0 Introduction In this paper, we give an algorithm to compute the following cohomology groups on U = C n n V (f) for any non-zero polynomial f 2 Qx 1 ; : : : ; x n ]: 1. H k (U; C U), where C U is the constant sheaf on U with stalk C. 2. H k (U; V), where V is the locally constant sheaf on U of rank one deened by a multi-valued function f a 1 1 f a d d with polynomials f 1 ; : : : ; f d 2 Qx] s...

If A is a bialgebra over a field k, a left-right Yetter-Drinfel’d module over A is a k-linear space M which is a left A-module, a right A-comodule and such that a certain compatibility condition between these two structures holds. YetterDrinfel’d modules were introduced by D. Yetter in [18] under the name of “crossed bimodules” (they are called “quantum Yang-Baxter modules” in [5]; the present ...

We present a simple proof of a precise version of the localization theorem in equivariant cohomology. As an application, we describe the cohomology algebra of any compact symplectic variety with a multiplicity-free action of a compact Lie group. This applies in particular to smooth, projective spherical varieties. 1 A precise version of the localization theorem Let X be a topological space with...

Let G be a group scheme of finite type over a field, and consider the cohomology ring H∗(G) with coefficients in the structure sheaf. We show that H∗(G) is a free module of finite rank over its component of degree 0, and is the exterior algebra of its component of degree 1. When G is connected, we determine the Hopf algebra structure of H∗(G).

In [8] we introduced the notion of a local torus actions modeled on the standard representation (we call it a local torus action for simplicity), which is a generalization of a locally standard torus action. In this note we define a lifting of a local torus action to a principal torus bundle, and show that there is an obstruction class for the existence of liftings in the first cohomology of th...

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for simple Lie group $E_{6(-14)}$, which is of Hermitian symmetric type. Each FS-scattered series $E_{6(-14)}$ realized as a composition factor certain $A_{\mathfrak{q}}(\lambda)$ module. Along way, we have also obtained fully supported integral infinitesimal characters.

Kumar described the Schubert classes which are the dual to the closures of the Bruhat cells in the flag varieties of the Kac-Moody groups associated to the infinite dimensional KacMoody algebras [17]. These classes are indexed by affine Weyl groups and can be chosen as elements of integral cohomologies of the homogeneous space L̂polGC/B̂ for any compact simply connected semisimple Lie groupG. Lat...

We study the algebraic Gauss-Manin system and the algebraic Brieskorn module associated to a polynomial mapping with isolated singularities. Since the algebraic GaussManin system does not contain any information on the cohomology of singular fibers, we construct a non quasi-coherent sheaf which gives the cohomology of every fiber. Then we study the algebraic Brieskorn module by comparing it wit...

Consider the ring R := Q[τ, τ−1] of Laurent polynomials in the variable τ . The Artin’s Pure Braid Groups (or Generalized Pure Braid Groups) act over R, where the action of every standard generator is the multiplication by τ . In this paper we consider the cohomology of such groups with corefficients in the module R (it is well known that such cohomology is strictly related to the untwisted int...

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q = −1. This allows us to define graded modules over the Hecke algebra at q = −1 that are ‘odd’ analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient...