Mirkovic, Ivan

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Professor, Department of Mathematics and Statistics
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Algebraic Geometry
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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Now showing 1 - 10 of 21
  • Publication
    (2008-01-01) Mirkovic, I; Vybornov, M
  • Publication
    Modules over the small quantum group and semi-infinite flag manifold
    (2005-01-01) Arkhipov, S; Bezrukavnikov, R; Braverman, A; Gaitsgory, D; Mirkovic, I
    We 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
    (1992) Mirkovic, I; UZAWA, T; VILONEN, K
  • Publication
    Intersection cohomology of Drinfeld‚s compactifications
    (2002-01-01) Braverman, A; Finkelberg, M; Gaitsgory, D; Mirkovic, I
    Let X be a smooth complete curve, G be a reductive group and a parabolic. Following Drinfeld, one defines a (relative) compactification of the moduli stack of P-bundles on X. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of . The description is given in terms of the combinatorics of the Langlands dual Lie algebra .
  • Publication
    Centers of reduced enveloping algebras
    (1999) Mirkovic, I; Rumynin, D
    We 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
    Equivariant homology and K-theory of affine Grassmannians and Toda lattices
    (2005-01-01) Bezrukavnikov, R; Finkelberg, M; Mirkovic, I
    For 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
    A note on a symplectic structure on the space of G-monopoles
    (1999) Finkelberg, M; Kuznetsov, A; Markarian, N; Mirkovic, I
    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=\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.
  • Publication
    On quiver varieties and affine Grassmanians of type A
    (2003-01-01) Mirkovic, I; Vybornov, M
    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. To cite this article: I. Mirkovi , M. Vybornov, C. R. Acad. Sci. Paris, Ser. I 336 (2003).