On q-deformed gl(l+1)-Whittaker function II

TL;DR

This paper links q-deformed gl(l+1)-Whittaker functions to cohomology of line bundles on quasi-maps, providing geometric interpretations and connections to semi-infinite periods and q-Gamma functions.

Contribution

It offers a geometric representation of q-deformed Whittaker functions via cohomology on quasi-maps and relates them to semi-infinite periods and q-Gamma functions.

Findings

Representation of q-deformed Whittaker functions in terms of cohomology groups.

Realization of Mellin-Barnes integral as a semi-infinite period map.

Connection to Givental-Lee J-function of q-deformed gl(2)-Toda chain.

Abstract

A representation of a specialization of a q-deformed class one lattice gl(\ell+1}-Whittaker function in terms of cohomology groups of line bundles on the space QM_d(P^{\ell}) of quasi-maps P^1 to P^{\ell} of degree d is proposed. For \ell=1, this provides an interpretation of non-specialized q-deformed gl(2)-Whittaker function in terms of QM_d(\IP^1). In particular the (q-version of) Mellin-Barnes representation of gl(2)-Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of Gamma-function as a substitute of topological genus in semi-infinite geometry. A relation with Givental-Lee universal solution (J-function) of q-deformed gl(2)-Toda chain is also discussed.