Integration on $p$-adic groups and crystal bases

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.