Combinatorics of Casselman-shalika Formula in Type A

In the recent works of Brubaker-Bump-Friedberg, Bump-Nakasuji, and others, the product in the Casselman-Shalika formula is written as a sum over a crystal. The coefficient of each crystal element is defined using the data coming from the whole crystal graph structure. In this paper, we adopt the tableau model for the crystal and obtain the same coefficients using data from each individual tableau; i.e., we do not need to look at the graph structure. We also show how to combine our results with tensor products of crystals to obtain the sum of coefficients for a given weight. The sum is a q-polynomial which exhibits many interesting properties. We use examples to illustrate these properties. 0. Introduction Let F be a p-adic field with a ring of integers oF and residue field of size q. We denote by a uniformizer of oF . Suppose N − is the maximal unipotent subgroup of GLr+1(F ) with maximal torus T , and f ◦ denotes the standard spherical vector corresponding to an unramified character χ of T . Let T (C) be the maximal torus in the dual group GLr+1(C) of GLr+1(F ), and let z ∈ T (C) be the element corresponding to χ via the Satake isomorphism. For a dominant integral weight λ = (λ1 ≥ λ2 ≥ · · · ≥ λr+1), we define ψλ ⎜⎜⎜⎝ 1 x2,1 1 .. . . . xr+1,1 · · · xr+1,r 1 ⎟⎟⎟⎠ = ψ0( 12xr+1,r + · · ·+ rr+1x2,1), where ψ0 is a fixed additive character on F which is trivial on oF but not on p −1. Let χλ(z) be the irreducible character of GLr+1(C) with highest weight λ. Then the Casselman-Shalika formula is