TY - JOUR
T1 - Differential operators on G/U and the Gelfand-Graev action
AU - Ginzburg, Victor
AU - Kazhdan, David
N1 - Publisher Copyright:
© 2022 Elsevier Inc.
PY - 2022/7/16
Y1 - 2022/7/16
N2 - 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.
AB - 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.
KW - Differential operators
KW - Gelfand-Graev action
KW - Semisimple group
UR - http://www.scopus.com/inward/record.url?scp=85128858504&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2022.108368
DO - 10.1016/j.aim.2022.108368
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AN - SCOPUS:85128858504
SN - 0001-8708
VL - 403
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 108368
ER -