Invariant distributions on projective spaces over local fields

Abstract:

Let $\Gamma$ be an $\widetilde A_n$ subgroup of $\operatorname {PGL}_{n+1}(\mathbb {K})$, with $n\ge 2$, where $\mathbb K$ is a local field with residue field of order $q$ and let $\mathbb P^n_{\mathbb K}$ be projective $n$-space over $\mathbb K$. The module of coinvariants $H_0(\Gamma ; C(\mathbb P^n_{\mathbb K},\mathbb Z))$ is shown to be finite. Consequently there is no nonzero $\Gamma$-invariant $\mathbb Z$-valued distribution on $\mathbb P^n_{\mathbb K}$.