On a question related to a basic convergence theorem of Harish-Chandra

In his first 1958 paper on zonal spherical functions Harish-Chandra proved an extremely delicate convergence theorem which was basic to his subsequent definition of his Schwartz space and his theory of cusp forms. This paper gives elementary proofs that a related integral converges for for groups of...

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Bibliographic Details
Main Author Wallach, Nolan R
Format Journal Article
LanguageEnglish
Published 29.06.2022
Subjects
Mathematics - Representation Theory
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Summary:In his first 1958 paper on zonal spherical functions Harish-Chandra proved an extremely delicate convergence theorem which was basic to his subsequent definition of his Schwartz space and his theory of cusp forms. This paper gives elementary proofs that a related integral converges for for groups of real rank one, several groups of real rank 2 (including $SO(n,2), Sp_4(R)$ and $Sp_4(C)$), $GL(n,R)$ and $GL(n,C)$. In fact, a stronger result has been proved in raphael. Applications of the question are also studied.
DOI:10.48550/arxiv.2206.14773