$q$-deformed Whittaker functions and Demazure modules

Whittaker functions are special functions on reductive groups, which are naturally arising in the theory of automorphic representations. The talk is devoted to recent results (joint with A. Gerasimov and D. Lebedev) on explicit construction of $q$-deformation of Whittaker functions for the group $GL(N,\mathbb{R})$. In the first part of my talk I will introduce two (integral) representations of $GL(N)$-Whittaker function, using two different limits of Macdonald polynomials. The first representation is a $q$-deformation of the classical Gelfand-Zetlin formula for the character. The other representation provides an identification of the constructed $q$-deformed Whittaker function with a character of certain Demazure module of the corresponding affine Lie algebra $\hat{\mathfrak{gl}}(N)$.