By the geometric Satake isomorphism of Mirkovic and Vilonen, decomposition numbers for reductive groups can be interpreted as decomposition numbers for equivariant perverse sheaves on the complex affine Grassmannian of the Langlands dual group. Using a description of the minimal degenerations of the affine Grassmannian obtained by Malkin, Ostrik and Vybornov, we are able to recover geometrically some decomposition numbers for reductive groups. In the other direction, we can use some decomposition numbers for reductive groups to prove geometric results, such as a new proof of non-smoothness results, and a proof that some singularities are not equivalent (a conjecture of Malkin, Ostrik and Vybornov). We also give counterexamples to a conjecture of Mirkovic and Vilonen stating that the stalks of standard perverse sheaves over the integers on the affine Grassmannian are torsion-free, and propose a modified conjecture, excluding bad primes.
The aim of this paper is to identify a certain tensor category of perverse sheaves on the real loop Grassmannian of a real form $G_{\mathbb R}$ of a connected reductive complex algebraic group $G$ with the category of finite-dimensional representations of a connected reductive complex algebraic subgroup $\check H$ of the dual group $\check G$. The root system of $\check H$ is closely related to the restricted root system of the real form $G_{\mathbb R}$. The fact that $\check H$ is reductive implies that an interesting family of real algebraic maps satisfies the conclusion of the Decomposition…
This is the third paper of a series (started by arXiv:0711.2083, arXiv:0908.3390) which describes a conjectural analog of the affine Grassmannian for affine Kac-Moody groups (also known as the double affine Grassmannian). The current paper is dedicated to describing a conjectural analog of the convolution diagram for the double affine Grassmannian and affine Zastava.
We study the connection between the affine degenerate Grassmannians in type $A$, quiver Grassmannians for one vertex loop quivers and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type $GL_n$ and identify it with semi-infinite orbit closure of type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex loop quiver provide finite-dimensional approximations of the degenerate affine Grassmannian. Finally, we give an explicit description of the degenerate affine Grassmannian of type $A_1^{(1)}$, propose a conjectural descrip…
This is the second paper of a series (started by arXiv:0711.2083) which describes a conjectural analog of the affine Grassmannian for affine Kac-Moody groups (also known as the double affine Grassmannian). The current paper is dedicated to describing a conjectural analog of the convolution diagram for the double affine Grassmannian. In the case when G=SL(n) our conjectures can be derived from arXiv:0809.2605.
We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew $(GL(m), GL(n))$ duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine…
In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic $p$ bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular version of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application…
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 study affine Grassmannians for ramified triality groups. These groups are of type ${}^3D_4$, so they are forms of the orthogonal or the spin groups in 8 variables. They can be given as automorphisms of certain twisted composition algebras obtained from the octonion algebra. Using these composition algebras, we give descriptions of the affine Grassmannians for these triality groups as functors classifying suitable lattices in a fixed space.
The minimal degeneration singularities in the affine Grassmannians of simple simply-laced algebraic groups are determined to be either Kleinian singularities of type A, or closures of minimal orbits in nilpotent cones. The singularities for non-simply-laced types are studied by intersection cohomology and equivariant Chow group methods.
We study spherical Schubert varieties in the affine Grassmannian. These Schubert varieties have a natural conjectural modular description due to Finkelberg-Mirković. This modular description is easily seen to be set-theoretically correct, but it is not obviously scheme-theoretically correct. We prove that this modular description is correct in many cases. We also link this modular description to the reducedness conjecture from Kamnitzer-Webster-Weekes-Yacobi for tranverse slices in the affine Grassmannian.
We categorify all the Reshetikhin-Turaev tangle invariants of type A. Our main tool is a categorification of the generalized Jones-Wenzl projectors (a.k.a. clasps) as infinite twists. Applying this to certain convolution product varieties on the affine Grassmannian we extend our earlier work with Kamnitzer from standard to arbitrary representations.
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.
