Related papers: A combinatorial geometric Satake equivalence

This is a companion paper of arXiv:1909.11492 and arXiv:1912.01930. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of twisted $Sp(2n,{\mathbb C}[\![t]\!])$-equivariant…

Let $G_\mathbb R$ be a connected real reductive group and let $X$ be the corresponding complex symmetric variety under the Cartan bijection. We construct a canonical equivalence between the relative Satake category of $G(\mathcal…

We lift the affine Matsuki correspondence between real and symmetric loop group orbits in affine Grassmannians to an equivalence of derived categories of sheaves. In analogy with the finite-dimensional setting, our arguments depend upon the…

Let G^\vee be a complex simple algebraic group. We describe certain morphisms of G^\vee(\calO)-equivariant complexes of sheaves on the affine Grassmannian \Gr of G^\vee in terms of certain morphisms of G-equivariant coherent sheaves on…

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…

For a reductive group $G$ we equip the category of $G_\mathcal{O}$-equivariant polarizable pure Hodge modules on the affine Grassmannian $\mathrm{Gr}_G$ with a structure of neutral Tannakian category. We show that it is equivalent to a…

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.

In this paper we prove, for G a connected reductive algebraic group satisfying a technical assumption, that the Satake category of G (with coefficients in a finite field, a finite extension of Q_l, or the ring of integers of such a field)…

We propose a geometric realization of the Feigin-Loktev fusion product of graded cyclic modules over the current algebra. This allows us to compute it in several new cases. We also relate the Feigin-Loktev fusion product to the convolution…

I extend the ramified geometric Satake equivalence of Zhu from tamely ramified groups to include the case of general connected reductive groups. As a prerequisite I prove basic results on the geometry of affine flag varieties.

We define a right Cartan-Eilenberg structure on the category of Kan's combinatorial spectra, and the category of sheaves of such spectra, assuming some conditions. In both structures, we use the geometric concept of homotopy equivalence as…

We introduce various affine Grassmannians, study their geometric properties, and give some applications. We also discuss the geometric Satake equivalence. These are the expanded lecture notes for a mini-course in 2015 PCMI summer school.…

We categorify the R-matrix isomorphism between tensor products of minuscule representations of U_q(sl(n)) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the…

For a variety $X$ separated over a perfect field of characteristic $p>0$ which admits an embedding into a smooth variety, we establish an anti-equivalence between the bounded derived categories of Cartier crystals on $X$ and constructible…

We construct the geometric Satake equivalence for quasi-split reductive groups over nonarchimedean local fields, using \'etale Artin-Tate motives with $\mathbb{Z}[\frac{1}{p}]$-coefficients. We consider local fields of both equal and mixed…

By the geometric Satake correspondence, the number of components of certain fibres of the affine Grassmannian convolution morphism equals the tensor product multiplicity for representations of the Langlands dual group. On the other hand, in…

These notes present an application of the geometric Satake equivalence to the description of characters of indecomposable tilting modules for reductive algebraic groups over fields of positive characteristic, obtained in joint work with G.…

We define 2-categories of microlocal perverse (resp. coherent) sheaves of categories on the skeleton of a hypertoric variety and show that the generators of these 2-categories lift the projectives (resp. simples) in hypertoric category…

For a possibly twisted loop group $LG$, and any character sheaf of its Iwahori subgroup, we identify the associated affine Hecke category with a combinatorial category of Soergel bimodules. In fact, we prove such results for affine Hecke…

We describe an isomorphism of categories conjectured by Kontsevich. If $M$ and $\widetilde{M}$ are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on $M$ and a suitable version of Fukaya's…