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

When is an irreducible cuspical representation of = GL(n, q) and is an orthogonal group associated to a symmetric matrix in then the space of -fixed vectors for is shown to have dimension at most one. Such a representation induces 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.