Let $G$ be a simple algebraic group of type $A$ or $D$ defined over $\C$ and $T$ be a maximal torus of $G$. For a dominant coweight $λ$ of $G$, the $T$-fixed point subscheme $(\bar{Gr}_G^λ)^T$ of the Schubert variety $\bar{Gr}_G^λ$ in the affine Grassmannian $Gr_G$ is a finite scheme. We prove that there is a natural isomorphism between the dual of the level one affine Demazure module corresponding to $λ$ and the ring of functions (twisted by certain line bundle on $Gr_G$) of $(\bar{Gr}_G^λ)^T$. We use this fact to give a geometric proof of the Frenkel-Kac-Segal isomorphism between basic repre…
The affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part of the ideal defining this embedding in terms of certain elements of the dual of Fock space called "shuffles", and they conjecture that these elements together with the Plücker relations suffice to cut out the affine Grassmannian. We give a proof of this conjecture in two steps: first we reinterpret the shuffles equations in terms of Frobenius twists of symmetr…
Let G be a reductive group; in this note we give an interpretation of the dynamical Weyl group of of the Langlands dual group $\check{G}$ defined by Etingof and Varchenko in terms of the geometry of the affine Grassmannian Gr of G. In this interpretation the dynamical parameters of Etingof and Varchenko correspond to equivariant parameters with respect to certain natural torus acting on Gr. We also present a conjectural generalization of our results to the case of affine Kac-Moody groups.
We construct for each choice of a quiver $Q$, a cohomology theory $A$ and a poset $P$ a "loop Grassmannian" $\mathcal{G}^P(Q,A)$. This generalizes loop Grassmannians of semisimple groups and the loop Grassmannians of based quadratic forms. The addition of a dilation torus $\mathcal{D}\subset\mathbb{G}_m^2$ gives a quantization $\mathcal{G}^P_\mathcal{D}(Q,A)$. The construction is motivated by the program of introducing an inner cohomology theory in algebraic geometry adequate for the Geometric Langlands program ['Some extensions of the notion of loop Grassmannians', Rad Hrvat. Akad. Znan. Umje…
The motivation of this work is to construct an analog of compactified moduli of abelian varieties and toric pairs in the case of non-commutative algebraic group G. We introduce a class of "stable reductive varieties" which contain connected reductive groups and their equivariant compactifications, and is closed under flat reduced degenerations. We classify them all, describe their degenerations, and establish a connection between these varieties and "reductive semigroups" which we also define. Finally, we construct a Hilbert scheme of embedded G-varieties by applying and generalizing a constru…
We introduce a notion of $ξ$-stability on the affine grassmannian $\xx$ for the classical groups, this is the local version of the $ξ$-stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient $\xx^ξ/T$ of the stable part $\xx^ξ$ by the maximal torus $T$ exists as an ind-$k$-scheme, and we introduce a reduction process analogous to the Harder-Narasimhan reduction for vector bundles. For the group $\mathrm{SL}_{d}$, we calculate the Poincaré series of the quotient $\xx^ξ/T$.
We present a version of the Matsuki correspondence for the affine Grassmannian $Gr=G(C((t)))/G(C[[t]])$ of a connected reductive complex algebraic group $G$. The main statement is an anti-isomorphism between the orbit posets of two subgroups of $G(C((t)))$ acting on $Gr$. The first is the polynomial loop group $LG_R$ of a real form $G_R$ of $G$; the second is the loop group $K(C((t)))$ of the complexification $K$ of a maximal compact subgroup $K_c$ of $G_R$. The orbit poset itself turns out to be simple to describe.
In this paper we study the derived category of sheaves on the affine Grassmannian of a complex reductive group G, contructible with respect to the stratification by G(C[[x]])-orbits. Following ideas of Ginzburg and Arkhipov-Bezrukavnikov-Ginzburg, we describe this category (and a mixed version) in terms of coherent sheaves on the nilpotent cone of its Langlands dual reductive group. We also show, in the mixed case, that restriction to the nilpotent cone of a Levi subgroup corresponds to hyperbolic localization on affine Grassmannians.
