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Article

Weyl modules and q-Whittaker functions

Mathematische Annalen. 2014. Vol. 359. No. 1. P. 45-59.
Michael Finkelberg, Braverman A.

Let G be a semisimple simply connected algebraic Lie group over complex numbers. Following Gerasimov, Lebedev and Oblezin, we use the q-Toda integrable system obtained by the quantum group version of the Kostant-Whittaker reduction to define the notion of q-Whittaker functions. This is a family of invariant polynomials on the maximal torus T in G depending on a dominant weight of G, whose coefficients are rational functions in the variable q. For a conjecturally the same (but a priori different) definition of the q-Toda system these functions were studied by I.Cherednik. For G=SL(N) these functions were extensively studied by Gerasivom, Lebedev and Oblezin. We show that when G is simply laced, the Whittaker function is equal to the character of the global Weyl module. When G is not simply laced a twisted version of the above result holds. Our proofs are algebro-geometric.