arXiv subjects

Ivan Mirković

Publications and source records attributed to Ivan Mirković.

15 recordsLinked to original sources

arXiv2026

Equivariant ($K$-)homology of affine Grassmannian and Toda lattice

For an almost simple complex algebraic group $G$ with affine Grassmannian $Gr_G= G(C((t)))/G(C[[t]])$ we consider the equivariant homology $H^{G(C[[t]])}(Gr_G)$, and $K$-theory $K^{G(C[[t]])}(Gr_G)$. They both have a commutative ring structure, with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group $\check G$, and we relate the spectrum of $K$-homology ring to the universal group-group centralizer of $G$ and of $\check G$. If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the ($K$)-homology ring, and thus a Poisson structure on its spectrum. We identify this structure with the standard one on the universal centralizer. The commutative subring of $G(C[[t]])$-equivariant homology of the point gives rise to a polarization which is related to Kostant's Toda lattice integrable system. We also compute the equivariant $K$-ring of the affine Grassmannian Steinberg variety. The equivariant $K$-homology of $Gr_G$ is equipped with a canonical base formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin-Loktev fusion product of $G(C[[t]])$-modules.

math.AG↗

arXiv2024

Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian and geometric Satake equivalence (with appendix by Dennis Gaitsgory)

Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\check{\mathfrak n})$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\check{\mathfrak g}$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.

math.AG↗

arXiv2019

An Iwahori-Whittaker model for the Satake category

In this paper we prove, for G a connected reductive algebraic group satisfying a technical assumption, that the Satake category of G (with coefficients in a finite field, a finite extension of Q_l, or the ring of integers of such a field) can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian. As an application, we confirm a conjecture of Juteau-Mautner-Williamson describing the tilting objects in the Satake category.

math.RT↗

arXiv2015

Linear Koszul Duality II - Coherent sheaves on perfect sheaves

In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general setting, and prove its compatibility with morphisms of vector bundles and base change.

math.RT↗

arXiv2015

A note on the symplectic structure on the space of G-monopoles

Let $G$ be a semisimple complex Lie group with a Borel subgroup $B$. Let $X=G/B$ be the flag manifold of $G$. Let $C=P^1\ni\infty$ be the projective line. Let $α\in H_2(X,{\Bbb Z})$. The moduli space of $G$-monopoles of topological charge $α$ (see e.g. [Jarvis]) is naturally identified with the space $M_b(X,α)$ of based maps from $(C,\infty)$ to $(X,B)$ of degree $α$. The moduli space of $G$-monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. We propose a simple explicit formula for the symplectic structure on $M_b(X,α)$. It generalizes the well known formula for $G=SL_2$ (see e.g. [Atiyah-Hitchin]). Let $P\supset B$ be a parabolic subgroup. The construction of the Poisson structure on $M_b(X,α)$ generalizes verbatim to the space of based maps $M=M_b(G/P,β)$. In most cases the corresponding map $T^*M\to TM$ is not an isomorphism, i.e. $M$ splits into nontrivial symplectic leaves. These leaves are explicilty described.

math.AG↗

arXiv2014

Iwahori-Matsumoto involution and linear Koszul Duality

We use linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras studied in previous papers of the authors to give a geometric realization of the Iwahori-Matsumoto involution of affine Hecke algebras. More generally we prove that linear Koszul duality is compatible with convolution in a general context related to convolution algebras.

math.RT↗

arXiv2013

Linear Koszul duality and affine Hecke algebras

In this paper we prove that the linear Koszul duality equivalence constructed in a previous paper provides a geometric realization of the Iwahori-Matsumoto involution of affine Hecke algebras.

math.RT↗

arXiv2009

Linear Koszul Duality

In this paper we construct, for F_1 and F_2 subbundles of a vector bundle E, a "Koszul duality" equivalence between derived categories of G_m-equivariant coherent (dg-)sheaves on the derived intersection of F_1 and F_2 inside E, and the corresponding derived intersection of the orthogonals of F_1 and F_2 inside the dual vector bundle E^*. We also propose applications to Hecke algebras.

math.RT↗

arXiv2008

Quiver varieties and Beilinson-Drinfeld Grassmannians of type A

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.

