Cuspidal representations associated to (GL(n), O(n)) over finite fields and p-adic fields

When (r) over bar is an irreducible cuspical representation of (G) over bar = GL(n, q) and (H) over bar is an orthogonal group associated to a symmetr ic matrix in (G) over bar then the space of (H) over bar-fixed vectors for (r) over bar is shown to have dimension at most one. Such a representation (r) over bar induces an irreducible supercuspidal representation pi of G = GL(n, E), where E is a p-adic field whose residue field has order q. The sp ace of those linear forms on the space of pi which are invariant under an o rthogonal group is computed. For the corresponding group of orthogonal simi litudes, it is shown that the dimension of the space of invariant linear fo rms is always at most one. (C) 1999 Academic Press.