Non-Archimedean Whittaker functions as characters: a probabilistic approach to the Shintani-Casselman-Shalika formula

For a reductive group $G$ over a non-Archimedean local field (e.g $GL_n( \mathbb{Q}_p )$ ), Jacquet's Whittaker function is essentially proportional to a character of an irreducible representation of the Langlands dual group $G^\vee( \mathbb{C} )$ ( a Schur function if $G = GL_n( \mathbb{Q}_p )$). 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 in the coweight lattice. Our presentation goes along the following lines. Thanks to a minuscule random walk $W^{(z)}$ on the coweight lattice and a related random walk on the Borel subgroup, we establish 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. A simple manipulation reduces the problem to evaluating the probability of $W^{(z)}$ never exiting the Weyl chamber. Then, an implementation of the reflection principle forces the appearance of the Weyl character formula and therefore retrieves characters of $G^\vee\left( \mathbb{C} \right)$. The construction of the random walk on the Borel subgroup requires some care. It is extracted from a spherical random walk whose increments have a distribution that can be understood as elements from the spherical Hecke algebra.