The spherical part of the local and global Springer actions

The affine Weyl group acts on the cohomology (with compact support) of affine Springer fibers (local Springer theory) and of parabolic Hitchin fibers (global Springer theory). In this paper, we show that in both situations, the action of the center of the group algebra of the affine Weyl group (the spherical part) factors through the action of the component group of the relevant centralizers. In the situation of affine Springer fibers, this partially verifies a conjecture of Goresky–Kottwitz–MacPherson and Bezrukavnikov–Varshavsky. We first prove this result for the global Springer action, and then deduce from it the result for the local Springer action. This gives an application of global Springer theory to more classical problems.     

Generalized Springer theory and weight functions

We show that each Weyl group which enters in the generalized Springer correspondence carries a natural weight function.     

Springer correspondence for disconnected groups

The Springer correspondence is a map from the set of unipotent conjugacy classes of a reductive algebraic group to the set of irreducible complex characters of the Weyl group. Here, we determine this map explicitly in the case of disconnected classical algebraic groups. Mathematics Subject Classification (2000): Primary 20G05; Secondary 20C33.     

Modules over the small quantum group and semi-infinite flag manifold

We 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.     

Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves

The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. It turns out to be related to irreducible objects in the heart of a certain t-structure on the derived category of equivariant coherent sheaves on the Springer resolution, and to equivariant coherent IC sheaves on the nil-cone. The support of the cohomology is described in terms of cells in affine Weyl groups. The basis in the Grothendieck group provided by the cohomology modules is shown to coincide with the Kazhdan-Lusztig basis, as predicted by J. Humphreys and V. Ostrik. The proof is based on the results of [ABG ], [AB] and [B], which allow us to reduce the question to purity of IC sheaves on affine flag varieties. To the memory of my father     

Minimal reduction type and the Kazhdan–Lusztig map

We introduce the notion of minimal reduction type of an affine Springer fiber, and use it to define a map from the set of conjugacy classes in the Weyl group to the set of nilpotent orbits. We show that this map is the same as the one defined by Lusztig in Lfromto, (2011) and that the Kazhdan–Lusztig map in Kazhdan and Lusztig, (1998) is a section of our map. This settles several conjectures in the literature. For classical groups, we prove more refined results by introducing and studying the “skeleta” of affine Springer fibers.     

Weyl groupoids with at most three objects

We adapt the generalization of root systems by the second author and H. Yamane to the terminology of category theory. We introduce Cartan schemes, associated root systems and Weyl groupoids. After some preliminary general results, we completely classify all finite Weyl groupoids with at most three objects. The classification yields the result that there exist infinitely many “standard”, but only 9 “exceptional” cases.     

Reflection functors for quiver varieties and Weyl group actions

We introduce reflectionfunctors on quiver varieties. They are hyper-Kähler isometries between quiver varieties with different parameters, related by elements in the Weyl group. The definition is motivated by the origial reflection functor given by Bernstein-Gelfand-Ponomarev [1], but they behave much nicely. They are isomorphisms and satisfy the Weyl group relations. As an application, we define Weyl group representations of homology groups of quiver varieties. They are analogues of Slodowys construction of Springer representations of the Weyl group. Mathematics Subject Classification (2000):Primary 53C26; Secondary 14D21, 16G20, 20F55, 33D80Supported by the Grant-in-aid for Scientific Research (No.11740011), the Ministry of Education, Japan.     

Equivariant sheaves on flag varieties

We show that the Borel-equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves. This proves a conjecture of Soergel and Lunts in the case of flag varieties.     

Springer’s work on unipotent classes and Weyl group representations

We discuss some of the contributions of T.A. Springer (1926–2011) to the theory of algebraic groups, with emphasis on his work on unipotent classes and representations of Weyl groups.     

On the Action of the Dual Group on the Cohomology of Perverse Sheaves on the Affine Grassmannian

It was proved by Ginzburg, Mirkovic and Vilonen that the G(O)-equivariant perverse sheaves on the affine Grassmannian of a connected reductive group G form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group G . In this paper we construct explicitly the action of G on the global cohomology of a perverse sheaf.     

Perverse Coherent t-Structures Through Torsion Theories

Bezrukavnikov, later together with Arinkin, recovered Deligne’s work defining perverse t-structures in the derived category of coherent sheaves on a projective scheme. We prove that these t-structures can be obtained through tilting with respect to torsion theories, as in the work of Happel, Reiten and Smalø. This approach allows us to define, in the quasi-coherent setting, similar perverse t-structures for certain noncommutative projective planes.     

Springer Correspondences for Dihedral Groups

Recent work by a number of people has shown that complex reection groups give rise to many representation-theoretic structures (e.g., generic degrees and families of characters), as though they were Weyl groups of algebraic groups. Conjecturally, these structures are actually describing the representation theory of as-yet undescribed objects called spetses, of which reductive algebraic groups ought to be a special case. In this paper we carry out the Lusztig–Shoji algorithm for calculating Green functions for the dihedral groups. With a suitable set-up, the output of this algorithm turns out to satisfy all the integrality and positivity conditions that hold in the Weyl group case, so we may think of it as describing the geometry of the “unipotent variety” associated to a spets. From this, we determine the possible “Springer correspondences,” and we show that, as is true for algebraic groups, each special piece is rationally smooth, as is the full unipotent variety. DOI: . Supported by NSF grant DMS-0500873.     

Exterior Tensor Product of Perverse Sheaves

Under certain assumptions, we prove that the Deligne tensor product of the categories of constructible perverse sheaves on pseudomanifolds X and Y is the category of constructible perverse sheaves on X×Y. The functor of the exterior Deligne tensor product is identified with the exterior geometric tensor product.     

Classification of graded Hecke algebras for complex reflection groups

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups. Received: July 25, 2001     

Exceptional sequences and clusters

We show that exceptional sequences for hereditary algebras are characterized by the fact that the product of the corresponding reflections is the inverse Coxeter element in the Weyl group. We use this result to give a new combinatorial characterization of clusters tilting sets in the cluster category in the case where the hereditary algebra is of finite type.     

Character sheaves on the semi-stable locus of a group compactification

We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the “ordinary” restriction of a character sheaf on the compactification to a “semi-stable stratum” is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer (2006) [21]. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture Lusztig (2004) [16, 12.6] inside the semi-stable locus of the wonderful compactification.