Weyl modules and $$q$$q-Whittaker functions

By Alexander Braverman, Michael Finkelberg. 2012

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Let $$G$$G be a semi-simple simply connected group over $$\mathbb {C}$$C. Following Gerasimov et al. (Comm Math Phys 294:97–119, 2010) we use the $$q$$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 $$q$$q-Whittaker functions $$\varPsi _{\check{\lambda }}(q,z)$$Ψλˇ(q,z). This is a family of invariant polynomials on the maximal torus $$T\subset G$$T⊂G (here $$z\in T$$z∈T) depending on a dominant weight $$\check{\lambda }$$λˇ of $$G$$G whose coefficients are rational functions in a variable $$q\in \mathbb {C}^*$$q∈C∗. For a conjecturally the same (but a priori different) definition of the $$q$$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 $$q$$q-Whittaker functions from Cherednik (Int Math Res Notices 20:3793–3842, 2009) by $$\varPsi '_{\check{\lambda }}(q,z)$$Ψλˇ′(q,z)]. For $$G=SL(N)$$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 $$G$$G is simply laced, the function $$\hat{\varPsi }_{\check{\lambda }}(q,z)=\varPsi _{\check{\lambda }}(q,z)\cdot {\prod \nolimits _{i\in I}\prod \nolimits _{r=1}^{\langle \alpha _i,\check{\uplambda }\rangle }(1-q^r)}$$Ψ^λˇ(q,z)=Ψλˇ(q,z)·∏i∈I∏r=1〈αi,λˇ〉(1-qr) (here $$I$$I denotes the set of vertices of the Dynkin diagram of $$G$$G) is equal to the character of a certain finite-dimensional $$G[[{\mathsf {t}}]]\rtimes \mathbb {C}^*$$G[[t]]⋊C∗-module $$D(\check{\lambda })$$D(λˇ) (the Demazure module). When $$G$$G is not simply laced a twisted version of the above statement holds. This result is known for $$\varPsi _{\check{\lambda }}$$Ψλˇ replaced by $$\varPsi '_{\check{\lambda }}$$Ψλˇ′ (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 $$K$$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 $$D(\check{\lambda })]$$D(λˇ)].

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/41829c0dc8070e6b73c3f80df1b6bdda13ba066e