Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0020
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
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 ...
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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.Less
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.
