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

Victor Ginzburg*, David Kazhdan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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 T(G/U), the cotangent bundle. A long time ago, S. Gelfand and M. 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 Grassmannian for the Langlands dual group.

Original languageEnglish
Article number108368
JournalAdvances in Mathematics
Volume403
DOIs
StatePublished - 16 Jul 2022

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.

Keywords

  • Differential operators
  • Gelfand-Graev action
  • Semisimple group

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