### 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 language | English (US) |
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Pages (from-to) | 41-68 |

Number of pages | 28 |

Journal | Journal of Number Theory |

Volume | 146 |

Issue number | C |

DOIs | |

State | Published - Jan 1 2015 |

### Keywords

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

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## Cite this

*Journal of Number Theory*,

*146*(C), 41-68. https://doi.org/10.1016/j.jnt.2014.01.001