Completing an extension of Tunnell's theorem

We look at a special case of a familiar problem: Given a locally compact group G, a subgroup H and a complex representation pi+ of G how does pi+ decompose on restriction to H. Here G is GL(2, F)(+), where F is a nonarchimedian local field of characteristic not two, K a separable quadratic extension of F, GL(2, F)(+) the subgroup of index 2 in GL(2, F) consisting of those matrices whose determinant is in N-K/F(K*), pi(+) is an irreducible, admissible supercuspidal representation of GL(2, F)(+) and H = K* under an embedding of K* into GL(2, F). (C) 2008 Elsevier Inc. All rights reserved.