Whittaker functions and Demazure operators

Ben Brubaker, Daniel Bump, Anthony Licata

Research output: Contribution to journalArticle

12 Scopus citations

Abstract

We show that elements of a natural basis of the Iwahori fixed vectors in a principal series representation of a reductive p-adic group satisfy certain recursive relations. The precise identities involve operators that are variants of the Demazure-Lusztig operators, with correction terms, which may be calculated by a combinatorial algorithm that is identical to the computation of the fibers of the Bott-Samelson resolution of a Schubert variety. This leads to an action of the affine Hecke algebra on functions on the maximal torus of the L-group. A closely related action was previously described by Lusztig using equivariant K-theory of the flag variety, leading to the proof of the Deligne-Langlands conjecture by Kazhdan and Lusztig. In the present paper, the action is applied to give a simple formula for the basis vectors of the Iwahori Whittaker functions. We also show that these Whittaker functions can be expressed as non-symmetric Macdonald polynomials.

Original languageEnglish (US)
Pages (from-to)41-68
Number of pages28
JournalJournal of Number Theory
Volume146
Issue numberC
DOIs
StatePublished - Jan 1 2015

Keywords

  • Bott-Samelson varieties
  • Demazure operators
  • Iwahori subgroup
  • Whittaker functions

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