Quantum affine Gelfand-Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces

TL;DR

This paper constructs actions of quantum loop and toroidal algebras on the equivariant K-theory of Laumon spaces, providing explicit formulas in the affine Gelfand-Tsetlin basis, linking geometric representation theory with algebraic structures.

Contribution

It introduces a novel geometric realization of quantum affine and toroidal algebras via Laumon spaces and provides explicit formulas in the Gelfand-Tsetlin basis.

Findings

Constructed quantum loop algebra action in K-theory of Laumon spaces.

Constructed quantum toroidal algebra action in affine Laumon spaces.

Derived explicit formulas in the affine Gelfand-Tsetlin basis.

Abstract

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL_n. We construct the action of the quantum loop algebra U_v(Lsl_n) in the equivariant K-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra U^{tor}_v(Lsl}_n) in the equivariant K-theory of the affine version of Laumon spaces. We write down explicit formulae for this action in the affine Gelfand-Tsetlin base, corresponding to the fixed point base in the localized equivariant K-theory.