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.
Bibliographical noteFunding Information:
We thank Bogdan Ion for pointing out that our recursion could be used to connect to a specialization of the non-symmetric Macdonald polynomial. We also thank Gautam Chinta, Solomon Friedberg, Paul Gunnells, David Kazhdan, Daniel Orr, Arun Ram, Mark Reeder, and Anne Schilling for helpful conversations. Schilling, Mark Shimozono, and Nicolas Thiéry wrote Sage code for non-symmetric Macdonald polynomials which was instrumental in helping us to refine our results. This work was supported by NSF grants DMS-0652817 , DMS-0844185 , and DMS-1001079 .
© 2014 Elsevier Inc.
- Bott-Samelson varieties
- Demazure operators
- Iwahori subgroup
- Whittaker functions