Non-Archimedean Whittaker Functions as Characters: A Probabilistic Approach to the Shintani–Casselman–Shalika Formula

Abstract

Let G be reductive group over a non-Archimedean local field (e.g GL n (Q p ) ) and G ∨ (C) its Langlands dual. Jacquet's Whittaker function on G is essentially proportional to the character of an irreducible representation of G ∨ (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.MSC 2010 subject classifications: 11F70, 11F85, 60B15, 60J45, 60J50

Point-to-line polymers and orthogonal Whittaker functions

“…The reason for this is that SO 2n+1 -Whittaker functions have an integral representation over an analogue of the Gelfand-Tsetlin patterns that correspond to the symplectic group Sp 2n (see Definition 2.4 and Figure 3b), which is dual to the orthogonal group SO 2n+1 . This is in agreement with the Casselman-Shalika [CS80] formula which describes the (unramified) Whittaker functions of a group G as characters of a finite dimensional representation of the dual group of G (see also [Ch16] for a probabilistic approach): the dual group of SO 2n+1 is Sp 2n , while the dual of GL n is itself, hence GL n -Whittaker functions are the analogue of Schur functions, while SO 2n+1 -Whittaker functions are the analogue of Sp 2n -Schur functions.…”

“…The latter reference provides also a comprehensive account of various algebraic properties and origins of Whittaker functions. Further extensions in this direction, in both the "Archimedean" and "non-Archimedean" cases have been achieved in [Ch15,Ch16]. In [Nte17] SO 2n+1 (R)-Whittaker functions also emerged in the description of the Markovian dynamics of systems of interacting particles restricted by a soft wall.…”

Limits and fluctuations of $p$-adic random matrix products

“…The family of probability distributions considered in [15] is more general than that of Theorem 1.3, but the latter covers cases when the matrices A i are not identically distributed and shows Gaussian fluctuations, which are not shown in [15]. A different family of random walks on GL n (Q p ) were studied by Chhaibi [19]; the perspective in this work is more similar to ours in that it is heavily based on special functions on p-adic groups, though the presentation and problems considered are quite different.…”