math.AG↗

arXiv2008

Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' Spaces

For a simple algebraic group $G$ we study the space $Q$ of Quasimaps from the projective line $C$ to the flag variety of $G$. We prove that the global Intersection Cohomology of $Q$ carries a natural pure Tate Hodge structure, and compute its generating function. We define an action of the Langlands dual Lie algebra $g^L$ on this cohomology. We present a new geometric construction of the universal enveloping algebra $U(n^L_+)$ of the nilpotent subalgebra of $g^L$. It is realized in the Ext-groups of certain perverse sheaves on Quasimaps' spaces, and it is equipped with a canonical basis numbered by the irreducible components of certain algebraic cycles, isomorphic to the intersections of semiinfinite orbits in the affine Grassmannian of $G$. We compute the stalks of the IC-sheaves on the Schubert strata closures in $Q$. They carry a natural pure Tate Hodge structure, and their generating functions are given by the generic affine Kazhdan-Lusztig polynomials. In the Appendix we prove that Kontsevich's space of stable maps provides a natural resolution of $Q$.

alg-geom↗

arXiv2006

Localization of modules for a semisimple Lie algebra in prime characteristic

We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on (generalized) Springer fibers. The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of $D$-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra. Thus the ``derived'' version of the Beilinson-Bernstein localization Theorem holds in sufficiently large positive characteristic. Next, the algebra of (``crystalline'') differential operators is an Azumaya algebra and its splittings on Springer fibers allow us to pass from D-modules to coherent sheaves. As an application we compute the rank of the Grothendieck group of the category of modules over the Lie algebra with a fixed central character.

math.RT↗

arXiv2002

On quiver varieties and affine Grassmannians of type A

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 Grassmannians.

math.AG↗

arXiv1998

Semiinfinite Flags. I. Case of global curve $P^1$

The Semiinfinite Flag Space appeared in the works of B.Feigin and E.Frenkel, and under different disguises was found by V.Drinfeld and G.Lusztig in the early 80-s. Another recent discovery (Beilinson-Drinfeld Grassmannian) turned out to conceal a new incarnation of Semiinfinite Flags. We write down these and other results scattered in folklore. We define the local semiinfinite flag space attached to a semisimple group $G$ as the quotient $G((z))/HN((z))$ (an ind-scheme), where $H$ and $N$ are a Cartan subgroup and the unipotent radical of a Borel subgroup of $G$. The global semiinfinite flag space attached to a smooth complete curve $C$ is a union of Quasimaps from $C$ to the flag variety of $G$. In the present work we use $C=P^1$ to construct the category $PS$ of certain collections of perverse sheaves on Quasimaps spaces, with factorization isomorphisms. We construct an exact convolution functor from the category of perverse sheaves on affine Grassmannian, constant along Iwahori orbits, to the category $PS$. Conjecturally, this functor should correspond to the restriction functor from modules over quantum group with divided powers to modules over the small quantum group.

alg-geom↗

arXiv1997

The Singular Supports of IC sheaves on Quasimaps' Spaces are Irreducible

Let $C$ be a smooth projective curve of genus 0. Let $B$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $α\in N[I]$ of positive integers one can consider the space $Q_α$ of algebraic maps of degree $α$ from $C$ to $B$. This space admits some remarkable compactifications $Q^D_α$ (Quasimaps), $Q^L_α$ (Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it was proved that the natural map $π: Q^L_α\to Q^D_α$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-MacPherson sheaf $IC_α$ on the Quasimaps' space $Q^D_α$. Namely, we prove that this singular support $SS(IC_α)$ is irreducible. The proof is based on the factorization property of Quasimaps' space and on the detailed analysis of Laumon's resolution $π: Q^L_α\to Q^D_α$.

alg-geom↗

arXiv1997

Perverse Sheaves on Loop Grassmannians and Langlands Duality

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 dual group $\check G$ 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 \cite{G} and some of the arguments are borrowed from \cite{G}. However, we use a more "natural" commutativity constraint for the convolution product, due to Drinfeld. Secondly, we give a direct geometric proof that the global cohomology functor is exact and decompose this cohomology functor into a direct sum of weights. We avoid the use of the decomposition theorem of \cite{BBD} which makes our techniques applicable to perverse sheaves with coefficients over arbitrary commutative rings. This note includes sketches of (some of the) proofs. The details, as well as the generalization to representations over arbitrary fields and commutative rings will appear elsewhere.

alg-geom↗