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     

The behavior of Nil-groups under localization and the relative assembly map

We study the behavior of the Nil-subgroups of K-groups under localization. As a consequence of our results, we obtain that the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups is rationally an isomorphism. Combined with the equivariant Chern character, we obtain a complete computation of the rationalized source of the K-theoretic assembly map that appears in the Farrell-Jones conjecture in terms of group homology and the K-groups of finite cyclic subgroups.Specifically we prove that under mild assumptions we can always write the Nil-groups and End-groups of the localized ring as a certain colimit over the Nil-groups and End-groups of the ring, generalizing a result of Vorst. We define Frobenius and Verschiebung operations on certain Nil-groups. These operations provide the tool to prove that Nil-groups are modules over the ring of Witt-vectors and are either trivial or not finitely generated as Abelian groups. Combining the localization results with the Witt-vector module structure, we obtain that Nil and localization at an appropriate multiplicatively closed set S commute, i.e. . An important corollary is that the Nil-groups appearing in the decomposition of the K-groups of virtually cyclic groups are torsion groups.     

Un théorème du point fixe de Lefschetz en géométrie d'Arakelov

We consider arithmetic varieties endowed with an action of the group scheme of n-th roots of unity and we define equivariant arithmetic K₀-theory for these varieties. We then state a Riemann-Roch theorem for the natural transformation of equivariant arithmetic K₀ -theory induced by the restriction to the fixed point scheme and we show that it implies a version of Bismut's conjecture of an equivariant arithmetic Riemann-Roch theorem.     

An equivariant Novikov conjecture

We discuss and formulate the correct equivariant generalization of the strong Novikov conjecture. This will be the statement that certain G-equivariant higher signatures (living in suitable equivariant K-groups) are invariant under G-maps of manifolds which, nonequivariantly, are homotopy equivalences preserving orientation. We prove this conjecture for manifolds modeled on a complete Riemannian manifold of nonpositive curvature on which G (a compact Lie group) acts by isometries. We also use the theory of harmonic maps to construct (in some cases) G-maps into such model spaces.Dedicated to Alexander GrothendieckPartially supported by NSF Grants DMS 84-00900 and 87-00551.Partially supported by NSF Grant DMS 86-02980, a Presidential Young Investigator Award, and a Sloan Foundation Fellowship.     

K-theory of equivariant quantization

Using an equivariant version of Connes? Thom isomorphism, we prove that equivariant K-theory is invariant under strict deformation quantization for a compact Lie group action.     

Δ-graphs of polytopes in Bruns and Gubeladze K-theory

W. Bruns and J. Gubeladze introduced a new version of algebraic K-theory where K-groups are additionally parameterized by polytopes of some type. In this paper we propose a concept of stable E-equivalence which can be used to calculate K-groups for high-dimensional polytopes. Polytopes which are stable E-equivalent have similar inner structures and isomorphic K-groups. In addition, for each polytope we define a Δ-graph which is an oriented graph being invariant under a stable E-equivalence.     

Some non-trivial Kazhdan-Lusztig coefficients of an affine Weyl group of type _n

In this paper we show that the leading coeficient μ(y,w) of some Kazhdan-Lusztig polynomials Py,w with y,w in an affine Weyl group of type n is n + 2.This fact has some consequences on the dimension of first extension groups of finite groups of Lie type with irreducible coefficients.     

Twisted K-theory of differentiable stacks

In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S¹-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological spaces, twisted equivariant K-theory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called “twisted vector bundles”.Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of K-theory (KK-theory) of C^∗-algebras.     

Algebraic <Emphasis Type="Italic">K</Emphasis>-theory of Gorenstein projective modules

We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the ‘resolution theorem’ in this context due to Quillen. We characterize the Gorenstein algebraic K-groups by two different algebraic K-groups and by the idempotent completeness of the Gorenstein singularity category of the ring. We compute the Gorenstein algebraic K-groups along a recollement of the bounded Gorenstein derived categories of CM-nite Gorenstein algebras.     

Equivariant homology and cohomology of groups

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ-equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.     

Dominant K-theory and integrable highest weight representations of Kac-Moody groups

We give a topological interpretation of the highest weight representations of Kac-Moody groups. Given the unitary form G of a Kac-Moody group (over C), we define a version of equivariant K-theory, KG on the category of proper G-CW complexes. We then study Kac-Moody groups of compact type in detail (see Section 2 for definitions). In particular, we show that the Grothendieck group of integrable highest weight representations of a Kac-Moody group G of compact type, maps isomorphically onto , where EG is the classifying space of proper G-actions. For the affine case, this agrees very well with recent results of Freed-Hopkins-Teleman. We also explicitly compute for Kac-Moody groups of extended compact type, which includes the Kac-Moody group E₁₀.     