The category of perverse sheaves on the affine Grassmannian of a complex reductive group $G$ gives a canonical geometric construction of the split form of the Langlands dual group $\check G_\bZ$ over the integers. Given a field $k$, we give a Tannakian construction of the quasi-split forms of $\check G_k$, as well as a construction of the gerbe associated to an inner form of $\check G_k$.
We prove in type A a conjecture which describes the ideal of transversal slices to spherical Schubert varieties in the affine Grassmannian. As a corollary, we prove a modular description (due to Finkelberg-Mirković) of the spherical Schubert varieties.
We propose a conjectural construction of various slices for double affine Grassmannians as Coulomb branches of 3-dimensional N=4 supersymmetric affine quiver gauge theories. It generalizes the known construction for the usual affine Grassmannian, and makes sense for arbitrary symmetric Kac-Moody algebras.
The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmanians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, …
Let $σ$ be a simple involution of an algebraic semisimple group $G$ and let $H$ be the subgroup of $G$ of points fixed by $σ$. If the restricted root system is of type $A$, $C$ or $BC$ and $G$ is simply connected or if the restricted root system is of type $B$ and $G$ is adjoint, then we describe a standard monomial theory and the equations for the coordinate ring $k[G/H]$ using the standard monomial theory and the Plücker relations of an appropriate (maybe infinite dimensional) Grassmann variety.
We endow the set of lattices in Q_p^n with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake correspondence in mixed characteristic. We also give an application of our theory to the study of Rapoport-Zink spaces.
Let $Q$ be a quiver of extended Dynkin type $D$. In this first of two papers, we show that the quiver Grassmannian $Gr_e(M)$ has a decomposition into affine spaces for every dimension vector $e$ and every indecomposable representation $M$ of defect $-1$ and defect $0$, with exception of the non-Schurian representations in homogeneous tubes. We characterize the affine spaces in terms of the combinatorics of a fixed coefficient quiver for $M$. The method of proof is to exhibit explicit equations for the Schubert cells of $Gr_e(M)$ and to solve this system of equations successively in linear term…
For a connected reductive group G and a finite-dimensional G-module V, we study the invariant Hilbert scheme that parameterizes closed G-stable subschemes of V affording a fixed, multiplicity-finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T-stable subscheme admitting a good quotient. The fibers of the quotient map classify affine G-schemes having a prescribed categorical quotient by a maximal unipotent subgroup of G. We show that V contains only finitely many multiplicity-free G-subva…
This is an expanded version of the text ``Perverse Sheaves on Loop Grassmannians and Langlands Duality'', AG/9703010. The main new result is a topological realization of algebraic representations of reductive groups over arbitrary rings. We outline a proof of a geometric version of the Satake isomorphism. Given a connected, complex algebraic reductive group G we show that the tensor category of representations of the Langlands dual group is naturally equivalent to a certain category of perverse sheaves on the loop Grassmannian of G. The above result has been announced by Ginsburg in and some o…
We give a complete list of smooth and rationally smooth normalized Schubert varieties in the twisted affine Grassmannian associated with a tamely ramified group and a special vertex of its Bruhat-Tits building. The particular case of the quasi-minuscule Schubert variety in the quasi-split but non-split form of ${\rm Spin}_8$ ("ramified triality") provides an input needed in the article by He-Pappas-Rapoport classifying Shimura varieties with good or semi-stable reduction.
We give a detailed description of the torsors that correspond to multiloop algebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Extended Affine Lie Algebras (which are higher nullity analogues of the affine Kac-Moody Lie algebras). The torsor approach that we take draws heavily for the theory of reductive group schemes developed by M. Demazure and A. Grothendieck. It also allows us to find a bridge between multiloop algebras and the work of J. Tits on reductive groups over complete local fields…
We construct Nakajima's quiver varieties of type A in terms of conjugacy classes of matrices and (non-Slodowy's) transverse slices naturally arising from affine Grassmannians. In full generality quiver varieties are embedded into Beilinson-Drinfeld Grassmannians of type A. Our construction provides a compactification of Nakajima's quiver varieties and a decomposition of an affine Grassmannian into a disjoint union of quiver varieties. As an application we provide a geometric version of skew and symmetric $(GL(m), GL(n))$ duality.
