Geometric Representation Theory

Tucson, AZ, March 24-27, 2005

Tucson, AZ, March 24-27, 2005

Main Conference Page

David Vogan, MIT

Signatures of invariant forms on infinite-dimensional representations

The most fundamental invariant of a finite-dimensional representation, ß, is its dimension. One natural generalization of this invariant is the character of the representation, which is the function sending a group element g to the trace of the operator ß(g). If ß preserves an invariant Hermitian form, then a second natural refinement of the dimension is the signature of the invariant form. For infinite-dimensional representations the operators ß(g) are rarely of trace class, but there are still (sometimes) generalizations both of the character and of the signature available. In the case of a real reductive group G, the Kazhdan-Lusztig conjecture (which has been proved) provides an algorithm for computing the character of any irreducible representation. I will describe a variant of the Kazhdan-Lusztig conjecture which provides (almost) an algorithm to compute the signature of an invariant Hermitian form on any irreducible representation of G. Such an algorithm in particular decides whether the representation is unitary. The "almost" part is that certain signs in the algorithm are not known, so what the algorithm actually provides is a finite collection of possible formulas for the signature. On the other hand, signatures are dimensions, and are therefore non-negative; so one can discard possible formulas that lead to negative numbers. I will explain this algorithm a little more carefully, and examine some examples of the ambiguity in it.

Dennis Gaitsgory, Chicago

Representations of affine Kac-Moody algebras at the critical level

The purpose of the talk is to present a conjecture that links representations of the affine algebra at the critical level with D-modules on the affine Grassmannian. This conjecture can

be thought of as an affine analogue of the well-known theorem of Beilinson-Bernstein that relates the category of representations of a semi-simple Lie algebra with a given central character to the category of twisted D-modules on the corresponding flag manifold.

Shrawan Kumar, North Carolina

Eigenvalue Problem and a New Product in the Cohomology of Flag Varieties

Click here for a PDF file with the abstract.

Hongyu He, Georgia State

Invariant Tensor Product and Unipotent Representations

Invariant Tensor Products are widely used and studied in representation theory. In this talk, I will introduce (algebraic) invariant tensor product and analytic invariant tensor product. I will then use invariant tensor product to produce certain induction functors. Finally, I will discuss how one can attach representations to nilpotent orbits (parametrized by certain signed Young diagrams). In particular, one obtains a set of unipotent representations attached to special rigid orbits. If time allows, I will state a general theorem concerning unitarity. I will also formulate serveral conjectures.

Matvei Libine, Amherst

Schmid-Vilonen Character Formulas

Recently W. Schmid and K. Vilonen proved two character formulas for certain representations of real reductive Lie groups G_{R}
-- the fixed
point character formula and the integral character formula. In the case
when G_{R} is compact, the former
reduces to the Weyl character formula
and the latter -- to Kirillov's character formula. These
representations were constructed by M. Kashiwara and W. Schmid. They
generalize the Borel-Weil-Bott construction, but instead of line
bundles on the flag variety they consider G_{R}-equivariant
constructible
sheaves F and, for each
integer p, they define
representations of G_{R} in Ext^{p}(F,O), where O is the sheaf
of functions on the flag
variety. In this talk I will explain these two character formulas and
outline my geometric proof of equivalence of these two formulas. The
corresponding problem for compact groups was solved by N. Berline and
M. Vergne using their famous integral localization formula for
equivariant cohomology. This new geometric argument leads to a
generalization of their localization formula to non-compact group
actions. Click
here for an extended verion of the abstract.

Xuhua He, MIT

The G-stable pieces of the wonderful group compactification

Let G be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification \bar{G} of G into finite many G-stable pieces, which were introduced by Lusztig. This talk consists of two parts. In the first part, we will investigate the closure of any G-stable piece in \bar{G}. We will show that the closure of each piece is a disjoint union of some other pieces. The closure relation can be described in terms of some relation on the Weyl group. The second part concerns the closure of unipotent variety in \bar{G}. It turns out the boundary of the closure is a union of some G-stable pieces and the irreducible components are related to certain Coxeter elements. I will explain how the Coxeter elements come into the picture. Based on the these results, some interesting questions about "parabolic character sheaves" will be formulated.

