Related papers: An example of the derived geometrical Satake corre…

In this paper we give a geometric version of the Satake isomorphism. Given a connected complex reductive algebraic group, we show that the category of representations of its Langlands dual is naturally equivalent to a certain category of…

For a simply-connected simple algebraic group $G$ over $\C$, we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of $G$, generalizing a well-known fact about $GL_n$. Using this variety, we…

The geometric Satake equivalence and the Springer correspondence are closely related when restricting to small representations of the Langlands dual group. We prove this result for \'etale sheaves, including the case of the mixed…

This is an informal expository article on geometric Satake correspondence for affine Kac-Moody Lie algebras of type $A$ given in arXiv:1810.04293. We emphasize formal analogies between this result and the author's earlier results on…

The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group $ G $ and the spherical perverse sheaves on the affine Grassmannian $Gr$ of its Langlands dual group.…

We calculate various categories of equivariant sheaves on the Beilinson-Drinfeld Grassmannian in Langlands dual terms. For one, we obtain the factorizable derived geometric Satake theorem. More generally, we calculate the categorical…

Fargues-Scholze developed a framework for the geometric Langlands program on the Fargues-Fontaine curve. In particular, they proved the geometric Satake equivalence on the moduli space of closed Cartier divisors on the curve. We prove the…

The geometric Satake isomorphism is an equivalence between the categories of spherical perverse sheaves on affine Grassmanian and the category of representations of the Langlands dual group. We provide a similar description for derived…

Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of…

We show that tilting modules and parity sheaves on the affine Grassmannian are related through the geometric Satake correspondence, when the characteristic is bigger than an explicit bound.

This is an expanded version of the text ``Perverse Sheaves on Loop Grassmannians and Langlands Duality'', AG/9703010. The main new result is a topological realization of algebraic representations of reductive groups over arbitrary rings. We…

The geometric Satake correspondence provides an equivalence of categories between the Satake category of spherical perverse sheaves on the affine Grassmannian and the category of representations of the dual group. In this note, we define a…

Suppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We…

The category of perverse sheaves on the affine Grassmannian of a complex reductive group $G$ gives a canonical geometric construction of the split form of the Langlands dual group $\check G_\bZ$ over the integers. Given a field $k$, we give…

A module over an affine Kac--Moody algebra g^ is called spherical if the action of the Lie subalgebra g[[t]] on it integrates to an algebraic action of the corresponding group G[[t]]. Consider the category of spherical g^-modules of…

The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant…

We discuss what is known about the structure of the bounded derived categories of coherent sheaves on Grassmannians of simple algebraic groups.

We formulate a Satake isomorphism for the integral spherical Hecke algebra of an unramified $p$-adic group $G$ and generalize the formulation to give a description of the Hecke algebra $H_G(V)$ of weight $V$, where $V$ is a lattice in an…

We show that the cotangent bundle $T^*(G/K)$ of a quasi-split symmetric space $G/K$ is isomorphic to the dual variety of the loop symmetric space for the Langlands dual group, providing instances of the relative Langlands duality for…

We outline a proof of a geometric version of the Satake isomorphism. Given a connected, complex algebraic reductive group G we show that the tensor category of representations of the dual group $\check G$ is naturally equivalent to a…