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

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 also quasi-classical counterpart: regular functions T?(G/U), cotangent bundle. A long time ago, S. Gelfand M. Graev have constructed an action Weyl by automorphisms. The Gelfand-Graev construction was not algebraic, it involved analytic methods in essential way. give new action, as well counterpart. Our approach is based Hamiltonian reduction involves ring Whittaker G/U, twisted analogue D(G/U). main result has interpretation, via geometric Satake, terms spherical perverse sheaves affine Grassmannian for Langlands dual group.