Some consequences of Harish-Chandra’s submersion principle

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.