Twisted geometric Satake equivalence

  title={Twisted geometric Satake equivalence},
  author={Michael Finkelberg and Sergey Lysenko},
  journal={Journal of the Institute of Mathematics of Jussieu},
  pages={719 - 739}
Abstract Let k be an algebraically closed field and O = k[[t]] ⊂ F = k((t)). For an almost simple algebraic group G we classify central extensions 1 → $\mathbb{G}_m\to E\to G(\bm{F})\$m → E → G(F) → 1; any such extension splits canonically over G(O). Fix a positive integer N and a primitive character ζ : μN(K) →$\mathbb{Q}}_\ell^*}$ (under some assumption on the characteristic of k). Consider the category of G(O)-bi-invariant perverse sheaves on E with $\mathbb{G}_m$m-monodromy ζ. We show that… 
Quantum geometric Langlands correspondence in positive characteristic: The $\mathrm{GL}_{N}$ case
We prove a version of quantum geometric Langlands conjecture in characteristic $p$. Namely, we construct an equivalence of certain localizations of derived categories of twisted crystalline $\mathcal
Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian
The geometric Satake equivalence of Ginzburg and Mirkovic– Vilonen, for a complex reductive group G, is a realization of the tensor category of representations of its Langlands dual group LG as a
Geometric Eisenstein series: twisted setting
Let G be a simple simply-connected group over an algebraically closed field k, X be a smooth connected projective curve over k. In this paper we develop the theory of geometric Eisenstein series on
A generalized Theta lifting, CAP representations, and Arthur parameters
  • Spencer Leslie
  • Mathematics
    Transactions of the American Mathematical Society
  • 2019
We study a new lifting of automorphic representations using the theta representation $\Theta$ on the $4$-fold cover of the symplectic group, $\overline{\mathrm{Sp}}_{2r}(\mathbb{A})$. This lifting
Betti Geometric Langlands
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
Twisted Satake Category
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
Doubling Constructions and Tensor Product $L$-Functions: coverings of the symplectic group
In this work we develop an integral representation for the partial $L$-function of a pair $\pi\times\tau$ of genuine irreducible cuspidal automorphic representations, $\pi$ of the $m$-fold covering
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
Twisted Whittaker models for metaplectic groups
Let G be a reductive group (over an algebraically closed field) equipped with the metaplectic data. In this paper we study the corresponding twisted Whittaker category for G. We construct and study a
Introduction to chtoucas for reductive groups and to the global Langlands parameterization
This is a translation in English of version 3 of the article arXiv:1404.3998, which is itself an introduction to arXiv:1209.5352. We explain all the ideas of the proof of the following theorem. For


Local shimura correspondence
It splits over the maximal compact subgroup. Let T be the corresponding lift of I, the Iwahori subgroup of G. Now fix ~, a faithful character of #n(F). Let R}(~) be the category of smooth
Geometric Langlands duality and representations of algebraic groups over commutative rings
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
Twisted Whittaker model and factorizable sheaves
Abstract.Let G be a reductive group. The geometric Satake equivalence realized the category of representations of the Langlands dual group Ğ in terms of spherical perverse sheaves (or D-modules) on
Monodromic systems on affine flag manifolds
  • G. Lusztig
  • Mathematics
    Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
  • 1994
There has been recently substantial progress in the programme of expressing the characters of irreducible modular representations of a semisimple group over a field of positive characteristic in
Hodge Cycles, Motives, and Shimura Varieties
General Introduction.- Notations and Conventions.- Hodge Cycles on Abelian Varieties.- Tannakian Categories.- Langlands's Construction of the Taniyama Group.- Motifs et Groupes de Taniyama.-
Electric-Magnetic Duality And The Geometric Langlands Program
The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are
Algebraic loop groups and moduli spaces of bundles
Abstract.We study algebraic loop groups and affine Grassmannians in positive characteristic. The main results are normality of Schubert-varieties, the construction of line-bundles on the affine
Tannakian Categories
1 Tensor Categories 4 Extending ̋ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Invertible objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7