Nil Hecke algebras and Whittaker D-modules

TL;DR

This paper establishes a deep connection between nil Hecke algebras, Whittaker D-modules, and geometric representation theory, revealing new equivalences and interpretations in the context of the Langlands program.

Contribution

It constructs an isomorphism linking the spherical subalgebra of the nil Hecke algebra with a Whittaker quantum Hamiltonian reduction, bridging different geometric and algebraic frameworks.

Findings

Isomorphism between spherical nil Hecke algebra and Whittaker quantum reduction

Equivalence of Whittaker D-modules with holonomic modules over nil Hecke algebra

Connection between equivariant homology of affine flag variety and affine Grassmannian

Abstract

Given a reductive 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 the 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.