Differential operators on G/U and the Gelfand-Graev action

TL;DR

This paper provides an algebraic construction of the Gelfand-Graev action on differential operators on G/U, connecting it to Hamiltonian reduction, Whittaker operators, and geometric Satake, advancing the understanding of these symmetries.

Contribution

It introduces a new algebraic approach to the Gelfand-Graev action using Hamiltonian reduction and Whittaker operators, replacing previous analytic methods.

Findings

Algebraic construction of Gelfand-Graev action

Connection to Hamiltonian reduction and Whittaker operators

Interpretation via geometric Satake and affine Grassmannian

Abstract

Let G be a complex semisimple group and U its maximal unipotent subgroup. We study the algebra D(G/U) of algebraic differential operators on G/U and also its quasi-classical counterpart: the algebra of regular functions on the cotangent bundle. A long time ago, Gelfand and Graev have constructed an action of the Weyl group on D(G/U) by algebra automorphisms. The Gelfand-Graev construction was not algebraic, it involved analytic methods in an essential way. We give a new algebraic construction of the Gelfand-Graev action, as well as its quasi-classical counterpart. Our approach is based on Hamiltonian reduction and involves the ring of Whittaker differential operators on G/U, a twisted analogue of D(G/U). Our main result has an interpretation, via geometric Satake, in terms of spherical perverse sheaves on the affine Grassmanian for the Langlands dual group.