An induction theorem for groups acting on trees

TL;DR

This paper proves an induction theorem for groups acting on trees, showing that certain irreducible representations of such groups can be obtained by induction from stabilizers of vertices or edges, with applications to p-adic groups.

Contribution

It establishes an induction theorem for representations arising from sheaves on trees, linking cohomology to subgroup induction in the context of groups acting on trees.

Findings

Irreducible cohomology representations are induced from stabilizers.

Application to supercuspidal representations of p-adic groups.

Every supercuspidal representation in rank-one cases arises from induction.

Abstract

If $G$ is a group acting on a tree $X$, and ${\mathcal S}$ is a $G$-equivariant sheaf of vector spaces on $X$, then its compactly-supported cohomology is a representation of $G$. Under a finiteness hypothesis, we prove that if $H_c^0(X, {\mathcal S})$ is an irreducible representation of $G$, then $H_c^0(X, {\mathcal S})$ arises by induction from a vertex or edge stabilizing subgroup. If $G$ is a reductive group over a nonarchimedean local field $F$, then Schneider and Stuhler realize every irreducible supercuspidal representation of $G(F)$ in the degree-zero cohomology of a $G(F)$-equivariant sheaf on its reduced Bruhat-Tits building $X$. When the derived subgroup of $G$ has relative rank one, $X$ is a tree. An immediate consequence is that every such irreducible supercuspidal representation arises by induction from a compact-mod-center open subgroup.