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 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.Publication LINEAR KOSZUL DUALITY AND AFFINE HECKE ALGEBRAS(2009-01-01) 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(2010-01-01) 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 Modules over the small quantum group and semi-infinite flag manifold(2005-01-01) 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 REPRESENTATIONS OF SEMI-SIMPLE LIE ALGEBRAS IN PRIME CHARACTERISTIC AND NONCOMMUTATIVE SPRINGER RESOLUTION(2011-01-01) Bezrukavnikov, R; Mirkovic, IPublication QUIVER VARIETIES AND BEILINSON-DRINFELD GRASSMANNIANS OF TYPE A(2008-01-01) Mirkovic, I; Vybornov, MPublication Singular localization and intertwining functors for reductive Lie algebras in prime characteristic(2006-01-01) Bezrukavnikov, R; Mirkovic, I; Rumynin, DIn [BMR] we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) {\em regular} central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters. The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character $\lambda$ as sheaves on the partial flag variety corresponding to the singularity of $\lambda$. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators. We discuss {\em translation functors} and {\em intertwining functors}. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of $\mathcal{D}$-modules and coherent sheaves.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 Geometric Langlands duality and representations of algebraic groups over commutative rings(2007-01-01) Mirkovic, I; Vilonen, KPublication PERVERSE SHEAVES ON LOOP GRASSMANNIANS AND LANGLANDS DUALITY(1997) Mirkovic, I; Vilonen, K