INTEGRATION ON p-ADIC GROUPS AND CRYSTAL BASES

Let G = GLr+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 ∏ (1− q−1zα)/(1− zα) 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 GLr+1 is deduced by using Kashiwara’s similarity of crystals.