2018/04/15 by Victor Ginzburg, David Kazhdan, Ginzburg, Victor +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1804.05295
openalex publication_date 2018/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.