Some remarks on representations of a division algebra and of the Galois group of a local field

Abstract

In this paper we prove a division algebra analogue of a theorem of Jacquet and Rallis about uniqueness of GL_n(k)×GL_n(k) invariant linear form on an irreducible admissible representation of GL_2n(k). We propose a conjecture about when this invariant form exists. We prove some results about self-dual representations of the invertible elements of a division algebra and of Galois groups of local fields. The Shalika model has been studied for principal series representations of GL₂(D) for Da division algebra and a conjecture made regarding its existence in general.