Unique functionals and representations of Hecke algebras

Benjamin Brubaker, Daniel Bump and Solomon Friedberg

Vol. 260 (2012), No. 2, 381–394

DOI: 10.2140/pjm.2012.260.381

Abstract

Rogawski (1985) used the affine Hecke algebra to model the intertwining operators of unramified principal series representations of p-adic groups. On the other hand, a representation of this Hecke algebra in which the standard generators act by Demazure–Lusztig operators was introduced by Lusztig (1989) and applied by Kazhdan and Lusztig (1987) to prove the Deligne–Langlands conjecture. These operators appear in various other contexts. Ion (2006) used them to express matrix coefficients of principal series representations in terms of nonsymmetric Macdonald polynomials, while Brubaker, Bump and Licata (2011) found essentially the same operators underlying recursive relationships for Whittaker functions. Here we explain the role of unique functionals and Hecke algebras in these contexts and revisit the results of Ion from the point of view of Brubaker et al.

Keywords

Hecke algebra, unramified principal series, Demazure–Lusztig operator, unique functional

Mathematical Subject Classification 2010

Primary: 22E50

Secondary: 22E35, 33D52

Milestones

Received: 13 September 2012

Accepted: 6 October 2012

Published: 30 November 2012