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One of the main problems of geometry is to find discrete invariants distinguishing geometric objects up to some equivalence. In algebraic geometry, the classical approach, based on ideas of Riemann, Hurwitz, Lefschetz, consists of representations of complex algebraic manifolds either as finite coverings of the projective spaces (generic coverings) or as codimension one fibrations over the projective line (Lefschetz pencils).

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.