We study the algebraic geometry and combinatorics of the central degeneration (the degeneration that shows up in local models of Shimura varieties and Gaitsgory's central sheaves) in type A. More specifically, we elucidate the central degeneration of semi-infinite orbits and explain its relations with Levi restriction. Also, we discuss the central degeneration of Mirkovi$\acute{\text{c}}$-Vilonen cycles in the affine Grassmannian, and the corresponding transformations of Mirkovi$\acute{\text{c}}$-Vilonen polytopes. In addition, we shed some light on the geometry of Iwahori MV cycles in the aff…
Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper, we prove that neighbouring generalized affine Grassmannian slices are related by Hamiltonian reduction by the action of the additive group. We also prove a weaker version of the same result for their quantizations, algebras known as truncated shifted Yangians.
For each integral dominant weight $λ$, we construct a twisted global section functor $Γ^λ$ from the category of critical twisted $D$-modules on affine Grassmannian to the category of $λ$-regular modules of affine Lie algebra at critical level. We proved that $Γ^λ$ is exact and faithful. This generalized the work of Frenkel and Gaitsgory in the case when $λ=0$.
This paper is the first in a series that describe a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. In this paper we construct a model for the singularities of some would-be Schubert varieties in the affine Grassmannian for an affine Kac-Moody group. We formulate a conjecture describing the (local) intersection cohomology of these varieties in terms of integrable representations of the Langlands dual affine Kac-Moody group and check this conjecture in a number of cases. Roughly speaking the above singularities are constructed by looking at the Uhlenbeck sp…
An important result of Arkhipov-Bezrukavnikov-Ginzburg relates constructible sheaves on the affine Grassmannian to coherent sheaves on the dual Springer resolution. In this paper, we prove a positive-characteristic analogue of this statement, using the framework of "mixed modular sheaves" recently developed by the first author and Riche. As an application, we deduce a relationship between parity sheaves on the affine Grassmannian and Bezrukavnikov's "exotic t-structure" on the Springer resolution.
In this paper, we introduce Schubert decompositions for quiver Grassmannians and investigate example classes of quiver Grassmannians with a Schubert decomposition into affine spaces. The main theorem puts the cells of a Schubert decomposition into relation to the cells of a certain simpler quiver Grassmannian. This allows us to extend known examples of Schubert decompositions into affine spaces to a larger class of quiver Grassmannians. This includes exceptional representations of the Kronecker quiver as well as representations of forests with block matrices of the form $\begin{pmatrix} 0&1\\ …
We consider the category of modules over the affine Kac-Moody algebra g^ of critical level with regular central character. In our previous paper math.RT/0508382 we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian G((t))/G[[t]]. This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding I^0 equivariant categories, where I^0 is the radical of the Iwahori subgroup of G((t)). Our result may be viewed as an affine analogue of the equivalence…
Invariant deformation theory of affine schemes with reductive group action
Institut für Algebraische Geometrie Gottfried Wilhelm Leibniz Universität Hannover（汉诺威莱布尼茨大学代数几何研究所）；Institut für Mathematik Johannes Gutenberg-Universität Mainz（美因茨约翰内斯·古腾堡大学数学研究所）
Let $G$ be a connected reductive group over $\CC$ and let $G^{\vee}$ be the Langlands dual group. Crystals for $G^{\vee}$ were introduced by Kashiwara as certain ``combinatorial skeletons'' of finite-dimensional representations of $G^{\vee}$. For every dominant integral weight of $G^{\vee}$ Kashiwara constructed a canonical crystal. Other (independent) constructions of those crystals were given by Lusztig and Littelmann. It was also shown by Kashiwara and Joseph that the above family of crystals is unique if certain reasonable conditions are imposed. The purpose of this paper is to give anothe…
Let $G$ be a reductive group and $U,U^-$ be the unipotent radicals of a pair of opposite parabolic subgroups $P,P^-$. We prove that the DG-categories of $U(\!(t)\!)$-equivariant and $U^-(\!(t)\!)$-equivariant D-modules on the affine Grassmannian $Gr_G$ are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian defined in arXiv:1805.07721. We study various properties of the mentioned nearby cycles, in particular compare them with the nearby cycles studied in arXiv:1411.4206 and a…
We describe the equivariant cohomology of cofibers of spherical perverse sheaves on the affine Grassmannian of a reductive algebraic group in terms of the geometry of the Langlands dual group. In fact we give two equivalent descriptions: one in terms of D-modules of the basic affine space, and one in terms of intertwining operators for universal Verma modules. We also construct natural collections of isomorphisms parametrized by the Weyl group in these three contexts, and prove that they are compatible with our isomorphisms. As applications we reprove some results of the first author and of Br…
We prove the Mirković-Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no p-torsion, as long as p is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when p is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau-Mautner-Williamson theory of parity sheaves.
