Tilting exercises

This is an application of the theory of tilting objects to the geometric setting of perverse sheaves. We show that this theory is a natural framework for Beilinson's gluing of perverse sheaves construction. In the special case of Schubert stratification of a flag variety we get a short proof of Soergel's "Struktursatz", and describe (following a conjecture of Kapranov) Serre functor for category O. Some of our results were obtained independently by Rouquier.Comment: This final version to appear in Moscow Math Journal differs very slightly from the previous on

Character sheaves on reductive Lie algebras

The paper develops a linearized notion of Lusztig\u27s character sheaves (on Lie algebras rather then on groups), which contains Lusztig\u27s class of character sheaves on Lie algebras. The theory is independent of the characteristic p of the field, and we use it to provide elementary proofs of some results of Lusztig (for instance, the observation that on groups all cuspidal sheaves are character sheaves)

LINEAR KOSZUL DUALITY AND AFFINE HECKE ALGEBRAS

n 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

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

Geometric Langlands duality and representations of algebraic groups over commutative rings

This is the pre-published version harvested from arXiv. The published version is located at http://annals.math.princeton.edu/2007/166-1/p0395-14

REPRESENTATIONS OF SEMI-SIMPLE LIE ALGEBRAS IN PRIME CHARACTERISTIC AND NONCOMMUTATIVE SPRINGER RESOLUTION

This is the pre-published version harvested from ArXiv.</p

On quiver varieties and affine Grassmanians of type A

We construct Nakajima\u27s 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\u27s 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)

SEMIINFINITE FLAGS. I. CASE OF GLOBAL CURVE P1.

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.Comment: References update

Some results about geometric Whittaker model

Let G be an algebraic reductive group over a field of positive characteristic. Choose a parabolic subgroup P in G and denote by U its unipotent radical. Let X be a G-variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of ℓ-adic sheaves on X with respect to a generic character commutes with Verdier duality. Namely, in the first example we take X to be an arbitrary G-variety and we prove the above property for all -equivariant sheaves on X where is the unipotent radical of an opposite parabolic subgroup; in the second example we take X=G and we prove the corresponding result for sheaves which are equivariant under the adjoint action (the latter result was conjectured by B. C. Ngo who proved it for G=GL(n)). As an application of the proof of the first statement we reprove a theorem of N. Katz and G. Laumon about local acyclicity of the kernel of the Fourier–Deligne transform