Samuel Evens, Notre Dame

Poisson geometry and Lie theory

I will discuss joint work with J-H Lu providing Poisson structures on homogeneous spaces and compactifications. I will survey some applications of this work to areas such as n-cohomology and Schubert varieties, harmonic analysis, and the Belavin-Drinfeld classification, and discuss some conjectural applications.

Ivan Mirkovic, Amherst

Critical quantization

This refers to a dream of treating in a parallel way various situations where in a family of algebras, the center jumps at a critical parameter. Examples include represntations of Lie algebras of positive characteristic (with Bezrukavnikov and Rumynin), and quantum groups at roots of unity (Backelin-Kremnizer).

Hadi Salmasian, Yale

Small Unitary Representations and Kirillov's Method of Coadjoint Orbits

We will introduce a notion of rank for unitary representations of semisimple groups based on Kirillov's method of coadjoint orbits for nilpotent groups. The definition applies Kostant's cascade construction and therefore is applicable to both classical and exceptional groups. The theory has consequences such as new bounds for the decay of matrix coefficients of non-minimal representations of exceptional groups and the isolatedness of the minimal representation in the unitary dual.

So Okada, Amherst

Stability Manifold of P^{1}

The notion of the stability manifold of a triangulated category was defined by T. Bridgeland as a moduli of refined t-structures. We show that the stability manifold of the bounded derived category of the coherent sheaves on P^{1} is C^{2}. This is the first
complete picture of a stability manifold for coherent sheaves on a
variety which is not Calabi-Yau.

Gizem Karaali, Santa Barbara

Super Solutions of the Dynamical Yang-Baxter Equation

We develop the super analog of the theory of dynamical r-matrices. We first discuss certain results which generalize the non-graded case and then concentrate on some examples which are peculiar to the super case.

Pramod Achar, Louisiana State

Hecke Algebras and Complex Reflection Groups

Complex reflection groups are in a natural way generalizations of Weyl groups and finite Coxeter groups. Recent work by a number of mathematicians has shown that the group algebras of these groups admit deformations, called "cyclotomic Hecke algebras," that behave very much like ordinary Hecke algebras. In particular, they give rise to such representation-theoretic objects as generic degrees, families of representations, and even Green functions, even though there is no underlying algebraic group. However, a uniform explanation for these phenomena has been lacking. I will discuss recent progress on this problem in rank-two case, as well as hints of a more general solution. This is joint work with A.-M. Aubert.

Kari Vilonen, Northwestern

Representations of Lie groups and Hodge theory

I will discuss a research program, joint with Wilfried Schmid, whose goal is to understand representation theory of reductive Lie groups using Hodge theoretic methods.

Back

Abstracts of Talks

David Vogan, MIT

Signatures of invariant forms on infinite-dimensional representations

The most fundamental invariant of a finite-dimensional representation, ß, is its dimension. One natural generalization of this invariant is the character of the representation, which is the function sending a group element g to the trace of the operator ß(g). If ß preserves an invariant Hermitian form, then a second natural refinement of the dimension is the signature of the invariant form. For infinite-dimensional representations the operators ß(g) are rarely of trace class, but there are still (sometimes) generalizations both of the character and of the signature available. In the case of a real reductive group G, the Kazhdan-Lusztig conjecture (which has been proved) provides an algorithm for computing the character of any irreducible representation. I will describe a variant of the Kazhdan-Lusztig conjecture which provides (almost) an algorithm to compute the signature of an invariant Hermitian form on any irreducible representation of G. Such an algorithm in particular decides whether the representation is unitary. The "almost" part is that certain signs in the algorithm are not known, so what the algorithm actually provides is a finite collection of possible formulas for the signature. On the other hand, signatures are dimensions, and are therefore non-negative; so one can discard possible formulas that lead to negative numbers. I will explain this algorithm a little more carefully, and examine some examples of the ambiguity in it.

Dennis Gaitsgory, Chicago

Representations of affine Kac-Moody algebras at the critical level

The purpose of the talk is to present a conjecture that links representations of the affine algebra at the critical level with D-modules on the affine Grassmannian. This conjecture can

be thought of as an affine analogue of the well-known theorem of Beilinson-Bernstein that relates the category of representations of a semi-simple Lie algebra with a given central character to the category of twisted D-modules on the corresponding flag manifold.

