Title : Cohomology of Cotangent Bundles of Flag Varieties and the BK-Filtration
Abstract: Let G be a complex algebraic group and let P be a parabolic subgroup of G. Let T*(G/P) denote the cotangent bundle of the flag variety G/P. In this talk I will describe results connecting cohomology of bundles on T*(G/P) to purely combinatorial objects such as filtrations on irreducible G-modules and generalizations of Lusztig's q-analog of weight multiplicity.
Title: Positivity for cluster algebras associated to surfaces
Abstract: An important class of cluster algebras is defined using triangulations of 2-dimensional surfaces with boundary. The simplest case are cluster algebras of Dynkin type A which correspond to triangulated polygons, but any surface with boundary gives rise to cluster algebra.
In this talk, I will explain recent progress on cluster expansion formulas and the positivity conjecture using certain paths on the triangulations on the surface.
Title: Highest weight theory for finite W-algebras
Abstract : To each nilpotent orbit O in a complex semisimple Lie algebra, one can associate a finite W-algebra. This algebra can be viewed as the enveloping algebra of the Slodowy slice through O, and has many connections to other areas of Lie theory. In this talk we outline an approach to highest weight representation theory of finite W-algebras from recent work of Brundan, G. and Kleshchev. We explain how this leads to a strategy for classifying the finite dimensional simple modules for finite W-algebras.
Title: Graph Theoretic Expansion Formula for Cluster Algebras of Classical Type
Title: Ad-nilpotent Ideals of Complex Lie algebras.
Abstract:
In the first part of the talk, we will talk about the relations between
ad-nilpotent ideals and affine Weyl groups. We will define an equivalence
relation for ad-nilpotent ideal based on its normalizer and generators, and
study its relation with Lusztig's star operator for type A.
Next, we will apply Jacobson-Morozov theorem and prove Sommers' conjecture
about the lower bounds of ad-nilpotent ideals with the same associated
orbit.
More precisely, we will construct some ideals with minimal dimension for
classical groups.
If time allows, I will briefly introduce the analogous notion for the real
groups and state some similar results.