Cuspidal Representations Associated to (GL(n), O(n)) over Finite Fields andp-Adic Fields

Abstract

Whenr̄is an irreducible cuspical representation ofḠ=GL(n,q) andH̄is an orthogonal group associated to a symmetric matrix inḠthen the space ofH̄-fixed vectors forr̄is shown to have dimension at most one. Such a representationr̄induces an irreducible supercuspidal representation π ofG=GL(n,E), whereEis ap-adic field whose residue field has orderq. The space of those linear forms on the space of π which are invariant under an orthogonal group is computed. For the corresponding group of orthogonal similitudes, it is shown that the dimension of the space of invariant linear forms is always at most one.