Formule de Plancherel Pour les Espaces Symetriques Reductifs

Delorme, Patrick

doi:10.2307/121014

Cited by 88 publications

(101 citation statements)

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“…Here we show that the space U(n)/O(n) can be realized as the space of unitary symmetric matrices and the space U(2n)/Sp(n) as the space of unitary skew-symmetric 2n × 2n-matrices 12 . We show that the groups Sp(2n, R) and SO * (4n) respectively act on these spaces by linear-fractional transformations.…”

mentioning

confidence: 89%

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

Neretin

2007

J Math Sci

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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 .

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confidence: 89%

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

Neretin

2007

J Math Sci

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“…In contrast to Riemannian symmetric spaces, it is known that "multiplicityfree property" in the Plancherel formula fails for (non-Riemannian) symmetric spaces G/H in general (see [3,8] for the description of the multiplicity of the most continuous series representations for G/H in terms of Weyl groups).…”

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

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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.

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“…More generally, we give, for any even integer p, a necessary and sufficient condition under which L 2 (G/H) is almost L p (see Theorem 4.1). Our criterion is new even when G/H is a reductive symmetric space where the disintegration of L 2 (G/H) was established up to the classification of discrete series representations for (sub)symmetric spaces ( [1,8,20]). Indeed irreducible unitary representations that contribute to L 2 (G/H) in the direct integral are obtained as a parabolic induction from discrete series for subsymmetric spaces, but a subtle point arises from discrete series with singular parameter.…”

Section: Introductionmentioning

confidence: 99%