On the restriction of supercuspidal representations to compact, open subgroups (Q1083560)

If G is a reductive group over a p-adic field F and K is a compact open subgroup, then an irreducible representation \(\tau\) of K is called (G,K)- principal if it does not occur in the restriction to K of any supercuspidal representation of G. This paper establishes some conditions which imply that a representation \(\tau\) is principal, using the \(\tau\)- spherical Hecke functions, an idea originated by Matsumoto when \(\tau\) is one-dimensional. Using these conditions, the author shows that any irreducible supercuspidal representation of GL(n,F) which has a fixed vector under the first congruence subgroup of GL(n,\({\mathfrak O}_ F)\) is induced from \(Z\cdot GL(n,{\mathfrak O}_ F)\). He also gives a criterion for a supercuspidal representation with a fixed vector under the kth congruence subgroup to be induced. These results are then used to give a new proof that every irreducible supercuspidal representation of GL(2,F) is induced.