Let G be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification Ḡ of G into finite many G-stable pieces, which was introduced by Lusztig. In this paper, we will investigate the closure of any G-stable piece in Ḡ. We will show that the closure is a disjoint union of some G-stable pieces, which was first conjectured by Luszti...

Bounded local G-shtuka are function field analogs for p-divisible groups with extra structure. We describe their deformations and moduli spaces. The latter are analogous to Rapoport-Zink spaces for p-divisible groups. The underlying schemes of these moduli spaces are affine DeligneLusztig varieties. For basic Newton polygons the closed Newton stratum in the universal deformation of a local G-sh...

In the present article, a basis of the coordinate algebra of the multi-parameter quantized matrix is constructed by using an elementary method due to Lusztig. The construction depends heavily on an antiautomorphism, the bar action. The exponential nature of the bar action is derived which provides an inductive way to compute the basis elements. By embedding the basis into the dual basis of Lusz...

We obtain a nonrecursive combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one, and which cannot be linearly simplified. Our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions. Résumé. On montre une formule combinatoire pour les polynômes de Kazhdan-Lusztig q...

For a representation of a matroid the combinatorially defined Kazhdan-Lusztig polynomial computes the intersection cohomology of the associated reciprocal plane. However, these polynomials are difficult to compute and there are numerous open conjectures about their structure. For example, it is unknown whether or not the coefficients are non-negative for non-representable matroids. The main res...

Using the Beilinson-Lusztig-MacPherson construction of the quantized enveloping algebra of gln and its associated monomial basis, we investigate q-Schur algebras Sq(n, r) as “little quantum groups”. We give a presentation for Sq(n, r) and obtain a new basis for the integral q-Schur algebra Sq(n, r), which consists of certain monomials in the original generators. Finally, when n > r, we interpre...

A homeomorphism f : X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x 6= y, then there is an integer n ∈ Z such that d(f(x), f(y)) > c. In this paper, we prove that if a homeomorphism f : X → X of a continuum X can be lifted to an onto map h : P → P of the pseudoarc P , then f is not expansive. As a corollary, we prove that there are no expansive ho...

Taylor, Dewey Terese. Fine Bruhat Intersections for Reductive Monoids. (Under the direction of Mohan S. Putcha.) Fine Bruhat intersections for reductive groups have been studied by several authors in connection with Kazhdan-Lusztig theory, canonical bases and Lie Theory. The purpose of this thesis is to study the analogous intersections for reductive monoids. We determine the conditions under w...

We study some conjectures on the endomorphism algebras of the cohomology of Deligne-Lusztig varieties which are a refinement of those of [BMi].