Local ε-Factors and Characters of GL(2)

Introduction. The representation theory of GL(2, K), for K a nonarchimedean local field, is somewhat more complicated than the corresponding theory for GL(2, R). In particular, the formulas for the characters of discrete series representation of GL(2, R) have a concise closed form, while the published character formulas for supercuspidal representations of GL(2, K) occupy several pages [10], [14]. The purpose of this paper is to give a concise formula for the character of certain infinite dimensional representations of GL(2, K). The representations considered are admissible in the sense that the stabilizer of a vector in the representation space is open in GL(2, K), and the subspace stabilized by an open compact subgroup of GL(2, K) is finite dimensional. Irreducible admissible representations ir of GL(2, K) satisfy Schur's lemma, so that the center K* of GL(2, K) acts by multiplication by a quasicharacter ,. There exists a locally constant character function ch , defined on a dense open subgroup of GL(2, K), which determines the isomorphism class of ir [7, Section 7]. Jacquet and Langlands [7, Section 2] introduced factors E(10 X) attached to the twist of ir by quasicharacters X of K* which describe the action of -r in a specific model. For each separable quadratic extension L of K, there exists an irreducible admissible representation BCL/K(1r) of GL(2, L), the base change of ir to L [9, Section 2]. This representation of GL(2, L) enters into the expression for the character. The main result of this paper is the following formula. Recall that conjugacy classes of nonsplit Cartan subgroups are parameterized by quadratic separable field extensions L of K, via embeddings of L* in GL(2, K).