Orthogonal polynomials, Whittaker functions and the local Langlands correspondence

Local Whittaker function is a special funciton on a group G over local
field F, defined as a matrix element in a principal series representation of G(F). In
the mid 70's Langlands conjectured, and Shintani and Casselman-Shalika proved, the
explicit formula, expressing the p-adic Whittaker function as a character of the dual
group G^. My talk is devoted to various aspects of Archimedean counterpart of the
Langlands-Shintani-Casselman-Shalika (LSCS) formula, discovered recently in
collaboration with A.Gerasimov and D.Lebedev.

In the first part of my talk I'll introduce basic constructions for
the Baxter Q-operator formalism for the Macdonald polynomials in type A. Next, I'll explain
special limits of Macdonald polynomials and outline its basic applications: the Archimedean
analog of the LSCS formula, and the Givental-Batyrev integral representations for (parabolic)
Whittaker functions, which are expected to describe Gromov-Witten invariants of partial
flag varieties GL(N)/P.