On q-Deformed {mathfrak{gl}_{ell+1}} -Whittaker Function I

We propose a new explicit form of q-deformed Whittaker functions solving q-deformed ${mathfrak{gl}_{ell+1}}A representation of a specialization of a q-deformed class one lattice mathfrakgl_l+1{mathfrak{gl}_{ell+1}}-Whittaker function in terms of cohomology groups of line bundles on the space QM_d(mathbbP^l){mathcal{QM}_d(mathbb{P}^{ell})} of quasi-maps mathbbP¹ ? mathbbP^l{mathbb{P}^1 to mathbb{P}^{ell}} of degree d is proposed. For ℓ = 1, this provides an interpretation of the non-specialized q-deformed mathfrakgl₂{mathfrak{gl}_{2}}-Whittaker function in terms of QM_d(mathbbP¹){mathcal{QM}_d(mathbb{P}^1)}. In particular the (q-version of the) Mellin-Barnes representation of the mathfrakgl₂{mathfrak{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 Γ-function as a topological genus in semi-infinite geometry. A relation with the Givental-Lee universal solution (J-function) of q-deformed mathfrakgl₂{mathfrak{gl}_2}-Toda chain is also discussed.