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Weyl Group Multiple Dirichlet SeriesType A Combinatorial Theory (AM-175)$
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Ben Brubaker, Daniel Bump, and Solomon Friedberg

Print publication date: 2011

Print ISBN-13: 9780691150659

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691150659.001.0001

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Crystals and p-adic Integration

Crystals and p-adic Integration

(p.132) Chapter Twenty Crystals and p-adic Integration
Weyl Group Multiple Dirichlet Series

Ben Brubaker

Daniel Bump

Solomon Friedberg

Princeton University Press

This chapter describes the properties of Kashiwara's crystal and its role in unipotent p-adic integrations related to Whittaker functions. In many cases, integrations of representation theoretic import over the maximal unipotent subgroup of a p-adic group can be replaced by a sum over Kashiwara's crystal. Partly motivated by the crystal description presented in Chapter 2 of this book, this perspective was advocated by Bump and Nakasuji. Later work by McNamara and Kim and Lee extended this philosophy yet further. Indeed, McNamara shows that the computation of the metaplectic Whittaker function is initially given as a sum over Kashiwara's crystal. The chapter considers Kostant's generating function, the character of the quantized enveloping algebra, and its association with Kashiwara's crystal, along with the Kostant partition function and the Weyl character formula.

Keywords:   p-adic integration, Whittaker function, p-adic group, Kashiwara's crystal, generating function, Kostant partition function, Weyl character formula

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