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

DOI10.1016/J.AIM.2022.108368zbMath1515.22005arXiv1804.05295WikidataQ114211517 ScholiaQ114211517MaRDI QIDQ6300358

Publication date: 14 April 2018

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.

Mathematics Subject Classification ID

Analysis on real and complex Lie groups (22E30) Rings of differential operators (associative algebraic aspects) (16S32)

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