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.Comment: v1: 29 pages, the present paper supersedes arXiv:0903.0678; v2: 26 pages, minor modifications; v3: 29 pages, final version, published in IMR

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.Comment: Amstex, 10 page

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 $\alpha\in N[I]$ of positive integers one can consider the space $Q_\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $B$. This space admits some remarkable compactifications $Q^D_\alpha$ (Quasimaps), $Q^L_\alpha$ (Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it was proved that the natural map $\pi: Q^L_\alpha\to Q^D_\alpha$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-MacPherson sheaf $IC_\alpha$ on the Quasimaps' space $Q^D_\alpha$. Namely, we prove that this singular support $SS(IC_\alpha)$ is irreducible. The proof is based on the factorization property of Quasimaps' space and on the detailed analysis of Laumon's resolution $\pi: Q^L_\alpha\to Q^D_\alpha$.Comment: 8 pages, AmsLatex 1.

Some extensions of the notion of loop Grassmannians

We report an ongoing attempt to establish in algebraic geometry certain analogues of topological ideas, The main goal is to associate to a scheme X over a commutative ring k its “relative motivic homology” which is again an algebro geometric object over the base k. This is motivated by Number Theory, so the Poincaré duality for this relative motivic homology should be an algebro geometric incarnation of Class Field Theory

Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution

We prove most of Lusztig’s conjectures on the canonical basis in homology of a Springer fiber. The conjectures predict that this basis controls numerics of representations of the Lie algebra of a semisimple algebraic group over an algebraically closed field of positive characteristic. We check this for almost all characteristics. To this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and Rumynin. On the other hand, it is compatible with the t-structure arising from an equivalence with the derived category of perverse sheaves on the affine flag variety of the Langlands dual group. This equivalence established by Arkhipov and the first author fits the framework of local geometric Langlands duality. The latter compatibility allows one to apply Frobenius purity theorem to deduce the desired properties of the basis. We expect the noncommutative counterpart of the Springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric Langlands duality.United States. Air Force Office of Scientific Research (Grant FA9550-08-1-0315)National Science Foundation (U.S.) (Grant DMS-0854764)National Science Foundation (U.S.) (Grant DMS-1102434

Technical Characteristics of Incunabulum in Europe

Incunabula are printed materials created in Europe from the time of Johann Gutenberg\u27s invention until 1500. Incunabula originate from the Latin language (lat. Incunabulum) and mean cradle or the beginning of something. In this paper, the representation of individual states and cities in the creation of incunabula is investigated and presented. The persons responsible for such development are also listed. Special attention is given to the presentation of Croatian incunabula. The mentioned works describe the characteristic features. Incunabula testify to a high level of culture, standards, and technological development of a particular area. The studied works reveal and confirm, as confirmed in this paper, the attitude of society towards literacy, education, and the national culture of each nation. This paper aims to comprehensively present the importance of incunabula for the development of European and Croatian culture, technological and comprehensive progress