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Some consequences of Harish-Chandra's submersion principle

Authors: Cary Rader and Allan Silberger
Journal: Proc. Amer. Math. Soc. 118 (1993), 1271-1279
MSC: Primary 22E50
MathSciNet review: 1169888
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Abstract: Let $ G$ be a reductive $ \mathfrak{p}$-adic group, $ K$ a good maximal compact subgroup, $ {K_1} \subset K$ any open subgroup, and $ \pi $ an admissible representation of $ G$ of finite type. In A submersion principle and its applications, Harish-Chandra proves the theorem that $ \int_K {\pi (kg{k^{ - 1}})\,dk} $ is a finite-rank operator for $ g$ in the regular set $ G'$ in order to show that the character $ {\Theta _\pi }(g)$ is a locally constant class function on $ G'$. From this, the authors derive the formula $ \theta (1){\Theta _\pi }(g) = d(\pi )\int_{G/Z} {\int_{{K_1}} {\theta (xkg{k^{ - 1}}{x^{ - 1}})\,dk\,d\dot x} \quad (g \in G')} $ for any $ K$-finite matrix coefficient $ \theta $ of a discrete series representation $ \pi $ with formal degree $ d(\pi )$. They use another technical result of the paper to prove that invariant integrals of Schwartz space functions converge absolutely. None of these results depends upon a characteristic zero assumption.

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Keywords: Character, discrete series, reductive $ \mathfrak{p}$-adic groups, Schwartz space, invariant integral
Article copyright: © Copyright 1993 American Mathematical Society

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