How to Come Up with Solvable Lattice Models?

Values of matrix coefficients of p-adic groups can be written in terms of solvable lattice models. But the usual argument for that is ad hoc -- you first know the model and then show that the partition functions match the values of matrix coefficients. In my talk, I'll show how one can start with the p-adic side and naturally come up with solvable lattice models. My results extend Peter J. McNamara's approach from 0907.2675.