Weyl modules and $q$-Whittaker functions

Abstract

Let $G$ be a semi-simple simply connected group over $\mathbb {C}$. 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 $\varPsi _{\check{\lambda }}(q,z)$. This is a family of invariant polynomials on the maximal torus $T\subset G$ (here $z\in T$) depending on a dominant weight $\check{\lambda }$ of $G$ whose coefficients are rational functions in a variable $q\in \mathbb {C}^*$. 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 $\varPsi '_{\check{\lambda }}(q,z)$]. 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 $\hat{\varPsi }_{\check{\lambda }}(q,z)=\varPsi _{\check{\lambda }}(q,z)\cdot {\prod \nolimits _{i\in I}\prod \nolimits _{r=1}^{\langle \alpha _i,\check{\uplambda }\rangle }(1-q^r)}$ (here $I$ denotes the set of vertices of the Dynkin diagram of $G$) is equal to the character of a certain finite-dimensional $G[[{\mathsf {t}}]]\rtimes \mathbb {C}^*$-module $D(\check{\lambda })$ (the Demazure module). When $G$ is not simply laced a twisted version of the above statement holds. This result is known for $\varPsi _{\check{\lambda }}$ replaced by $\varPsi '_{\check{\lambda }}$ (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 $D(\check{\lambda })]$.