Mirkovic, Ivan
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Professor, Department of Mathematics and Statistics
Last Name
Mirkovic
First Name
Ivan
Discipline
Algebraic Geometry
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Introduction
The motivating dream is to understand the geometry that underlies Number Theory and QFT, for instance zeta functions, Langlands conjectures and string dualities. In this direction I study geometric objects that contain representation theoretic-information, such as flag varieties and loop Grassmannians.
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Publication QUIVER VARIETIES AND BEILINSON-DRINFELD GRASSMANNIANS OF TYPE A(2008-01) Mirkovic, I; Vybornov, MPublication REPRESENTATIONS OF SEMI-SIMPLE LIE ALGEBRAS IN PRIME CHARACTERISTIC AND NONCOMMUTATIVE SPRINGER RESOLUTION(2011-01) Bezrukavnikov, R; Mirkovic, IPublication LINEAR KOSZUL DUALITY AND AFFINE HECKE ALGEBRAS(2009-01) Mirkovic, I; Richie, Sn 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.Publication Linear Koszul duality(2010-01) Mirkovic, I; Riche, SIn this paper we construct, for F1 and F2 subbundles of a vector bundle E, a ‘Koszul duality’ equivalence between derived categories of m-equivariant coherent(dg-)sheaves on the derived intersection , and the corresponding derived intersection . We also propose applications to Hecke algebras.Publication MATSUKI CORRESPONDENCE FOR SHEAVES(1992) Mirkovic, I; UZAWA, T; VILONEN, KPublication Characteristic cycles for the loop Grassmannian and nilpotent orbits(1999) Evens, S; Mirkovic, IPublication Equivariant homology and K-theory of affine Grassmannians and Toda lattices(2005-01) Bezrukavnikov, R; Finkelberg, M; Mirkovic, IFor an almost simple complex algebraic group G with affine Grassmannian $\text{Gr}_G=G(\mathbb{C}(({\rm t})))/G(\mathbb{C}[[{\rm t}]])$, we consider the equivariant homology $H^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$ and K-theory $K^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$. 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 relate this structure to the standard one on the universal centralizer. The commutative subring of $G(\mathbb{C}[[{\rm t}]])$-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 GrG is equipped with a canonical basis 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 $G(\mathbb{C}[[{\rm t}]])$-modules.Publication Centers of reduced enveloping algebras(1999) Mirkovic, I; Rumynin, DWe compute the inverse image of a functional in the Zassenhaus variety. We apply this computation to describe the category of representations for a regular functional.Publication Modules over the small quantum group and semi-infinite flag manifold(2005-01) Arkhipov, S; Bezrukavnikov, R; Braverman, A; Gaitsgory, D; Mirkovic, IWe develop a theory of perverse sheaves on the semi-infinite flag manifold G((t))/N((t)) · T[[t]], and show that the subcategory of Iwahori-monodromy perverse sheaves is equivalent to the regular block of the category of representations of the small quantum group at an even root of unity.Publication A note on a symplectic structure on the space of G-monopoles(1999) Finkelberg, M; Kuznetsov, A; Markarian, N; Mirkovic, ILet G be a semisimple complex Lie group with a Borel subgroup B. Let X=G/B be the flag manifold of G. Let C=\PP1 ' ¥Unknown control sequence '\PP' be the projective line. Let a Î H2(\bX,\Bbb Z)Unknown control sequence '\bX'. The moduli space of G-monopoles of topological charge f is naturally identified with the space \CMb(\bX,a)Unknown control sequence '\CM' of based maps from (C,X) to (X,B) of degree f. The moduli space of G-monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. It was explicitly computed by R. Bielawski in case G=SLn. We propose a simple explicit formula for another natural symplectic structure on \CMb(\bX,a)Unknown control sequence '\CM' . It is tantalizingly similar to R. Bielawski's formula, but in general (rank >1) the two structures do not coincide. Let P´G be a parabolic subgroup. The construction of the Poisson structure on \CMb(\bX,a)Unknown control sequence '\CM' generalizes verbatim to the space of based maps \CM = \CMb(\bG/\bP,b)Unknown control sequence '\CM'. In most cases the corresponding map T*\CM® T\CMUnknown control sequence '\CM' is not an isomorphism, i.e. \CMUnknown control sequence '\CM' splits into nontrivial symplectic leaves. These leaves are explicilty described.