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 ofGLn(k)×GLn(k) invariant linear form on an irreducible admissible representation ofGL2n(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 ofGL2(D) forDa division algebra and a conjecture made regarding its existence in general.