Non-Archimedean Whittaker Functions as Characters: A Probabilistic Approach to the Shintani–Casselman–Shalika Formula
Résumé
Let G be reductive group over a non-Archimedean local field (e.g., GL(n)(Q(p))) and G(boolean OR)(C) its Langlands dual. Jacquet's Whittaker function on G is essentially proportional to the character of an irreducible representation of G(boolean OR)(C) (a Schur function if G = GL(n)(Q(p))). We propose a probabilistic approach to this claim, known as the Shintani-Casselman-Shalika formula, when the group G has at least one minuscule cocharacter. Thanks to random walks on the group, we start by establishing a Poisson kernel formula for the non-Archimedean Whittaker function. The expression and its ingredients are similar to the one previously obtained by the author in the Archimedean case, hence a unified point of view.