Satake Correspondence Research Articles

Equivariant Schubert calculus and geometric Satake

The main classical result of Schubert calculus is that multiplication rules for the basis of Schubert cycles inside the cohomology ring of the Grassmannian G(n,m) are the same as multiplication rules for the basis of Schur polynomials in the ring of symmetric polynomials. In this paper, we explain how to recover this somewhat mysterious connection by using the geometric Satake correspondence to put the structure of a representation of GLm on H•(G(n,m)) and comparing it to the Fock space representation on symmetric polynomials. This proof also extends to equivariant Schubert calculus, and gives an explanation of the relationship between torus-equivariant cohomology of Grassmannians and double Schur polynomials.

Towards geometric Satake correspondence for Kac-Moody algebras --- Cherkis bow varieties and affine Lie algebras of type $A$

Towards geometric Satake correspondence for Kac-Moody algebras --- Cherkis bow varieties and affine Lie algebras of type $A$

Sixfolds of generalized Kummer type and K3 surfaces

We prove that any hyper-Kähler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm {K}3^{[3]}$ type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic involutions acting trivially on its second cohomology. When $K$ is projective, the variety $Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective $\mathrm {K}3$ surface $S_K$. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces $S_K$, producing infinitely many new families of $\mathrm {K}3$ surfaces of general Picard rank $16$ satisfying the Kuga–Satake Hodge conjecture.

Lie algebra actions on module categories for truncated shifted yangians

AbstractWe develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof’s induction and restriction functors for Cherednik algebras, but their definition uses different tools.After this general definition, we focus on quiver gauge theories attached to a quiver$\Gamma $. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra$\mathfrak {g}_{\Gamma }$on category$ \mathcal {O}$for these Coulomb branch algebras. When$ \Gamma $is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence.To establish this categorical action, we define a new class of ‘flavoured’ KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.

Bases of tensor products and geometric Satake correspondence

The geometric Satake correspondence can be regarded as a geometric construction of rational representations of a complex connected reductive group G . In their study of this correspondence, Mirković and Vilonen introduced algebraic cycles that provide a linear basis in each irreducible representation. Generalizing this construction, Goncharov and Shen define a linear basis in each tensor product of irreducible representations. We investigate these bases and show that they share many properties with the dual canonical bases of Lusztig.

Hyper–Kähler Manifolds of Generalized Kummer Type and the Kuga–Satake Correspondence

We first describe the construction of the Kuga–Satake variety associated to a (polarized) weight-two Hodge structure of hyper-Kähler type. We describe the classical cases where the Kuga–Satake correspondence between a hyper-Kähler manifold and its Kuga–Satake variety has been proved to be algebraic. We then turn to recent work of O’Grady and Markman which we combine to prove that the Kuga–Satake correspondence is algebraic for projective hyper-Kähler manifolds of generalized Kummer deformation type.

Kazhdan–Lusztig conjecture via zastava spaces

Abstract We deduce the Kazhdan–Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category 𝒪 {\mathcal{O}} from the equivariant geometric Satake correspondence and the analysis of torus fixed points in zastava spaces. We make similar speculations for the affine Lie algebras and 𝒲 {\mathscr{W}} -algebras.

Equivariant multiplicities of simply-laced type flag minors

Let g \mathfrak {g} be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism D ¯ \overline {D} on the coordinate ring C [ N ] \mathbb {C}[N] related to Brion’s equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram’s strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when g \mathfrak {g} is of type A n A_n or D 4 D_4 , the map D ¯ \overline {D} takes similar distinguished values on the set of all flag minors of C [ N ] \mathbb {C}[N] , raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of D ¯ \overline {D} on flag minors belonging to the same standard seed, and we show that in any A D E ADE type these relations are preserved under cluster mutations from one standard seed to another. The proofs of these results partly rely on Kang-Kashiwara-Kim-Oh’s monoidal categorification of the cluster structure of C [ N ] \mathbb {C}[N] via representations of quiver Hecke algebras.

