The Representation theory of p-adic GL(n) and Deligne-Langlands parameters

Abstract

In this article we cover an episode in the representation theory of GL(n) defined over a p-adic field with finite residue class field. We concentrate on the irreducible tempered representations admitting non-zero Iwahori-fixed vectors. We describe the space of these representations in terms of Deligne-Langlands parameters. In [6], Kazdhan and Lusztig prove the Deligne-Langlands conjecture for split reductive p-adic groups with connected centre. For GL(n), this conjecture amounts to a parametrization of such representations by certain pairs (s, N) satisfying the equation sNs⁻¹ = qN where q is the cardinality of the residue field. We discuss these parameters in §3. In §4 and §5 we discuss the theory of Zelevinsky segments and prove results concerning the form of irreducible representations of GL(n) admitting non-zero Iwahori-fixed vectors. In the final section we define the Brylinski quotient Bryl(n) for the space Tⁿ equipped with the natural action of the symmetric group S _n and prove that the space of Deligne-Langlands parameters of these representations is homeomorphic to Bryl(n).