A combinatorial description of the Gindikin-Karpelevich formula

Abstract

The Gindikin-Karpelevich formula expresses the value of a certain p-adic integral over an algebraic group G as a product over the positive root system corresponding to the Langlands dual group G^v of G. We use the Young tableaux realization of the crystal basis B(∞) for the negative part of the quantum group corresponding to G^v to expand the latter product as a sum over the crystal B(∞). In other words, we define a statistic on tableaux which yields the appropriate co-efficient in the sum. This expansion is achieved when the crystal B(∞) is of non-exceptional finite type or type G ₂. We also interpret our statistic on tableaux in terms of Kamnitzer's MV polytopes and the Kashiwara-Saito geometric construction of B(∞) when the underlying Lie algebra is of type A_r.