A combinatorial description of the affine Gindikin-Karpelevich formula of type A_{n}(1)

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 (1) n . The coefficients of the terms in the sum are determined explicitly by the combinatorial data from Young walls. 0. Introduction The classical Gindikin-Karpelevich formula originated from a certain integration on real reductive groups [GK62]. When Langlands calculated the constant terms of Eisenstein series on reductive groups [Lan71], he considered a p-adic analogue of the integration and called the resulting formula the Gindikin-Karpelevich formula. In the case of GLn+1, the formula can be described as follows: let F be a p-adic field with residue field of q elements and let N− be the maximal unipotent subgroup of GLn+1(F ) with maximal torus T . Let f ◦ denote the standard spherical vector corresponding to an unramified character χ of T , let T (C) be the maximal torus in the L-group GLn+1(C) of GLn+1(F ), and let z ∈ T (C) be the element corresponding to χ via the Satake isomorphism. Then the Gindikin-Karpelevich 2010 Mathematics Subject Classification. Primary 17B37; Secondary 05E10.