Integration on 𝑝-adic groups and crystal bases

Bump, Daniel; Nakasuji, Maki

doi:10.1090/S0002-9939-09-10206-X

Integration on $p$-adic groups and crystal bases
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by Daniel Bump and Maki Nakasuji PDF

Proc. Amer. Math. Soc. 138 (2010), 1595-1605 Request permission

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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