Whittaker limits of difference spherical functions

TL;DR

This paper introduces the q-Whittaker function as a limit of q,t-spherical functions, providing new formulas and connections to Macdonald polynomials, Demazure characters, and Harish-Chandra asymptotics.

Contribution

It constructs a convergent series representation of the q-Whittaker function and extends classical formulas to arbitrary root systems with a q,t-counterpart of Harish-Chandra asymptotics.

Findings

Series converges everywhere as a generating function for q-Hermite polynomials

Generalizes the Shintani-Casselman-Shalika formula to arbitrary root systems

Provides a q,t-counterpart of the Harish-Chandra asymptotic formula

Abstract

We introduce the (global) q-Whittaker function as the limit at t=0 of the q,t-spherical function extending the symmetric Macdonald polynomials to arbitrary eigenvalues. The construction heavily depends on the technique of the q-Gaussians developed by the author (and Stokman in the non-reduced case). In this approach, the q-Whittaker function is given by a series convergent everywhere, a kind of generating function for multi-dimensional q-Hermite polynomials (closely related to the level 1 Demazure characters). One of the applications is a q-version of the Shintani- Casselman- Shalika formula, which appeared directly connected with q-Mehta- Macdonald identities in terms of the Jackson integrals. This formula generalizes that of type A due to Gerasimov et al. to arbitrary reduced root systems. At the end of the paper, we obtain a q,t-counterpart of the Harish-Chandra asymptotic formula for the spherical functions, including the Whittaker limit.