Footnotes to papers of O’Grady and Markman

In this paper, we first generalize to any hyper-Kähler manifold X with \(b_3(X)\not =0\) results proved by O’Grady for hyper-Kähler manifolds of generalized Kummer type. In the second part, we restrict to hyper-Kähler manifolds of generalized Kummer type and prove, using results of Markman, that their Kuga–Satake correspondence is algebraic.

Topological-antitopological fusion and the quantum cohomology of Grassmannians

We suggest an explanation for the part of the Satake Correspondence which relates the quantum cohomology of complex Grassmannians and the quantum cohomology of complex projective space, as well as their respective Stokes data, based on the original physics approach using the tt* equations. We also use the Stokes data of the tt* equations to provide a Lie-theoretic link between particles in affine Toda models and solitons in certain sigma-models. Along the way, we review some old ideas from supersymmetric field theory, whose mathematical manifestations are becoming increasingly widespread.

The Mirković–Vilonen basis and Duistermaat–Heckman measures

Using the geometric Satake correspondence, the Mirkovic-Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group G . We prove that these bases are perfect, i.e. compatible with the action of the Chevelley generators of the positive half of the Lie algebra g. We compute this action in terms of intersection multiplicities in the affine Grassmannian. We prove that these bases stitch together to a basis for the algebra C[N] of regular functions on the unipotent subgroup. We compute the multiplication in this MV basis using intersection multiplicities in the Beilinson-Drinfeld Grassmannian, thus proving a conjecture of Anderson. In the third part of the paper, we define a map from C[N] to a convolution algebra of measures on the dual of the Cartan subalgebra of g. We characterize this map using the universal centralizer space of G. We prove that the measure associated to an MV basis element equals the Duistermaat-Heckman measure of the corresponding MV cycle. This leads to a proof of a conjecture of Muthiah. Finally, we use the map to measures to compare the MV basis and Lusztig's dual semicanonical basis. We formulate conjectures relating the algebraic invariants of preprojective algebra modules (which underlie the dual semicanonical basis) and geometric invariants of MV cycles. In the appendix, we use these ideas to prove that the MV basis and the dual semicanonical basis do not coincide in SL_6.

A Proof of the Refined PRV Conjecture via the Cyclic Convolution Variety

In this brief note we illustrate the utility of the geometric Satake correspondence by employing the cyclic convolution variety to give a simple proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture, along with Kumar’s refinement. The proof involves recognizing certain MV-cycles as orbit closures of a group action, which we make explicit by unique characterization. In an Appendix, joint with P. Belkale, we discuss how this work fits in a more general framework.

Trihedral Soergel bimodules

The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum $\mathfrak{sl}_2$ representation category. It also establishes a precise relation between the simple transitive $2$-representations of both monoidal categories, which are indexed by bicolored $\mathsf{ADE}$ Dynkin diagrams. Using the quantum Satake correspondence between affine $\mathsf{A}_{2}$ Soergel bimodules and the semisimple quotient of the quantum $\mathfrak{sl}_3$ representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke algebras and Soergel bimodules. These have their own Kazhdan-Lusztig combinatorics, simple transitive $2$-representations corresponding to tricolored generalized $\mathsf{ADE}$ Dynkin diagrams.

Smith theory and geometric Hecke algebras

In 1960 Borel proved a “localization” result relating the rational cohomology of a topological space X to the rational cohomology of the fixed points for a torus action on X. This result and its generalizations have many applications in Lie theory. In 1934, Smith proved a similar localization result relating the mod p cohomology of X to the mod p cohomology of the fixed points for a $${\mathbb {Z}}/p$$ -action on X. In this paper we study $${\mathbb {Z}}/p$$ -localization for constructible sheaves and functions. We show that $${\mathbb {Z}}/p$$ -localization on loop groups is related via the geometric Satake correspondence to some special homomorphisms that exist between algebraic groups defined over a field of small characteristic.

