Branching Coefficients of Holomorphic Representations and Segal–Bargmann Transform

Zhang, Genkai

doi:10.1006/jfan.2002.3957

Cited by 30 publications

(33 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For type (D/B) we are likewise considering invariance under the full, not necessarily connected, group K. Hence the results of [27], [28] imply that a K C -signature (of length r C = r) is even if and only if it has the form…”

Section: Theorem 14 For Domains Of Type (A) the Formal Power Seriesmentioning

confidence: 99%

See 1 more Smart Citation

Toeplitz quantization and asymptotic expansions for real bounded symmetric domains

Engliš

Upmeier

2010

Math. Z.

View full text Add to dashboard Cite

Abstract. An analogue of the star product, familiar from deformation quantization, is studied in the setting of real bounded symmetric domains. The analogue turns out to be a certain invariant operator, which one might call star restriction, from functions on the complexification of the domain into functions on the domain itself. In particular, we establish the usual (i.e. semiclassical) asymptotic expansion of this star restriction, and further obtain a real-variable analogue of a theorem of Arazy and Ørsted concerning the analogous expansion for the Berezin transform. Covariant Calculi on Complex and Real Symmetric DomainsLet D = G/K be an irreducible bounded symmetric domain in C d in the HarishChandra realization, with G the identity connected component of the group of all biholomorphic self-maps of D and K the stabilizer of the origin; K can also be realized as the automorphism group Aut(Z) of the Hermitian Jordan triple Z ≈ C where dz stands for the Lebesgue measure, K(z, w) is the ordinary (unweighted) Bergman kernel of D, and c ν is a normalizing constant to make dµ ν a probability measure. The space, where J g 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 J g −1 (z) ν/p .) This situation will henceforth be called the complex bounded case.In addition to bounded symmetric domains, we will also consider the complex flat case of a Hermitian vector space In most cases, such calculi can be built by the recipewhere dµ 0 is a G-invariant measure on D, and A ζ is a family of operators in(One calls such a family a covariant operator field on D. One also usually normalizes the measure dµ 0 so that A 1 is the identity operator.) Note that in view of the transitivity of the action of G on D, any covariant operator field is uniquely determined by its value A 0 at the origin ζ = 0. The best known examples of such calculi are the Toeplitz calculus T and the Weyl calculus W, corresponding to T 0 = ·, 1 1 (the projection onto the constants) and W 0 f (z) = f (−z) (the reflection operator), respectively [2]. For the complex flat case, W is just the well-known Weyl calculus from the theory of pseudodifferential operators, see e.g. [12].Given a covariant operator calculus A, the associated star product * on functions on D is defined byWhile f * g is a well-defined object for some calculi (e.g. for A = W, at least on C d and rank one symmetric domains, see [4]), in most cases (e.g. for A = T , the Toeplitz calculus), it makes sense only for very special functions f, g and (7) 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 [6] that for any f, g ∈ C ∞ (D) with compact support,as ν → ∞, for some bilinear differential operators C j (not depending on f, g and ν).(The assumption of compact support can be relaxed, cf.[9].) We can thus define f * g as the formal power seriesInterpreting ν as th...

show abstract

Section: Theorem 14 For Domains Of Type (A) the Formal Power Seriesmentioning

confidence: 99%

“…For all other types, note first of all that by holomorphy, any polynomial on Z C is uniquely determined by its restriction to Z; and by Chevalley's theorem (cf. [27], Proposition 6.2) and the "polar decomposition" [11, Section VI.2]…”

Section: Theorem 14 For Domains Of Type (A) the Formal Power Seriesmentioning

confidence: 99%

Toeplitz quantization and asymptotic expansions for real bounded symmetric domains

Engliš

Upmeier

2010

Math. Z.

