# Equivariant Satake category and Kostant-Whittaker reduction

@article{Bezrukavnikov2007EquivariantSC, title={Equivariant Satake category and Kostant-Whittaker reduction}, author={Roman Bezrukavnikov and Michael Finkelberg}, journal={arXiv: Representation Theory}, year={2007} }

We explain (following V. Drinfeld) how the equivariant derived category of the affine Grassmannian can be described in terms of coherent sheaves on the Langlands dual Lie algebra equivariant with respect to the adjoint action, due to some old results of V. Ginzburg. The global cohomology functor corresponds under this identification to restricti on to the Kostant slice. We extend this description to loop rotation equivariant derived category, linking it to Harish-Chandra bimodules for the…

## 61 Citations

Twisted Satake Category

- Mathematics
- 2012

We extend Bezrukavnikov and Finkelberg's description of the G(\C[[t]])-equivariant derived category on the affine Grassmannian to the twisted setting of Finkelberg and Lysenko. Our description is in…

Nil-Hecke Algebras and Whittaker -Modules

- Mathematics
- 2018

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…

Equivariant Coherent Sheaves , Soergel Bimnodules , and Categorification of Affine Hecke Algebras by MASSACHUSETTS INSTITUTE

- Mathematics
- 2011

In this thesis, we examine three different versions of "categorification" of the affine Hecke algebra and its periodic module: the first is by equivariant coherent sheaves on the Grothendieck…

Dynamical Weyl groups and equivariant cohomology of transversal slices on affine Grassmannians

- Mathematics
- 2010

Let G be a reductive group; in this note we give an interpretation of the dynamical Weyl group of of the Langlands dual group $\check{G}$ defined by Etingof and Varchenko in terms of the geometry of…

Loop Spaces and Langlands Parameters

- Mathematics
- 2007

We apply the technique of S^1-equivariant localization to sheaves on loop spaces in derived algebraic geometry, and obtain a fundamental link between two families of categories at the heart of…

SOME RESULTS RELATED TO THE QUANTUM GEOMETRIC LANGLANDS PROGRAM

- Mathematics
- 2013

One of the fundamental results in geometric
representation theory is the geometric Satake equivalence, between
the category of spherical perverse sheaves on the affine
Grassmannian of a reductive…

On two geometric realizations of an affine Hecke algebra

- Mathematics
- 2012

The article is a contribution to the local theory of geometric Langlands duality. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra associated to a…

Betti Geometric Langlands

- MathematicsAlgebraic Geometry: Salt Lake City 2015
- 2018

We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of…

CHIRAL ALGEBRAS OF CLASS S AND MOORE-TACHIKAWA SYMPLECTIC VARIETIES

- Mathematics
- 2018

We give a functorial construction of the genus zero chiral algebras of class S, that is, the vertex algebras corresponding to the theory of class S associated with genus zero punctured Riemann…

Kazhdan–Lusztig conjecture via zastava spaces

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2022

Abstract We deduce the Kazhdan–Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category 𝒪 {\mathcal{O}} from the equivariant…

## References

SHOWING 1-10 OF 43 REFERENCES

Loop Spaces and Langlands Parameters

- Mathematics
- 2007

We apply the technique of S^1-equivariant localization to sheaves on loop spaces in derived algebraic geometry, and obtain a fundamental link between two families of categories at the heart of…

Perverse sheaves on affine flags and langlands dual group

- Mathematics
- 2002

This is the first in a series of papers devoted to describing the category of sheaves on the affine flag manifold of a simple algebraic group in terms of the Langlands dual group. In the present…

Quantum groups, the loop Grassmannian, and the Springer resolution

- Mathematics
- 2003

We establish equivalences of derived categories of the following 3 categories:
(1) Principal block of representations of the quantum at a root of 1;
(2) G-equivariant coherent sheaves on the…

Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves

- Mathematics
- 2004

The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. It turns out to be related to…

Loop Grassmannian cohomology, the principal nilpotent and Kostant theorem

- Mathematics
- 1998

Given a complex projective algebraic variety, write H(X) for its cohomology with complex coefficients and IH(X) for its Intersection cohomology. We first show that, under some fairly general…

Perverse sheaves and *-actions

- Mathematics
- 1991

where t * x denotes the action of t on x. The set Xw is known to be a locally-closed C* -stable algebraic subvariety of X isomorphic to an affine space. The pieces Xw form a cell decomposition X =…

Koszul Duality Patterns in Representation Theory

- Mathematics
- 1996

The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to repre- sentation theory. The paper consists of three parts…

Quantum Groups

- Mathematics
- 1993

This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions…

Geometric Langlands duality and representations of algebraic groups over commutative rings

- Mathematics
- 2004

As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the…

Quantum cohomology of flag manifolds G/B and quantum Toda lattices

- Mathematics
- 1996

Let G be a connected semi-simple complex Lie group, B its Borel subgroup, T a maximal complex torus contained in B, and Lie (T ) its Lie algebra. This setup gives rise to two constructions; the…