The spherical transform of a Schwartz function on the Heisenberg group
type: Item
instance of: scholarly article
title: The spherical transform of a Schwartz function on the Heisenberg group
zbMATH Open document ID: 0914.22013
author: Chal Benson
author: J. W. Jenkins
author: Gail Ratcliff
published in: Journal of Functional Analysis
publication date: 1999-01-24T00:00:00Z
review text: Let \(K\subset U(n)\) be a compact Lie group acting on the \((2n+1)\)-dimensional Heisenberg group \(H_n=V\times\mathbb{R}\). We say that \((H_n, K)\) is a Gelfand pair if the convolution algebra \(L^1_K(H_n)\) of integrable \(K\)-invariant functions on \(H_n\) is commutative. In this case, the Gelfand space \(\Delta=\Delta(K,H_n)\) is equipped with the Plancherel measure, and the spherical transform \(^\wedge\): \(L^2_K(H_n)\to L^2_K(\Delta)\) is an isometry. The main result of this paper provides a characterization of the set \({\mathcal S}_K(H_n)^\wedge=\{\widehat f| f\in{\mathcal S}_K(H_n)\}\) of spherical transforms of \(K\)-invariant Schwartz functions on \(H_n\). The authors show that a function \(F\) on \(\Delta\) belongs to \({\mathcal S}_K(H_n)^\wedge\) if and only if the functions obtained from \(F\) via application of certain derivatives and difference operators satisfy decay conditions. In order to describe these conditions let \(\mathbb{C}[V]=\sum_{\alpha\in\Lambda}P_\alpha\subset{\mathcal F}\) be the decomposition of \(\mathbb{C}[V]\) into \(K\)-irreducible subspaces \(P_\alpha\). Here \({\mathcal F}\) is the Fock space on which the Heisenberg group acts via the irreducible representation \(\pi\). For every \(\alpha\in\Lambda\), \(\lambda\in \mathbb{R}^*\), let \(\Phi_{\alpha,\lambda}\) denote the corresponding \(K\)-spherical function \[ \Phi_{\alpha,\lambda}(z,t)={1\over d_\alpha} \sum^{d_\alpha}_{j=1} \biggl \langle\pi\bigl(\sqrt{|\lambda|}z, \lambda t\bigr) v_j,v_j\biggr \rangle, \] where the \(v_j\)'s denote an orthogonal basis of \(P_\alpha\). These \(\Phi_{\alpha,\lambda}\) form a dense open subset of \(\Delta\). The other elements \(\Phi_w\), \(w\in V/K\), of \(\Delta\) are coming from the one-dimensional representations of \(H_n\). Hence \(\Delta=\Lambda\times\mathbb{R}^*\cup V/K\). The \(K\)-spherical transform \(^\wedge\) for \(f\in L^1_K(H_n)\) is the function \[ \widehat f(\Phi)=\int_{H_n}f(z,t)\overline{\Phi(z,t)}dzdt,\;\Phi\in\Delta. \]…
reviewed by: Jean Ludwig
Mathematics Subject Classification ID: 22E27
zbMATH DE Number: 1216766
zbMATH Keywords: compact Lie group
zbMATH Keywords: Heisenberg group
zbMATH Keywords: Gelfand pair
zbMATH Keywords: spherical transforms
zbMATH Keywords: Schwartz functions
MaRDI profile type: Publication
OpenAlex ID: W2063208367
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DOI: 10.1006/JFAN.1997.3197
description: scientific article; zbMATH DE number 1216766