Vol. 260, No. 2, 2012

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Unique functionals and representations of Hecke algebras

Benjamin Brubaker, Daniel Bump and Solomon Friedberg

Vol. 260 (2012), No. 2, 381–394

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.

Hecke algebra, unramified principal series, Demazure–Lusztig operator, unique functional
Mathematical Subject Classification 2010
Primary: 22E50
Secondary: 22E35, 33D52
Received: 13 September 2012
Accepted: 6 October 2012
Published: 30 November 2012
Benjamin Brubaker
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
United States
Daniel Bump
Department of Mathematics
Stanford University
Building 380
Stanford, CA 94305-2125
United States
Solomon Friedberg
Department of Mathematics
Boston College
Chestnut Hill MA 02467-3806
United States