Let $G:=\widehat{SL_2}$ denote the affine Kac-Moody group associated to $SL_2$ and $\bar{\mathcal{X}}$ the associated affine Grassmannian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant Grothendieck group of $\bar{\mathcal{X}}$. In the case of ordinary (non-equivariant) $K$-theory we find an explicit closed form for the structure constants. We also determine an inductive formula for the structure constants in the torus-equivariant cohomology ring, and use this formula to find closed forms for some of the structure constants.
Let $G$ be a reductive affine algebraic group, and let $X$ be an affine algebraic $G$-variety. We establish a (poly)stability criterion for points $x\in X$ in terms of intrinsically defined closed subgroups $H_{x}$ of $G$, and relate it with the numerical criterion of Mumford, and with Richardson and Bate-Martin-Röhrle criteria, in the case $X=G^{N}$. Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson-Millson and Sikora about complex representation varieties of finitely …
We generalize a construction of Freudenburg and Moser-Jauslin in order to obtain an example of a non-linearizable action of a commutative reductive algebraic group on the affine space for every field of characteristic zero which admits a quadratic field extension.
Let $G$ be a reductive algebraic group and $H$ its reductive subgroup. Fix a Borel subgroup $B\subset G$ and a maximal torus $T\subset B$. The Cartan space $\a_{G,G/H}$ is, by definition, the subspace of $\Lie(T)^*$ generated by the weights of $B$-semiinvariant rational functions on $G/H$. We compute the spaces $\a_{G,G/H}$.
Braverman, Finkelberg and Nakajima introduced the generalized affine Grassmannian slices $\overline{\mathcal W}^λ_μ$ and showed that they are Coulomb branches of $3d$ $\mathcal N=4$ gauge theories. We prove a conjecture of theirs that a transversal slice of $\mathcal W^ν_μ$ in $\overline{\mathcal W}^λ_μ$ is isomorphic to $\overline{\mathcal W}^λ_ν$. In addition, they conjecture that Coulomb branches of $3d$ $\mathcal N=4$ gauge theories have symplectic singularities, and we confirm this conjecture for generalized affine Grassmannian slices $\overline{\mathcal W}^λ_μ$. Along the way we give a n…
The generalized affine Grassmannian slices $\overline{\mathcal{W}}_μ^λ$ are algebraic varieties introduced by Braverman, Finkelberg, and Nakajima in their study of Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories. We prove a conjecture of theirs by showing that the dense open subset $\mathcal{W}_μ^λ\subseteq \overline{\mathcal{W}}_μ^λ$ is smooth. An explicit decomposition of $\overline{\mathcal{W}}_μ^λ$ into symplectic leaves follows as a corollary. Our argument works over an arbitrary ring and in particular implies that the complex points $\mathcal{W}_μ^λ(\mathbb{C})$ are a smoo…
