Geometry & Representation Theory (May 10-14)
Title: "Applications of the Deconcini-Procesi Compactification"
Speaker: Sam Evens (University of Notre Dame)
Abstract: In these talks, I will introduce the DeConcini-Procesi compactification of a semisimple algebraic group and explain some of its applications to geometry, Schubert calculus, and representation theory.
Title: "Combinatorics, unitary representations, and algebraic geometry related to Hermitian symmetric spaces"
Speaker: Markus Hunziker (Baylor University)
Abstract: Hermitian symmetric spaces have played a distinguished role in the history of Lie theory and they continue to be a source of beautiful and often surprising results. In this series of lectures, a natural generalization of Young diagrams for Hermitian symmetric spaces is used to give a concrete and uniform approach to a wide variety of interconnected topics including noncompact roots and canonical reduced expressions, abelian ideals in a Borel subalgebra, combinatorial invariance of parabolic Kazhdan-Lusztig polynomials, rational smoothness of Schubert varieties in cominuscule flag varieties and affine Grassmannians, equivalences of categories of highest weight modules, BGG resolutions of unitary highest weight representations, and syzygies of determinantal ideals.
Title: "Infinite Grassmannians, Moduli Spaces of G-Bundles and Verlinde Formula"
Speaker: Shrawan Kumar (University of North Carolina)
Abstract: Let C be a smooth projective irreducible algebraic curve over C of any genus and G a connected simply-connected simple affine algebraic group over C. In this series of talks, we elucidate the relationship between (1) the space of vacua (“conformal blocks”) defined in Conformal Field Theory, using an integrable highest weight representation of the affine Kac-Moody algebra associated to G and (2) the space of regular sections (“generalized theta functions”) of a line bundle on the moduli spaceMof semistable principal G-bundles on C. We will use this correspondence and the Decomposition Theorem for the conformal blocks due to Tsuchiya-Ueno-Yamada to give a sketch of the proof of the celebrated Verlinde formula. Some familiarity with the affine Kac-Moody Lie algebras and the corresponding groups will be helpful, though not necessary since I will recall the relevant results during my lectures. Here is some background material on affine Lie algebras and groups that will be useful for Shrawan Kumar's lectures.
Title: "Geometry of representations of GL(n)"
Speaker: Peter Trapa (University of Utah)
Abstract: Let G_R be a real form of a connected complex reductive algebraic group G with Lie algebra \g. These lectures will introduce the geometric theory of three categories of representations: category O for \g, the category of Harish-Chandra modules for G_R, and the category of finite-dimensional representations of the graded affine Hecke algebra attached to \g. (The relevant geometry in each case may be interpreted as arising from spaces of Langlands parameters.) When G=GL(n), the three geometries are naturally related. This leads to functorial relationships between the categories themselves.
Title: "The Geometry of Springer Fibers"
Speaker: Roger Zierau (Oklahoma State University)
Abstract: Springer's resolution of singularities of the nilpotent cone plays a role in a number of topics in representation theory. The lectures will give a brief description of this resolution of singularities (and its fibers) and will hint at the role it plays in the theory of Harish-Chandra modules and their invariants. Then we will show how to give a geometric description of some components of the fibers and use this description to compute multiplicities in the associated cycles of some representations.
Homological Methods in Representation Theory (May 17-21)
Title: "Representation theory of symmetric groups and related Hecke algebras"
Speaker: Alexander Kleshchev (University of Oregon)
Abstract: We will explain some fundamental results in representation theory of symmetric groups and related objects which were mainly obtained in the last fifteen years. The emphasis will be on connections with Lie theory via categorification. We will present results on branching rules and crystal graphs, decomposition numbers and canonical bases, graded representation theory, connections with cyclotomic and affine Hecke algebras, Khovanov-Lauda-Rouquier algebras. (Time permitting (which is unlikely!) we might also touch upon connections to the category ${\mathcal O}$, $W$-algebras,\dots)
Title: "Working with the Lusztig character formula"
Speaker: Brian Parshall (University of Virginia)
Abstract: These talks center on the modular representation theory of reductive groups, an active area of contemporary research. The Lusztig character formula (LCF) predicts the characters of the irreducible rational representations of a semisimple group G in positive characteristic p. It is conjectured to be true if p > h (the Coxeter number of G), and it is known to hold if p large enough, or in some specific smaller rank cases if p > h. A very large
lower bound on p sufficient for its validity is provided for each root system by recent work of Peter Fiebig. There is a similar version for quantum enveloping algebras at lth roots of unity (which is known to hold with mild restrictions on l).
We will discuss the combinatorial and homological underpinnings of the LCF, aiming to see how it can be used to explicitly calculate Ext-groups, support varieties, understand the structure of important finite dimensional algebras attached to G, etc. We will begin with background material on cohomology of algebraic groups and Kazhdan-Lusztig polynomials. The lectures will be coordinated with the afternoon AIM-style workshops on related issues and open problems.
Title: "\Pi Points and Applications to Cohomology and Modular Representation Theory"
Speaker: Julia Pevtsova (University of Washington)
Abstract: We shall describe the notion of a $\pi$-point (a generalization of a cyclic shifted subgroup for a finite group scheme, and
develop the theory of support varieties from both cohomological and representation-theoretic prospective. In particular, we shall cover the Quillen stratification and generalizations, detection of nilpotents in cohomology and criteria for projectivity of modules. We shall then move to more recent developments and discuss non-maximal support varieties, modules of constant Jordan type and connections with vector bundles on
projective varieties.
View program schedule here