View full text Add to dashboard Cite

show abstract

“…On the one hand, this is sufficiently general to produce many interesting consequences, some of which are new and some others may be regarded as prototypes of various multiplicity-free branching theorems (e.g. [5,10,46,54,58,66,68,81,90,92]). On the other hand, the line bundle case is sufficiently simple, so that we can illustrate the essence of our main ideas without going into technical details.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…Some special cases have been worked on in this direction so far, for G = SL(2, R) by V. F. Molchanov [64]; for G = SU (2, 2) by B. Ørsted and G. Zhang [70]; for G = SU (n, 1) by G. van Dijk and S. Hille [10]; for G = SU (p, q) by Y. Neretin and G. Ol'shanskiȋ [66,67]. See also G. van Dijk-M. Pevzner [11], M. Pevzner [72] and G. Zhang [92]. Their results show that a different family of irreducible unitary representations (sometimes, spherical complementary series representations) can occur in the same branching laws and each multiplicity is not greater than one.…”

Section: Analysis On Multiplicity-free Representationsmentioning

confidence: 99%

Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs

Kobayashi

Progress in Mathematics

View full text Add to dashboard Cite

Summary. The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known results such as the Clebsh-Gordan-Pieri formula for tensor products, the Plancherel theorem for Hermitian symmetric spaces (also for line bundle cases), the Hua-Kostant-Schmid K-type formula, and the canonical representations in the sense of Vershik-GelfandGraev. Our method works in a uniform manner for both finite and infinite dimensional cases, for both discrete and continuous spectra, and for both classical and exceptional cases.

show abstract

“…5 Some modifications of proofs of results of [54] were obtained later in [15] and [89], see also [13], [31]. 6 Apparently the Plancherel formula for all the groups can be reduced to some single identity with the Heckman-Opdam spherical hypergeometric transform (see, for instance, [28]), as far as I know, this possibility is yet not realized.…”

mentioning

confidence: 99%

Notes on Stein-Sahi representations and some problems of non-L 2 harmonic analysis

Neretin

2007

J Math Sci

View full text Add to dashboard Cite

We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.Recently, Oshima [67] published his formula for c-function for L 2 on pseudo-Riemannian symmetric spaces (see also works of Delorm [12] and van den Ban-Schlichtkrull [3], [4]). After this, there arises a natural question about other solvable problems of non-commutative harmonic analysis.In the Appendix to the paper [57], the author proposed a series of non L 2inner products in spaces of functions on pseudo-Riemannian symmetric spaces and conjectured that this object is reasonable and admits an explicit harmonic analysis.In this work, we discuus the problem in more details, in particular, we obtain the Plancherel formula for these kernel for Riemannian symmetric spaces U(n), U(n)/O(n), U(2n)/Sp(n). We also give a new proof of Sahi's results [76].our inner product is non-negative definite.The latter statement is a reformulation of the well-known theorem (Berezin-Gindikin-Rossi-Vergne-Wallach) on unitarizability of scalar highest weight representations ([6], [83], see a relatively recent exposition in [18]).1 The first work that can be attributed to this subject, is Vershik, Gelfand, Graev,[84], [20]. These authors apply the representations ρσ of the groups U(1, q) to construct representations of current groups; for a collection of other early references, see [56].2 Compare the Stein inner product (0.6) and the Berezin inner product (0.11). In the first case, the kernel has a singularity on the surface det(x − y) = 0. This surface itself is singular, and its most singular strate is the diagonal x = y. On the contary, the Berezin kernel has no singularities in Bp,q × Bp,q at all. We can substitute to (0.11) arbitrary compactly supported distributions unstead of functions F 1 , F 2 .

show abstract

Branching Coefficients of Holomorphic Representations and Segal–Bargmann Transform

Cited by 30 publications

References 28 publications

Toeplitz quantization and asymptotic expansions for real bounded symmetric domains

Toeplitz quantization and asymptotic expansions for real bounded symmetric domains

Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs

Notes on Stein-Sahi representations and some problems of non-L 2 harmonic analysis

Contact Info

Product

Resources

About