Differential operators on G/U and the Gelfand-Graev action
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.
Cite
@article{arxiv.1804.05295,
title = {Differential operators on G/U and the Gelfand-Graev action},
author = {Victor Ginzburg and David Kazhdan},
journal= {arXiv preprint arXiv:1804.05295},
year = {2022}
}
Comments
Several parts rewritten. Final version to appear in Advances in Math.(2022) https://doi.org/10.1016/j.aim.2022.108368