We generalize the classical Satake equivalence as follows. Let k be an algebraically closed field, set O=k[[t]] and F=k((t)). For an almost simple algebraic group G we classify central extensions of G(F) by the multiplicative group. Any such extension E splits canonically over G(O). Consider the category of G(O)-biinvariant perverse sheaves on E with a given Gm-monodromy . We show that this is a tensor category, which is tensor equivalent to the category of representations of a reductive group. We compute the root datum of this group.
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. References updated and more details added.
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 terms of coherent sheaves on the twisted dual Lie algebra. We also extend their computation of the corresponding loop rotation equivariant derived category, which is described in terms of Harish-Chandra bimodules for the twisted dual Lie algebra. To carry this out, we have to find a substitute for the functor of global equivariant cohomology. We describe such a functor, and show as in Bezrukavnikov…
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 twisted tensor product of the category of representations of the Langlands dual group and the category of pure polarizable Hodge structures.
For an arbitrary split adjoint group we identify the unramified Whittaker space with the space of skew-invariant functions on the lattice of coweights and deduce from it the Casselman-Shalika formula.
These notes are devoted to a detailed exposition of the proof of the Geometric Satake Equivalence for general coefficients, following Mirkovic-Vilonen.
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 combinatorial version of the Satake category using irreducible components of fibres of the convolution morphism. We then prove an equivalence of coboundary categories between this combinatorial Satake category and the category of crystals of the dual group.
We prove the Casselman-Shalika formula for unramified groups over a non-archimedean local field by studying the action of the spherical Hecke algebra on the space of compact spherical Whittaker functions via the twisted Satake transform. This method provides a conceptual explanation of the appearance of characters of a dual group in the Casselman-Shalika formula.
We establish a derived geometric Satake equivalence for the quaternionic general linear group GL_n(H). By applying the real-symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the symmetric variety GL_2n/Sp_2n. We explain how these equivalences fit into the general framework of a geometric Langlands correspondence for real groups and the relative Langlands duality conjecture. As an application, we compute the stalks of the IC-complexes for spherical orbit closures in the quaternionic affine Grassmannian and the loop space of GL_2n/Sp_2n. We s…
Let $k$ be an algebraically closed field of characteristic $p$. Denote by $W(k)$ the ring of Witt vectors of $k$. Let $F$ denote a totally ramified finite extension of $W(k)[1/p]$ and $\mathcal{O}$ the its ring of integers. For a connected reductive group scheme $G$ over $\mathcal{O}$, we study the category $P_{L^+G}(Gr_G,Λ)$ of $L^+G$-equivariant perverse sheaves in $Λ$-coefficient on the affine Grassmannian $Gr_G$ where $Λ=\mathbb{Z}_{\ell}$ and $\mathbb{F}_{\ell}$ and prove it is equivalent as a tensor category to the category of finitely generated $Λ$-representations of the Langlands dual …
We prove analogues of fundamental results of Kostant on the universal centralizer of a connected reductive algebraic group for algebraically closed fields of positive characteristic (with mild assumptions), and for integral coefficients. As an application, we use these results to obtain a "mixed modular" analogue of the derived Satake equivalence of Bezrukavnikov-Finkelberg.
We prove a generalization of the twisted geometric Satake equivalence of Finkelberg--Lysenko in the context of the factorizable grassmannian of a reductive group G relative to a smooth curve X, similar to Gaitsgory's generalization in "On de Jong's conjecture" of the Satake equivalence itself. In order to express the dependence of this result on factorizability, we consider a certain 2-category of gerbes on this grassmannian, which we call "sf gerbes"; the theorem then concerns perverse sheaves twisted by such gerbes. This notion of twisting differs from, but in special cases is equivalent to,…
This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of $SO(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $SO_N$. We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.
In this note we explicitly construct top-dimensional components of the cyclic convolution varieties. These components correspond (via the geometric Satake equivalence) to irreducible summands $V(λ+μ-Nβ) \subset V(λ) \otimes V(μ)$ for $G^\vee=SL_{n+1}$, where $N\ge 1$ and $β$ is a positive root. Furthermore, we deduce from these constructions a nontrivial lower bound on the multiplicity of these subrepresentations when $β$ is not a simple root. Finally, we demonstrate that not all such top-dimensional components can be realized as closures of orbits.
