscripts needed in order to make the table entries non-redundant, can be found e.g. in the cited chapter in Loos [30]. As a matter of notation, we use Gn(K) and Up,q(K) for the identity component of the general linear (resp. pseudo-unitary) group over K = R,C,H (= quaternions). Sp2r(K) is the 2r × 2r-symplectic group over K = R,C, whereas On(H) is the quaternion analogue of On(C) (usually denoted by SO∗(2n)). The genus of BC is given in terms of the domain parameters by p = (rC − 1)aC + bC + 2, while the dimension d = dimRBR = dimCBC equals d = rR(rR − 1) 2 aR + rR = rC(rC − 1) 2 aC + rC for type A, and d = rR(rR − 1)aR + rRbR + rRcR + rR = rC(rC − 1) 2 aC + rCbC + rC for all other types. Domains of type D2 turn out to have, in some sense, two multiplicities a instead of one. Note that the unit interval corresponds to IR 1,1, the unit ball of Rm, m > 1, to IR 1,m, the unit ball of Cm to I1,m, the unit ball of the algebra of quaternions H to IH 2,2m, the unit ball of Hm, m > 1, to IH 2,2m, and the unit ball of the Cayley plane O2 to V O. In the “complex” cases, the root type does not quite make sense (“BC×BC”) and nor do the parameters aC, bC, rC, while BC is just the product B ×B; so these columns are left empty. 28 M . E nglǐs and H . U pm eier BR GR/KR Σ rR aR bR cR d rC aC bC BC IR r,r+b Ur,r+b(R)/Ur(R)× Ur+b(R) Dr/Br r 1 b 0 r(r + b) r 2 b Ir,r+b Ir,r+b Ur,r+b(C)/Ur(C)× Ur+b(C) r 2 2b 1 2r(r + b) (product case) IH 2r,2r+2b Ur,r+b(H)/Ur(H)× Ur+b(H) Cr/BCr r 4 4b 3 4r(r + b) 2r 2 2b I2r,2r+2b V O0 U2,2(H)/U2(H)× U2(H) B2 2 3 4 0 16 2 6 4 V IIIR r Gr(R)/Ur(R) Ar r 1 − − 1 2r(r + 1) r 1 0 IIIr IC r,r Gr(C)/Ur(C) Ar r 2 − − r2 r 2 0 Ir,r IIH 2r Gr(H)/Ur(H) Ar r 4 − − r(2r − 1) r 4 0 II2r V IO0 G4(H)/U4(H) D3 3 4 0 0 27 3 8 0 V I IIIr Sp2r(R)/Ur(C) r 1 0 1 r(r + 1) (product case) IIIH 2r Sp2r(C)/Ur(H) Cr r 2 0 2 r(2r + 1) 2r 1 0 III2r IIR 2r+ε O2r+ε(C)/U2r+ε(R) Dr/Br r 2 2ε 0 r(2(r + ε)− 1) r 4 2 II2r+ε II2r+ε O2r+ε(H)/U2r+ε(C) r 4 4ε 1 2r(2(r + ε)− 1) (product case) IV R,q p+q SOp,1 × SO1,q/SOp,0 × SO0,q D2/A2 2 n/a 0 0 p+ q 2 p+ q − 2 0 IVp+q IVn SOn,2/SOn,0 × SO0,2 2 n− 2 0 1 2n (product case) V E6(−14)/Spin(10)× SO(2) 2 6 8 1 32 (product case) IV R,0 n SOn,1/SOn,0 C1 1 − 0 n− 1 n 2 n− 2 0 IVn V O F4(−20)/SO(9) BC1 1 − 8 7 16 2 6 4 V V I E7(−25)/E6 × SO(2) 3 8 0 1 54 (product case) V IO E6(−26) ×O(2)/F4 ×O(1) A3 3 8 − − 27 3 8 0 V I Toeplitz Quantization and Asymptotic Expansions: Geometric Construction 29 Acknowledgements Research supported by the German-Israeli Foundation (GIF), I-696-17.6/2001; the Academy of Sciences of the Czech Republic institutional research plan no. 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Anal. 195 (2002), 306–349, math.RT/0110212. http://arxiv.org/abs/math.QA/0006063 http://arxiv.org/abs/math.RT/9911020 http://arxiv.org/abs/math.QA/9907171 http://arxiv.org/abs/math.QA/9910137 http://arxiv.org/abs/math.RT/0110212 1 Introduction 2 Preliminaries 3 Invariant retractions and Moyal restrictions 4 Asymptotic expansion: the complex case 5 Examples A Table of parameters of real bounded symmetric domains References

Toeplitz Quantization and Asymptotic Expansions: Geometric Construction

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