On distinguishedness

Abstract

Let F be a finite extension of Q_p and K a quadratic extension of F. If (Π,V ) is a representation of GL₂(K), H a subgroup of GL_p(K) and μ a character of the image subgroup det(H) of K∗, then Π is said to be μ-distinguished with respect to H if there exists a nonzero linear form l on V such that l(Π(g)v) = μ(detg)l(v) for g ∈ H and v ∈ V . We provide new proofs, using entirely local methods, of some well-known results in the theory of non-archimedean distinguished representations for GL(2).