Integration on $p$-adic groups and crystal bases

Abstract

Let G = GL r+1 over a nonarchimedean local field F. The Kashiwara crystal B(∞) 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 B(∞). 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 Π(1 ― q ― z α )/(1 ― z α ) where z is the Langlands parameter and the product is over positive roots. This may also be expressed as a sum over B(∞). The corresponding equivalence over a metaplectic cover of GL r+1 is deduced by using Kashiwara's similarity of crystals.