Dynamical Weyl groups and equivariant cohomology of transversal slices in affine Grassmannians

TL;DR

This paper interprets the dynamical Weyl group of the Langlands dual group using the geometry of affine Grassmannians, linking algebraic parameters to geometric equivariant parameters, and suggests a possible extension to affine Kac-Moody groups.

Contribution

It provides a geometric interpretation of the dynamical Weyl group in terms of equivariant cohomology of affine Grassmannian slices, connecting algebraic and geometric perspectives.

Findings

Interpretation of dynamical Weyl group via affine Grassmannian geometry

Identification of dynamical parameters with equivariant parameters

Conjectural extension to affine Kac-Moody groups

Abstract

Let G be a reductive group; in this note we give an interpretation of the dynamical Weyl group of of the Langlands dual group $\check{G}$ defined by Etingof and Varchenko in terms of the geometry of the affine Grassmannian Gr of G. In this interpretation the dynamical parameters of Etingof and Varchenko correspond to equivariant parameters with respect to certain natural torus acting on Gr. We also present a conjectural generalization of our results to the case of affine Kac-Moody groups.