Whittaker functions and Demazure operators

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 b...

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Bibliographic Details
Published in:Journal of number theory 2015-01, Vol.146, p.41-68
Main Authors: Brubaker, Ben, Bump, Daniel, Licata, Anthony
Format: Article
Language:English
Subjects:
Bott–Samelson varieties
Demazure operators
Iwahori subgroup
Whittaker functions
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Summary: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.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2014.01.001