On the Self-dual Representations of Division Algebras over Local Fields

Let k be a non-Archimedean local field of characteristic 0. Let D be a division algebra with center k and index n. The group D∗ is a locally compact group which is compact modulo the center. Its complex irreducible representations are finite dimensional. If π is an irreducible representation of D∗ which is self-dual, i.e., if π∨ denotes the dual of π, π∨ ∼= π, then there exists a D∗-invariant, nondegenerate, bilinear form B : π × π → C which is unique up to scaling. It is either symmetric, or skew-symmetric. The representation π is said to be orthogonal if π carries a symmetric bilinear form, and π is said to be symplectic if it carries a skew-symmetric bilinear form. The aim of this work is to understand which of these two possibilities occurs for a given self-dual representation π. Before we come to the proposed answer, we must fix some notation. We recall that according to the Jacquet-Langlands correspondence (proved by Jacquet-Langlands for n = 2, by Deligne-Kazhdan-Vigneras, and independently by Rogawski, for general n), there exists a natural bijective correspondence, denoted by π → JL(π), between irreducible representations of D∗ and irreducible discrete series representations of GLn(k). This correspondence is characterised by the character identity