Integration on $p$-adic groups and crystal bases
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- by Daniel Bump and Maki Nakasuji PDF
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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.References
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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
- Received by editor(s): August 12, 2009
- Published electronically: December 29, 2009
- Communicated by: Wen-Ching Winnie Li
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 1595-1605
- MSC (2010): Primary 17B37; Secondary 22E35, 11F85
- DOI: https://doi.org/10.1090/S0002-9939-09-10206-X
- MathSciNet review: 2587444