Geometric Satake Research Articles

A geometric proof of the Feigin-Frenkel theorem

We reprove the theorem of Feigin and Frenkel relating the center of the critical level enveloping algebra of the Kac-Moody algebra for a semisimple Lie algebra to opers (which are certain de Rham local systems with extra structure) for the Langlands dual group. Our proof incorporates a construction of Beilinson and Drinfeld relating the Feigin-Frenkel isomorphism to (more classical) Langlands duality through the geometric Satake theorem.

Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian

The geometric Satake equivalence of Ginzburg and Mirković- Vilonen, for a complex reductive group G G , is a realization of the tensor category of representations of its Langlands dual group L G {}^L G as a category of “spherical” perverse sheaves on the affine grassmannian Gr G = G ( C ( ( t ) ) / G ( C [ [ t ] ] ) \operatorname {Gr}_G = G(\mathbb {C}(\mspace {-3.5mu}(t)\mspace {-3.5mu})/G(\mathbb {C}[\mspace {-2mu}[t]\mspace {-2mu}]) . Since its original statement it has been generalized in two directions: first, by Gaitsgory, to the Beilinson–Drinfeld or factorizable grassmannian, which for a smooth complex curve X X is a collection of spaces over the powers X n X^n whose general fiber is isomorphic to Gr G n \operatorname {Gr}_G^n but with the factors “fusing” as they approach points with equal coordinates, allowing a more natural description of the structures and properties even of the Mirković–Vilonen equivalence. The second generalization, due recently to Finkelberg–Lysenko, considers perverse sheaves twisted in a suitable sense by a root of unity, and obtains the category of representations of a group other than the Langlands dual. This latter result can be considered as part of “Langlands duality for quantum groups”. In this work we obtain a result simultaneously generalizing all of the above. We consider the general notion of twisting by a gerbe and define the natural class of “factorizable” gerbes by which one can twist in the context of the Satake equivalence. These gerbes are almost entirely described by the quadratic forms on the weight lattice of G G . We show that a suitable formalism exists such that the methods of Mirković–Vilonen can be applied directly in this general context virtually without change and obtain a Satake equivalence for twisted perverse sheaves. In addition, we present new proofs of the properties of their structure as an abelian tensor category.

Generating basis webs for SLn

Given a simple algebraic group G, a web is a directed trivalent graph with edges labelled by dominant minuscule weights. There is a natural surjection of webs onto the invariant space of tensor products of minuscule representations. Following the work of Westbury, we produce a set of webs for SLn which form a basis for the invariant space via the geometric Satake correspondence. In fact, there is an upper unitriangular change of basis to the Satake basis. This set of webs agrees with previous work in the cases n=2,3 and generalizes the work of Westbury in the case n⩾4.

A Brylinski filtration for affine Kac–Moody algebras

Braverman and Finkelberg have recently proposed a conjectural analogue of the geometric Satake isomorphism for untwisted affine Kac–Moody groups. As part of their model, they conjecture that (at dominant weights) Lusztig's q-analog of weight multiplicity is equal to the Poincare series of the principal nilpotent filtration of the weight space, as occurs in the finite-dimensional case. We show that the conjectured equality holds for all affine Kac–Moody algebras if the principal nilpotent filtration is replaced by the principal Heisenberg filtration. The main body of the proof is a Lie algebra cohomology vanishing result. We also give an example to show that the Poincare series of the principal nilpotent filtration is not always equal to the q-analog of weight multiplicity. Finally, we give some partial results for indefinite Kac–Moody algebras.

Pursuing the double affine Grassmannian, I: Transversal slices via instantons on Ak-singularities

This article is the first in a series that describes a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. (For simplicity, we only consider the untwisted and simply connected case here.) The usual geometric Satake isomorphism for a reductive group G identifies the tensor category Rep(G∨) of finite-dimensional representations of the Langlands dual group G∨ with the tensor category PervG(O)(GrG) of G(O)-equivariant perverse sheaves on the affine Grassmannian GrG=G(K)/G(O) of G. (Here K=C((t)) and O=C[[t]].) As a by-product one gets a description of the irreducible G(O)-equivariant intersection cohomology (IC) sheaves of the closures of G(O)-orbits in GrG in terms of q-analogs of the weight multiplicity for finite-dimensional representations of G∨. The purpose of this article is to try to generalize the above results to the case when G is replaced by the corresponding affine Kac-Moody group Gaff. (We refer to the (not yet constructed) affine Grassmannian of Gaff as the double affine Grassmannian.) More precisely, in this article we construct certain varieties that should be thought of as transversal slices to various Gaff(O)-orbits inside the closure of another Gaff(O)-orbit in GrGaff. We present a conjecture that computes the intersection cohomology sheaf of these varieties in terms of the corresponding q-analog of the weight multiplicity for the Langlands dual affine group Gaff∨, and we check this conjecture in a number of cases. Some further constructions (such as convolution of the corresponding perverse sheaves, analog of the Beilinson-Drinfeld Grassmannian, and so forth) will be addressed in another publication

Spherical Varieties and Langlands Duality

Let G be a connected reductive complex algebraic group. This paper is devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical G-variety X. The space Z may be thought of as an algebraic model for the loop space of X. In this paper, we associate to X a connected reductive complex algebraic subgroup $\check H$ of the dual group $\check G$. The construction of $\check H$ is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of $\check H$. Combinatorial shadows of the group $\check H$ govern many aspects of the geometry of X such as its compactifications and invariant differential operators. When X is a symmetric variety, the group $\check H$ coincides with that associated to the corresponding real form of G via the (real) geometric Satake correspondence.

Quiver Varieties and Branching

Braverman and Finkelberg recently proposed the geometric Satake correspon- dence for the affine Kac-Moody groupGaff (Braverman A., Finkelberg M., arXiv:0711.2083). They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of Gcpt-instantons on R 4 /Zr correspond to weight spaces of rep- resentations of the Langlands dual group G _ at level r. When G = SL(l), the Uhlenbeck compactification is the quiver variety of type sl(r)aff, and their conjecture follows from the author's earlier result and I. Frenkel's level-rank duality. They further introduce a con- volution diagram which conjecturally gives the tensor product multiplicity (Braverman A., Finkelberg M., Private communication, 2008). In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for G = SL(l).

Knot homology via derived categories of coherent sheaves II, $\mathfrak{sl}_{m}$ case

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu's by homological mirror symmetry.

Twisted Whittaker model and factorizable sheaves

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 the affine Grassmannian GrG = 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 Ğ in terms of the geometry of GrG. The idea of the construction belongs to Jacob Lurie.

Hives and the fibres of the convolution morphism

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 the case of GLn, combinatorial objects called hives also count tensor product multiplicities. The purpose of this paper is to give a simple bijection between hives and the components of these fibres. In particular, we give a description of the individual components. We also describe a conjectural generalization involving the octahedron recurrence.

Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de jacquet et ye

We give a geometric interpretation of the base change homomorphism between the Hecke algebra of GL( n) for an unramified extension of local fields of positive characteristic. For this, we use some results of Ginzburg, Mirkovic and Vilonen related to the geometric Satake isomorphism. We give a new proof of these results in the positive characteristic case. By using that geometric interpretation of the base change homomorphism, we prove the fundamental lemma of Jacquet and Ye for arbitrary Hecke function in the equal characteristic case.

Geometric Satake, categorical traces, and arithmetic of Shimura varieties

We survey some recent work on the geometric Satake of p-adic groups and its applications to some arithmetic problems of Shimura varieties. We reformulate a few constructions appeared in the previous works more conceptually.