Nil-Hecke Algebras and Whittaker 𝔇-Modules

  • Victor Ginzburg
Part of the Progress in Mathematics book series (PM, volume 326)


Given a semisimple group G, Kostant and Kumar defined a nil-Hecke algebra that may be viewed as a degenerate version of the double affine nil-Hecke algebra introduced by Cherednik. In this paper, we construct an isomorphism of the spherical subalgebra of the nil-Hecke algebra with a Whittaker type quantum Hamiltonian reduction of the algebra of differential operators on G. This result has an interpretation in terms of geometric Satake and the Langlands dual group. Specifically, the isomorphism provides a bridge between very differently looking descriptions of equivariant Borel-Moore homology of the affine flag variety (due to Kostant and Kumar) and of the affine Grassmannian (due to Bezrukavnikov and Finkelberg), respectively.

It follows from our result that the category of Whittaker 𝔇-modules on G, considered by Drinfeld, is equivalent to the category of holonomic modules over the nil-Hecke algebra, and it is also equivalent to a certain subcategory of the category of Weyl group equivariant holonomic 𝔇-modules on the maximal torus.


Hamiltonian reduction differential operators Hecke algebra 

Mathematics Subject Classification (2010):

20G20 17B20 


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

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