An induction theorem for groups acting on trees

Abstract

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