2008/07/14 by Ivan Cherednik, Cherednik, Ivan
Mathematics · #FOS: Mathematics #Mathematical functions and polynomials #Point processes and geometric inequalities #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.0807.2155
openalex publication_date 2008/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.