# Mathematics Colloquia and Seminars

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, Brubaker-Bump-Friedberg, Bump-Nakasuji, 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, mention generalizations to other root systems, and allude to possible connections with special functions from algebraic combinatorics.