Spherical generalized functions induced on \(G_{\mathbb{C}}/G_{\mathbb{R}}\) and applications
type: Item
instance of: scholarly article
title: Spherical generalized functions induced on \(G_{\mathbb{C}}/G_{\mathbb{R}}\) and applications
zbMATH Open document ID: 0755.43005
DOI: 10.1016/0022-1236(92)90137-8
published in: Journal of Functional Analysis
publication date: 1992-06-28T00:00:00Z
review text: Let \(G\) be a complex semi-simple Lie group, and \(H\) a connected subgroup which is a real form of \(G\). By induction one constructs a spherical generalized function \(p-\text{ind}^ G_ L\theta\) on the symmetric space \(G/H\) from a spherical generalized function \(h\) on the smaller symmetric space \(L/L \cap H\), where \(L\) is the centralizer of the split part of a Cartan subalgebra of the Lie algebra \(\mathfrak h\) of \(H\). The spherical generalized function \(p -\text{ind}^ G_ L\theta\) is an analytic function on the set of regular elements, and one gives an explicit formula for it. When \(\theta\) is the coefficient of a representation of \(L\), then \(p - \text{ind}^ G_ L\theta\) is a coefficient of a representation induced from a parabolic subgroup. The example of the symmetric space \(GL(2,\mathbb{C})/GL(2,\mathbb{R})\) is studied in detail. As an application one proves that the inversion formula for \(GL(n,\mathbb{C})/GL(n,\mathbb{R})\) due to Sano is actually a Plancherel formula, by proving that the spherical generalized functions involved in Sano's inversion formula are positive definite and extremal.
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 35783
zbMATH Keywords: complex semi-simple Lie group
zbMATH Keywords: induction
zbMATH Keywords: symmetric space
zbMATH Keywords: inversion formula
zbMATH Keywords: Plancherel formula
zbMATH Keywords: spherical generalized functions
author: Pascale Harinck
reviewed by: Jacques Faraut
MaRDI profile type: Publication
full work available at URL: 0022-1236(92)90137-8
OpenAlex ID: W1986840472
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description: scientific article; zbMATH DE number 35783