Lattice Models for Double Whittaker Polynomials and Motivic Chern Classes

Abstract

We will describe solvable lattice models whose partition functions depend on two sets of variables, $x_1,\cdots,x_n$ and $y_1, y_2, \cdots $ that have different connections with the representation theory of $\text{GL}(n,F)$ where $F$ is a nonarchimedean local field. If the boundary conditions are chosen in one way, they are essentially the Motivic Chern classes that were used very effectively by Aluffi, Mihalcea, Sch\"urmann and Su (AMSS) to study such problems. In particular, using this specialization we can obtain deformations $r_{u,v}$ of the Kazhdan-Lusztig R-polynomials that were used by Bump, Nakasuji and Naruse to study matrix coefficients of intertwining operators (introduced by Casselman). Thus we are able see that the recursion formula for the $r_{u,v}$ is a reflection of the Yang-Baxter equation. On the other hand, with more general boundary conditions, specializing the parameters $y_i\to 0$ we recover colored lattice models that were previously used by Brubaker, Buciumas, Bump and Gustafsson to represent Iwahori Whittaker functions on $GL(n,F)$. Thus we term the resulting two-variable-set family of functions as ``double Whittaker polynomials.''