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

In this paper we prove a division algebra analogue of a theorem of Jacquet and Rallis about uniqueness of GL(n)(k) x 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 GL(2)(D) for D a division algebra and a conjecture made regarding its existence in general. (C) 1999 Academic Press.