FROBENIUS-SCHUR INDICATORS FOR p-ADIC SPECIAL LINEAR GROUPS

Abstract

Let F be a non-Archimedean local field, and let G be the special linear group SL(n,F). This dissertation studies irreducible smooth complex self-dual representations of G. Such a representation admits a unique non-degenerate G-invariant bilinear form up to scaling, and this form is either symmetric or skew-symmetric. The main goal is to determine whether this form is symmetric or skew-symmetric. The dissertation gives a nearly complete answer in terms of the congruence class of n modulo 4. It proves that the invariant bilinear form is symmetric when n is odd or when n is congruent to 0 modulo 4. It also proves that when n is congruent to 2 modulo 4 and F contains a square root of -1, the answer is determined by the value of the central character at the negative identity matrix.