Separation of series for hyperboloids
type: Item
instance of: scholarly article
title: Separation of series for hyperboloids
zbMATH Open document ID: 0924.43011
DOI: 10.1007/BF02465785
publication date: 1998-08-26T00:00:00Z
review text: By \textit{I. M. Gelfand} and \textit{S. G. Gindikin} [Funkt. Anal. Appl. 11, 258-265 (1978); translation from Funkts. Anal. Prilozh. 11, 19-27 (1978; Zbl 0449.22018)] a program for developing harmonic analysis on homogeneous spaces \(G/H\) was proposed (the so-called Gelfand-Gindikin program). The first step of this program is to decompose \(L^2(G/H)\) into subspaces \(L_j\) on each of which irreducible unitary representations act that belong to a single series. The subsequent analysis must provide a natural description of functions in \(L_j\) (for example, as boundary values of functions holomorphic on complex manifolds), etc. In this paper, the problem of the separation of series for real hyperboloids \(X=SO_0 (p,q)/\) \(SO_0 (p,q-1)\) is solved. As is known, the quasiregular representation on the hyperboloid can be decomposed into two series of irreducible unitary representations, namely, the continuous and discrete series. Therefore, it suffices to indicate the projection onto the subspace of \(L^2(X)\) on which one of these series acts. We give explicit formulas for the projection on the subspace \(L^2_d(X)\) on which the discrete series acts. For the case \(q=2\), the hyperboloid \(X\) is the Shilov boundary of some complex manifolds, and the discrete series splits into two series, namely, the analytic and antianalytic series. In [J. Funct. Anal. 147, 26-50 (1997; Zbl 0881.43005)] we obtained the projections related to each of these series, together with the corresponding Cauchy-Szegö kernels. For completeness we write out the formulas for these projections in the present paper. Note that in the general case (for arbitrary \(q)\), the discrete series is related to a certain analyticity, see formulas (1.1) and (3.4), (3.3) for the projection and for the spherical functions. The problem on the separation of series for hyperboloids was considered by \textit{S. G. Gindikin} [J. Geom. Phys. 10, 175-184 (1993; Zbl 0787.22014)], where, in another form the projection…
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 1188092
zbMATH Keywords: harmonic analysis
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: Gelfand-Gindikin program
zbMATH Keywords: real hyperboloids
zbMATH Keywords: irreducible unitary representations
zbMATH Keywords: discrete series
zbMATH Keywords: Shilov boundary
zbMATH Keywords: complex manifolds
zbMATH Keywords: spherical functions
author: V. F. Molchanov
MaRDI profile type: Publication
cites work: Q5820721
cites work: Q5568743
cites work: SPHERICAL FUNCTIONS ON HYPERBOLOIDS
cites work: Q3881053
cites work: Conformal harmonic analysis on hyperboloids
cites work: Discrete series for an affine symmetric space
cites work: Holomorphic discrete series for hyperboloids of Hermitian type
cites work: The holomorphic discrete series for affine symmetric spaces. I
cites work: Analysis on real hyperbolic spaces
cites work: Harmonic analysis on hyperboloids
cites work: Distributions Invariant under an Orthogonal Group of Arbitrary Signature.
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1188092