The Gindikin-Karpelevich formula, the Casselman-Shalika formula, and crystals of tableaux
Abstract: Each of the Gindikin-Karpelevich formula and the Casselman-Shalika formula is the evaluation of a certain p-adic integral as a product over a positive root system. Loosely speaking, the former may be thought of as a Verma module identity, while the latter may be thought of as its irreducible highest weight module analogue. In recent work, Bump-Nakasuji, Brubaker-Bump-Friedberg, Kim-Lee, and McNamara have succeeded in expressing these same formulas as sums over the corresponding crystal bases (or corresponding canonical bases) using various methods. In joint work with K.-H. Lee and Lee-Lombardo, respectively, we use the explicit realization of the relevant crystals in terms of Young tableaux to obtain refined expressions for both formulas when the underlying root system is of type A. In this talk, I will present our results for both formulas in the type A case.