On two geometric realizations of an affine Hecke algebra

The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant functions on a semi-simple group over a local non-Archimedian field, and Grothendieck group of equivariant coherent sheaves on Steinberg variety of the Langlands dual group; this isomorphism due to Kazhdan--Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures. The paper is a continuation of an earlier joint work with S. Arkhipov, it relies on technical material developed in a paper with Z. Yun.

Wreath Macdonald polynomials and categorical McKay correspondence

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_nltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert scheme, as a byproduct he obtained a derived equivalence between coherent sheaves on the Hilbert scheme and coherent sheaves on the orbifold quotient of ${mathbb A}^{2n}$ by the symmetric group $S_n$. A short proof of a similar derived equivalence for any symplectic quotient singularity has been obtained by the first author and Kaledin via quantization in positive characteristic. In the present note we show the properties of the derived equivalence which imply the generalized Macdonald positivity for wreath products.

On the size and structure of finite linear groups

This is a nearly complete manuscript left behind by Boris Weisfeiler before his disappearance during a hiking trip in Chile in 1985. It is posted on a request from the authors sister, Olga Weisfeiler.

Geometry of second adjointness for p-adic groups

We present a geometric proof of Bernsteins second adjointness for a reductive $p$-adic group. Our approach is based on geometry of the wonderful compactification and related varieties. Considering asymptotic behavior of a function on the group in a neighborhood of a boundary stratum of the compactification, we get a co-specialization map between spaces of functions on various varieties with $Gtimes G$ action. These maps can be viewed as maps of bimodules for the Hecke algebra, and the corresponding natural transformations of functors lead to the second adjointness. We also get a formula for the co-specialization map expressing it as a composition of the orishperic transform and inverse intertwining operator; a parallel result for $D$-modules was obtained in arXiv:0902.1493. As a byproduct we obtain a formula for the Plancherel functional restricted to a certain commutative subalgebra in the Hecke algebra, generalizing a result by Opdam.

Highest weight modules at the critical level and noncommutative Springer resolution

In arXiv:1001.2562 a certain non-commutative algebra $A$ was defined starting from a semi-simple algebraic group, so that the derived category of $A$-modules is equivalent to the derived category of coherent sheaves on the Springer (or Grothendieck-Springer) resolution. Let $hat{g}$ be the affine Lie algebra corresponding to the Langlands dual Lie algebra. Using results of Frenkel and Gaitsgory arXiv:0712.0788 we show that the category of $hat{g}$ modules at the critical level which are Iwahori integrable and have a fixed central character, is equivalent to the category of modules over a quotient of $A$ by a central character. This implies that numerics of Iwahori integrable modules at the critical level is governed by the canonical basis in the $K$-group of a Springer fiber, which was conjecturally described by Lusztig and constructed in arXiv:1001.2562.

Stability conditions for Slodowy slices and real variations of stability

We provide examples of an explicit submanifold in Bridgeland stabilities space of a local Calabi-Yau, and propose a new variant of definition of stabilities on a triangulated category, which we call a real variation of stability conditions. We discuss its relation to Bridgelands definition; the main theorem provides an illustration of such a relation. We also state a conjecture by the second author and Okounkov relating this structure to quantum cohomology of symplectic resolutions and establish its validity in some special cases. More precisely, let X be the standard resolution of a transversal slice to an adjoint nilpotent orbit of a simple Lie algebra over C. An action of the affine braid group on the derived category of coherent sheaves on X and a collection of t-structures on this category permuted by the action have been constructed in arXiv:1101.3702 and arXiv:1001.2562 respectively. In this note we show that the t-structures come from points in a certain connected submanifold in the space of Bridgeland stability conditions. The submanifold is a covering of a submanifold in the dual space to the Grothendieck group, and the affine braid group acts by deck transformations. In the special case when dim (X)=2 a similar (in fact, stronger) result was obtained in arXiv:math/0508257.

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

We prove most of Lusztigs conjectures from the paper Bases in equivariant K-theory II, including the existence of a canonical basis in the Grothendieck group of a Springer fiber. The conjectures also predict that this basis controls numerics of representations of the Lie algebra of a semi-simple algebraic group over an algebraically closed field of positive characteristic. We check this for almost all characteristics. To this end we construct a non-commutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is shown to be compatible with the positive characteristic version of Beilinson-Bernstein localization equivalences. On the other hand, it is compatible with the t-structure arising from the equivalence of Arkhipov-Bezrukavnikov with the derived category of perverse sheaves on the affine flag variety of the Langlands dual group, which was inspired by local geometric Langlands duality. This 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.