Related papers: Sheaves on affine Schubert varieties, modular repr…

In math.RT/0201073 we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves…

We prove the (graded) Jordan--H\"{o}lder multiplicities of (mixed) tilting sheaves on flag varieties admit a geometric interpretation as the hypercohomology of certain sheaves on Richardson varieties in the Langlands dual flag variety.…

We revisit sheaves on locales by placing them in the context of the theory of quantale modules. The local homeomorphisms $p:X\to B$ are identified with the Hilbert $B$-modules that are equipped with a natural notion of basis. The…

We compute the value of finitary localizing invariants, including algebraic $K$-theory, on categories of sheaves over stably locally compact spaces $X$. Our formula simultaneously generalizes the cases of locally compact Hausdorff and…

For a field of characteristic $\ne 2$ we study vector spaces that are graded by the weight lattice of a root system, and are endowed with linear operators in each simple root direction. We show that these data extend to a graded semisimple…

Let $G$ be a simple simply connected complex algebraic group and let $\mathfrak{g}_*$ be a $\mathbf{Z}/m$-grading on its Lie algebra $\mathfrak{g}$. In a recent series of articles, G. Lusztig and Z. Yun, studied the classification of simple…

Building on Simpson's original definition over the complex numbers, we introduce the notion of restricted sheaf $\Lambda$ of rings of differential operators on a variety defined over a field of positive characteristic. We define the notion…

Let g be a semi-simple Lie algebra. In this paper we study the spaces of based quasi-maps from the projective line P^1 to the flag variety of g (it is well-known that their singularities are supposed to model the singularities of the so…

For a non-isotrivial family of surfaces of general type over a complex projective curve, we give upper bounds for the degree of the direct images of powers of the relative dualizing sheaf. They imply that, fixing the curve and the possible…

The Kazhdan Lusztig isomorphism, relating the affine Hecke algebra of a $p$-adic group to the equivariant $K$ theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne Langlands conjectures…

We review some recent advances in modular representation theory of symmetric groups and related Hecke algebras. We discuss connections with Khovanov-Lauda-Rouquier algebras and gradings on the blocks of the group algebras $F\Sigma_n$, which…

In this paper, we look at the problem of determining the composition factors for the affine graded Hecke algebra via the computation of Kazhdan-Lusztig type polynomials. We review the algorithms of \cite{L1,L2}, and use them in particular…

Suppose $\mathfrak{g}=\mathfrak{g}_{\bar 0}+\mathfrak{g}_{\bar 1} is a Lie superalgebra of queer type or periplectic type over an algebraically closed field $\textbf{k}$ of characteristic $p>2$. In this article, we initiate preliminarily to…

This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type $\mathrm{A}^{(1)}_\ell$). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also…

Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring $R$, $\,R$-modules built from the rings of functions on principal affine open subschemes in…

We address the problem of defining Schubert classes independently of a reduced word in equivariant elliptic cohomology, based on the Kazhdan-Lusztig basis of a corresponding Hecke algebra. We study some basic properties of these classes,…

Let $U$ and $A$ be algebras over a field $k$. We study algebra structures $H$ on the underlying tensor product $U{\otimes}A$ of vector spaces which satisfy $(u{\otimes}a)(u'{\otimes}a') = uu'{\otimes}aa'$ if $a = 1$ or $u' = 1$. For a pair…

Let $K$ be a $C_1$-field of any characteristic and $X$ a projective variety over $K$. In this article we prove that for a finite Galois extension $L$ of $K$, a simple sheaf with covering datum on $X \times_K L$ descends to a simple sheaf on…

The main goal of this paper is to establish close relations among sheaves of modules on atomic sites, representations of categories, and discrete representations of topological groups. We characterize sheaves of modules on atomic sites as…

We prove the decomposition conjecture of Leclerc and Thibon for the Schur algebra. We also give a new approach to the Lusztig conjecture for the dimension of the simple U(sl_k)-modules at roots of unity via canonical bases of the Hall…