Invariant distributions on projective spaces over local fields

TL;DR

This paper proves that for certain subgroups of projective linear groups over local fields, the module of coinvariants is finite, implying no nonzero invariant distributions exist on projective space.

Contribution

It establishes the finiteness of coinvariants for $ ilde{A}_n$ subgroups of $PGL_{n+1}(K)$ acting on projective space over local fields.

Findings

The module of coinvariants is finite.

No nonzero invariant distributions exist on projective space.

Results apply to subgroups of $PGL_{n+1}(K)$ over local fields.

Abstract

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