A family of cubic fourfolds with finite-dimensional motive

We prove that cubic fourfolds in a certain 10-dimensional family have finite-dimensional motive. The proof is based on the van Geemen–Izadi construction of an algebraic Kuga–Satake correspondence for these cubic fourfolds, combined with Voisin’s method of “spread”. Some consequences are given.

Quantum K-theoretic geometric Satake: the case

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 and 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 define a convolution category$K\operatorname{Conv}(Gr)$whose morphism spaces are given by the$G^{\vee }\times \mathbb{C}^{\times }$-equivariant algebraic K-theory of certain fibre products. We conjecture that$K\operatorname{Conv}(Gr)$is equivalent to a full subcategory of the category of$U_{q}\mathfrak{g}$-equivariant${\mathcal{O}}_{q}(G)$-modules. We prove this conjecture when$G=\operatorname{SL}_{n}$. A key tool in our proof is the$\operatorname{SL}_{n}$spider, which is a combinatorial description of the category of$U_{q}\mathfrak{sl}_{n}$representations. By applying horizontal trace, we show that the annular$\operatorname{SL}_{n}$spider describes the category of$U_{q}\mathfrak{sl}_{n}$-equivariant${\mathcal{O}}_{q}(\operatorname{SL}_{n})$-modules. Then we use quantum loop algebras to relate the annular$\operatorname{SL}_{n}$spider to$K\operatorname{Conv}(Gr)$. This gives a combinatorial/diagrammatic description of both categories and proves our conjecture.

Total positivity, Schubert positivity, and geometric Satake

Let G be a simple, simply-connected complex algebraic group, and let X⊂G∨ be the centralizer of a principal nilpotent. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian GrG. Peterson furthermore connected X to the quantum cohomology rings of partial flag varieties G/P.In this paper we study three notions of positivity for X: (1) Schubert positivity arising via Peterson's work, (2) Lusztig's total positivity and (3) Mirković–Vilonen positivity obtained from the MV-cycles in GrG. The first main theorem establishes that these three notions of positivity coincide. Our second main theorem proves a parametrization of the totally nonnegative part of X, confirming a conjecture of the second author.In type A the parametrization and relationship with Schubert positivity were proved earlier by the second author. Here we tackle the general type case and also introduce a crucial new connection with the affine Grassmannian and geometric Satake correspondence.

A combinatorial geometric Satake equivalence

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.

Parity sheaves and tilting modules

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.

Geometry of canonical bases and mirror symmetry

A decorated surface S is an oriented surface with boundary and a finite, possibly empty, set of special points on the boundary, considered modulo isotopy. Let G be a split reductive group over Q. A pair (G, S) gives rise to a moduli spaceAG,S , closely related to the moduli space of G-local systems on S. It is equipped with a positive structure (Fock and Goncharov, Publ Math IHES 103:1–212, 2006). So a set AG,S(Z ) of its integral tropical points is defined. We introduce a rational positive function W on the space AG,S , called the potential. Its tropicalisation is a function W t : AG,S(Z ) → Z. The condition W t ≥ 0 defines a subset of positive integral tropical points AG,S(Zt ). For G = SL2, we recover the set of positive integralA-laminations on S from Fock and Goncharov (Publ Math IHES 103:1–212, 2006). We prove that when S is a disc with n special points on the boundary, the set AG,S(Zt ) parametrises top dimensional components of the fibers of the convolution maps. Therefore, via the geometric Satake correspondence (Lusztig, Asterisque 101–102:208– 229, 1983; Ginzburg,1995; Mirkovic and Vilonen, Ann Math (2) 166(1):95– 143, 2007; Beilinson and Drinfeld, Chiral algebras. American Mathematical Society Colloquium Publications, vol. 51, 2004) they provide a canonical basis in the tensor product invariants of irreducible modules of the Langlands dual group GL : (Vλ1 ⊗ . . .⊗ Vλn )G L . (1) A. Goncharov (B) · L. Shen Mathematics Department, Yale University, New Haven, CT 06520, USA e-mail: alexander.goncharov@yale.edu