The Gindikin-Karpelevich formula

Abstract

The Gindikin-Karpelevich formula expresses a certain power series involving cardinalities of intersections of double cosets in a p-adic Lie group (or loop group) as a product over the associated root system. It is closely related to Macdonald's celebrated formula for the spherical function on a p-adic Lie group. We discuss the combinatorics of the Gindikin-Karpelevich formula and the connections with the geometry of the associated affine building. This is the initial stage of a work in progress with Joel Kamnitzer (University of Toronto). One aim is to find a Hecke-algebraic proof of the affine Gindikin-Karpelevich formula of Braverman-Finkelberg-Kazhdan (which is an analogous formula for loop groups of affine Kac-Moody groups).