A combinatorial description of the affine Gindikin-Karpelevich formula of type A_n^(1)

TL;DR

This paper provides a combinatorial description of the affine Gindikin-Karpelevich formula of type A_n^(1) by expressing it as a sum over generalized Young walls, with explicitly determined coefficients.

Contribution

It introduces a novel combinatorial framework using generalized Young walls to express the affine Gindikin-Karpelevich formula for type A_n^(1).

Findings

Formula expressed as a sum over generalized Young walls.

Coefficients explicitly determined by Young wall combinatorics.

Connects geometric and combinatorial approaches in affine Kac-Moody theory.

Abstract

The classical Gindikin-Karpelevich formula appears in Langlands' calculation of the constant terms of Eisenstein series on reductive groups and in Macdonald's work on p-adic groups and affine Hecke algebras. The formula has been generalized in the work of Garland to the affine Kac-Moody case, and the affine case has been geometrically constructed in a recent paper of Braverman, Finkelberg, and Kazhdan. On the other hand, there have been efforts to write the formula as a sum over Kashiwara's crystal basis or Lusztig's canonical basis, initiated by Brubaker, Bump, and Friedberg. In this paper, we write the affine Gindikin-Karpelevich formula as a sum over the crystal of generalized Young walls when the underlying Kac-Moody algebra is of affine type A_n^(1). The coefficients of the terms in the sum are determined explicitly by the combinatorial data from Young walls.