English

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

Representation Theory 2022-04-05 v4 Algebraic Geometry

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.

Keywords

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

R2 v1 2026-06-23T01:23:52.011Z