Shrawan Kumar, North Carolina

Eigenvalue Problem and a New Product in the Cohomology of Flag Varieties

Click here for a PDF file with the abstract.

Hongyu He, Georgia State

Invariant Tensor Product and Unipotent Representations

Invariant Tensor Products are widely used and studied in representation theory. In this talk, I will introduce (algebraic) invariant tensor product and analytic invariant tensor product. I will then use invariant tensor product to produce certain induction functors. Finally, I will discuss how one can attach representations to nilpotent orbits (parametrized by certain signed Young diagrams). In particular, one obtains a set of unipotent representations attached to special rigid orbits. If time allows, I will state a general theorem concerning unitarity. I will also formulate serveral conjectures.

Matvei Libine, Amherst

Schmid-Vilonen Character Formulas

Recently W. Schmid and K. Vilonen proved two character formulas for certain representations of real reductive Lie groups G

Xuhua He, MIT

The G-stable pieces of the wonderful group compactification

Let G be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification \bar{G} of G into finite many G-stable pieces, which were introduced by Lusztig. This talk consists of two parts. In the first part, we will investigate the closure of any G-stable piece in \bar{G}. We will show that the closure of each piece is a disjoint union of some other pieces. The closure relation can be described in terms of some relation on the Weyl group. The second part concerns the closure of unipotent variety in \bar{G}. It turns out the boundary of the closure is a union of some G-stable pieces and the irreducible components are related to certain Coxeter elements. I will explain how the Coxeter elements come into the picture. Based on the these results, some interesting questions about "parabolic character sheaves" will be formulated.

Samuel Evens, Notre Dame

Poisson geometry and Lie theory

I will discuss joint work with J-H Lu providing Poisson structures on homogeneous spaces and compactifications. I will survey some applications of this work to areas such as n-cohomology and Schubert varieties, harmonic analysis, and the Belavin-Drinfeld classification, and discuss some conjectural applications.

Ivan Mirkovic, Amherst

Critical quantization

This refers to a dream of treating in a parallel way various situations where in a family of algebras, the center jumps at a critical parameter. Examples include represntations of Lie algebras of positive characteristic (with Bezrukavnikov and Rumynin), and quantum groups at roots of unity (Backelin-Kremnizer).

Hadi Salmasian, Yale

Small Unitary Representations and Kirillov's Method of Coadjoint Orbits

We will introduce a notion of rank for unitary representations of semisimple groups based on Kirillov's method of coadjoint orbits for nilpotent groups. The definition applies Kostant's cascade construction and therefore is applicable to both classical and exceptional groups. The theory has consequences such as new bounds for the decay of matrix coefficients of non-minimal representations of exceptional groups and the isolatedness of the minimal representation in the unitary dual.

So Okada, Amherst

Stability Manifold of P

The notion of the stability manifold of a triangulated category was defined by T. Bridgeland as a moduli of refined t-structures. We show that the stability manifold of the bounded derived category of the coherent sheaves on P

Gizem Karaali, Santa Barbara

Super Solutions of the Dynamical Yang-Baxter Equation

We develop the super analog of the theory of dynamical r-matrices. We first discuss certain results which generalize the non-graded case and then concentrate on some examples which are peculiar to the super case.

Pramod Achar, Louisiana State

Hecke Algebras and Complex Reflection Groups

Complex reflection groups are in a natural way generalizations of Weyl groups and finite Coxeter groups. Recent work by a number of mathematicians has shown that the group algebras of these groups admit deformations, called "cyclotomic Hecke algebras," that behave very much like ordinary Hecke algebras. In particular, they give rise to such representation-theoretic objects as generic degrees, families of representations, and even Green functions, even though there is no underlying algebraic group. However, a uniform explanation for these phenomena has been lacking. I will discuss recent progress on this problem in rank-two case, as well as hints of a more general solution. This is joint work with A.-M. Aubert.

Kari Vilonen, Northwestern

Representations of Lie groups and Hodge theory

I will discuss a research program, joint with Wilfried Schmid, whose goal is to understand representation theory of reductive Lie groups using Hodge theoretic methods.

Back