Let G be an algebraic reductive group over a an algebraically closed 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 of \(U\) commutes with Verdier duality. In the first example we take \(\chi\) to be an arbitrary \(G\)-variety and we prove the above property for all \(\overline{P}\)-equivariant sheaves on \(X\) where \(\overline{P}\) is an opposite parabolic subgroup assuming \(\chi\) satisfies a strong nondegeneracy condition (such a \(\chi\) exists for some but not all choices of \(P\)). In the case when \(P\) is a Borel subgroup it is enough to require that the sheaf in question is \(\overline{U}\) equivariant where \(\overline{U}\) is the unipotent radical of \(\overline{P}\). In the second example we take \(X = G\) where \(G\) acts by left translations and we prove the corresponding result when \(P\) is a Borel subgroup for sheaves equivariant under the adjoint action of \(G\) (the latter result was conjectured by B. C. Ngo who proved it for \(G = GL(n)\)). As an application we reprove a theorem of N. Katz and G. Laumon about local acyclicity of the kernel of the Fourier-Deligne transform.
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 Bridgeland's 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.
This paper is an introduction, in a simplified setting, to Lusztig's theory
of character sheaves. It develops a notion of character sheaves on reductive
Lie algebras which is more general then such ...notion of Lusztig, and closer to
Lusztig's theory of character sheaves on groups. The development is self
contained and independent of the characteristic $p$ of the ground field. The
results for Lie algebras are then used to give simple and uniform proofs for
some of Lusztig's results on groups.
Let \(G\) be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian \(Gr_G\). We prove Simon ...Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with \(U(\check{\mathfrak n})\) (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra \(\check{\mathfrak g}\)). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.
This paper is an introduction, in a simplified setting, to Lusztig's theory of character sheaves. It develops a notion of character sheaves on reductive Lie algebras which is more general then such ...notion of Lusztig, and closer to Lusztig's theory of character sheaves on groups. The development is self contained and independent of the characteristic \(p\) of the ground field. The results for Lie algebras are then used to give simple and uniform proofs for some of Lusztig's results on groups.
We prove most of Lusztig's 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.
In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We ...construct this duality in a very general setting, and prove its compatibility with morphisms of vector bundles and base change.
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