Weyl modules and q-Whittaker functions

Let G be a semi-simple simply connected group over ℂ. Following Gerasimov et al. (Comm Math Phys 294:97-119, 2010) we use the q-Toda integrable system obtained by quantum group version of the Kostant-Whittaker reduction (cf. Etingof in Am Math Soc Trans Ser 2:9-25, 1999, Sevostyanov in Commun Math Phys 204:1-16, 1999) to define the notion of q-Whittaker functions Ψ_λ(q,z). This is a family of invariant polynomials on the maximal torus T⊂G(here z∈ T) depending on a dominant weight λ of G whose coefficients are rational functions in a variable q∈ ℂ*. For a conjecturally the same (but a priori different) definition of the q-Toda system these functions were studied by Ion (Duke Math J 116:1-16, 2003) and by Cherednik (Int Math Res Notices 20:3793-3842, 2009) [we shall denote the q-Whittaker functions from Cherednik (Int Math Res Notices 20:3793-3842, 2009) by Ψ'_λ(q,z)]. For G=SL(N) these functions were extensively studied in Gerasimov et al. (Comm Math Phys 294:97-119, 2010; Comm Math Phys 294:121-143, 2010; Lett Math Phys 97:1-24, 2011). We show that when G is simply laced, the function (Formula presented.) (here I denotes the set of vertices of the Dynkin diagram of G) is equal to the character of a certain finite-dimensional G[[t]]⋊ ℂ*-module D(λ) (the Demazure module). When G is not simply laced a twisted version of the above statement holds. This result is known for Ψ_λ replaced by Ψ '_λ (cf. Sanderson in J Algebraic Combin 11:269-275, 2000 and Ion in Duke Math J 116:1-16, 2003); however our proofs are algebro-geometric [and rely on our previous work (Braverman, Finkelberg in Semi-infinite Schubert varieties and quantum K-theory of flag manifolds, arXiv/1111.2266, 2011)] and thus they are completely different from Sanderson (J Algebraic Combin 11:269-275, 2000) and Ion (Duke Math J 116:1-16, 2003) [in particular, we give an apparently new algebro-geometric interpretation of the modules D(λ)].