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arXiv · 2206.14773

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

Nolan R. Wallach
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Abstract

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.

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Nolan R. Wallach. 2022-06-29. On a question related to a basic convergence theorem of Harish-Chandra. https://arxiv.org/abs/2206.14773

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