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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Integration on $ p$-adic groups and crystal bases


Authors: Daniel Bump and Maki Nakasuji
Journal: Proc. Amer. Math. Soc. 138 (2010), 1595-1605
MSC (2010): Primary 17B37; Secondary 22E35, 11F85
Published electronically: December 29, 2009
MathSciNet review: 2587444
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Abstract: Let $ G=\mathrm{GL}_{r+1}$ over a nonarchimedean local field $ F$. The Kashiwara crystal $ \mathcal{B}(\infty)$ is the quantized enveloping algebra of the lower triangular maximal unipotent subgroup $ N_-$. Examples are given where an integral over $ N_-(F)$ may be replaced by a sum over $ \mathcal{B}(\infty)$. Thus the Gindikin-Karpelevich formula evaluates the integral of the standard spherical vector in the induced model of a principal series representation as a product $ \prod(1-q^{-1}\mathbf{z}^\alpha)/(1-\mathbf{z}^\alpha)$ where $ \mathbf{z}$ is the Langlands parameter and the product is over positive roots. This may also be expressed as a sum over $ \mathcal{B}(\infty)$. The corresponding equivalence over a metaplectic cover of $ \mathrm{GL}_{r+1}$ is deduced by using Kashiwara's similarity of crystals.


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Additional Information

Daniel Bump
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305-2125
Email: bump@math.stanford.edu

Maki Nakasuji
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305-2125
Email: nakasuji@math.stanford.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10206-X
PII: S 0002-9939(09)10206-X
Received by editor(s): August 12, 2009
Published electronically: December 29, 2009
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.