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. Bezrukavnikov-Finkelberg developed a derived version of this equivalence which relates the derived category of $ G^\vee$-equivariant constructible sheaves on $ Gr $ with the category of $G$-equivariant ${\mathcal O}(\mathfrak g)$-modules. In this paper, we develop a K-theoretic version of the derived geometric Satake which involves the quantum group $ U_q \mathfrak g $. We d…
We give an interpretation of sl_n webs as morphisms between certain singular Soergel bimodules. We explain how this is a combinatorial, algebraic version of the geometric Satake equivalence (in type A). We then q-deform the construction, giving an equivalence between representations of U_q(sl_n) and certain singular Soergel bimodules for a q-deformed Cartan matrix. In this paper, we discuss the general case but prove only the case n=2,3. In the sequel we will prove the case n >= 4.
In a 1983 paper the author has established a (decategorified) Satake equivalence for affine Hecke algebras. In this paper we give new proofs for some results of that paper, one based on the theory of J-rings and one based on the known character formula for rational representations of a reductive group in positive, large, characteristic. We also give an extension of that character formula to disconnected groups.
We construct (cohomological) correspondences between mod $p$ fibers of different Shimura varieties and describe the fibers of these correspondences by studying irreducible components of affine Deligne-Lusztig varieties. In particular, in the case of correspondences from a Shimura set to a Shimura variety, we obtain a description of the basic Newton stratum of the latter, and show that the irreducible components of the basic Newton stratum generate all the Tate classes in the middle cohomology of the Shimura variety, under a certain genericity condition. Along the way, we also determine the set…
Let $\mathbf{G}$ be a connected reductive group over a {non-archimedean local field} $F$. Let $K_\mathcal{F}$ be the parahoric subgroup attached to a facet $\mathcal{F}$ in the Bruhat--Tits building of $\mathbf{G}$. The ultimate goal of the present paper is to describe the center of the parahoric--Hecke algebra $\mathcal{H}(\mathbf{G}(F)//K_{\mathcal{F}}, \mathbb{Z}[q^{-1}])$ with level $K_{\mathcal{F}}$ and prove the compatibility of generalized (twisted) Bernstein and Satake homomorphisms.
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 construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group.
We give an exposition of three geometric constructions of the irreducible representations of GL_n. In particular, we discuss Borel-Weil theory, the Ginzburg construction, and the geometric Satake construction. We also explain how to deduce the Ginzburg construction from the geometric Satake equivalence. These are lecture notes for a lecture series at the Summer School on geometric representation theory and extended affine Lie algebras held at University of Ottawa in June 2009.
For a split reductive group scheme $G$ over a commutative ring $k$ with Weyl group $W$, there is an important functor $Rep(G,k) \to Rep(W,k)$ defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representations has an alternative geometric description, in terms of the affine Grassmannian and the nilpotent cone of the Langlands dual group to $G$. The translation from representation theory to geometry is via the Satake equivalence and the Springer correspondence. This generalizes the result for the $k=\mathbb{C}$ case proved by the fi…
Let G be a reductive group. The geometric Satake equivalence realized the category of representations of the Langlands dual group ^LG in terms of spherical perverse sheaves (or D-modules) on the affine Grassmannian Gr_G=G((t))/G[[t]] of the original group G. In the present paper we perform a first step in realizing the category of representations of the quantum group corresponding to ^LG in terms of the geometry of Gr_G. The idea of the construction belongs to Jacob Lurie.
We discuss recent developments in the Langlands program for function fields, and in the geometric Langlands program. In particular we explain a canonical decomposition of the space of cuspidal automorphic forms for any reductive group G over a function field, indexed by global Langlands parameters. The proof uses the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence.
Shifted twisted Yangians and affine Grassmannian islices
Shenzhen International Center for Mathematics and Department of Mathematics, Southern University of Science and Technology（深圳国际数学中心及南方科技大学数学系）；Department of Mathematics, University of Virginia（弗吉尼亚大学数学系）；Département de mathématiques, Université de Sherbrooke（舍布鲁克大学数学系）
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 categories of l-adic sheaves on an affine flag variety which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra; of the anti-spherical module over this algebra; and of the space of Iwahori-invariant Whitakker functions.
We introduce and study the category of twisted modules over a triangular differential graded bocs. We show that in this category idempotents split, that it admits a natural structure of a Frobenius category, that a twisted module is homotopically trivial if and only if its underlying complex is acyclic, and that any homotopy equivalence of differential graded bocses determines an equivalence of the corresponding homotopy categories of twisted modules. The category of modules over an $A_{\infty}$-algebra is equivalent to the category of twisted modules over a triangular differential graded bocs…