Twisted Equivariant Matter

We show how general principles of symmetry in quantum mechanics lead to twisted notions of a group representation. This framework generalizes both the classical threefold way of real/complex/ quaternionic representations as well as a corresponding tenfold way which has appeared in condensed matter and nuclear physics. We establish a foundation for discussing continuous families of quantum systems. Having done so, topological phases of quantum systems can be defined as deformation classes of continuous families of gapped Hamiltonians. For free particles, there is an additional algebraic structure on the deformation classes leading naturally to notions of twisted equivariant K-theory. In systems with a lattice of translational symmetries, we show that there is a canonical twisting of the equivariant K-theory of the Brillouin torus. We give precise mathematical definitions of two invariants of the topological phases which have played an important role in the study of topological insulators. Twisted equivariant K-theory provides a finer classification of topological insulators than has been previously available.     

REDUCTIVE COMPACT HOMOGENEOUS CR MANIFOLDS

We define and investigate a class of compact homogeneous CR manifolds, that we call $ \mathfrak{n} $ -reductive. They are orbits of minimal dimension of a compact Lie group K ₀ in algebraic affine homogeneous spaces of its complexification K. For these manifolds we obtain canonical equivariant fibrations onto complex flag manifolds, generalizing the Hopf fibration $ {S^3}\to \mathbb{C}{{\mathbb{P}}^1} $ . These fibrations are not, in general, CR submersions, but satisfy the weaker condition of being CR-deployments; to obtain CR submersions we need to strengthen their CR structure by lifting the complex stucture of the base.     

Affine Jacquet functors and Harish-Chandra categories

We define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan on the structure of Verma modules in the Bernstein-Gelfand-Gelfand categories O for Kac-Moody algebras. This is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine Harish-Chandra modules of Lian and Zuckerman. The latter provides an extension of the works of Rocha-Caridi and Wallach [A. Rocha-Caridi, N.R. Wallach, Projective modules over infinite dimensional graded Lie algebras, Math. Z. 180 (1982) 151-177] and Deodhar, Gabber and Kac [V. Deodhar, O. Gabber, V. Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. Math. 45 (1982) 92-116] on block decompositions of BGG categories for Kac-Moody algebras. We also derive a compatibility relation between the affine Jacquet functor and the Kazhdan-Lusztig tensor product and apply it to prove that the affine Harish-Chandra category is stable under fusion tensoring with the Kazhdan-Lusztig category. This compatibility will be further applied in studying translation functors for the affine Harish-Chandra category, based on the fusion tensor product.     

Poisson Morphisms and Reduced Affine Poisson Group Actions

We establish the concept of a quotient affine Poisson group, and study the reduced Poisson action of the quotient of an affine Poisson group G on the quotient of an affine Poisson-G-variety V. The Poisson morphisms (including equivariant cases) between Poisson affine varieties are also discussed. Received April 5, 1999, Accepted March 5, 2001     

Clean unital ℓ-groups

A ring with identity is said to be clean if every element can be written as a sum of a unit and an idempotent. The study of clean rings has been at the forefront of ring theory over the past decade. The theory of partially-ordered groups has a nice and long history and since there are several ways of relating a ring to a (unital) partially-ordered group it became apparent that there ought to be a notion of a clean partially-ordered group. In this article we define a clean unital lattice-ordered group; we state and prove a theorem which characterizes clean unital ?-groups. We mention the relationship of clean unital ?-groups to algebraic K-theory. In the last section of the article we generalize the notion of clean to the non-unital context and investigate this concept within the framework of W-objects, that is, archimedean ?-groups with distinguished weak order unit.     

On the strongly closed subgroups or H-subgroups of finite groups

Let G be a finite group. Goldschmidt, Flores, and Foote investigated the concept: Let K ≤ G. A subgroup H of K is called strongly closed in K with respect to G if H ^g ∩ K ≤ H for all g ∈ G. In particular, when H is a subgroup of prime-power order and K is a Sylow subgroup containing it, H is simply said to be a strongly closed subgroup. Bianchi and the others called a subgroup H of G an H-subgroup if N _G (H) ∩ H ^g ≤ H for all g ∈ G. In fact, an H-subgroup of prime power order is the same as a strongly closed subgroup. We give the characterizations of finite non-T-groups whose maximal subgroups of even order are solvable T-groups by H-subgroups or strongly closed subgroups. Moreover, the structure of finite non-T-groups whose maximal subgroups of even order are solvable T-groups may be difficult to give if we do not use normality.     

Real, complex and quaternionic equivariant vector fields on spheres

The equivariant real, complex and quaternionic vector fields on spheres problem is reduced to a question about the equivariant J-groups of the projective spaces. As an application of this reduction, we give a generalization of the results of Namboodiri [U. Namboodiri, Equivariant vector fields on spheres, Trans. Amer. Math. Soc. 278 (2) (1983) 431-460], on equivariant real vector fields, and Önder [T. Önder, Equivariant cross sections of complex Stiefel manifolds, Topology Appl. 109 (2001) 107-125], on equivariant complex vector fields, which avoids the restriction that the representation containing the sphere has enough orbit types.