spring 2003
This semester we meet on Fridays in 406 Malott Hall. Starting 28 March, our meeting time will be 1:30-2:30.
may
2: no seminar because of Topology Festival.
april
25: Reyer Sjamaar (Cornell), Kaehler potentials on orbits of Borel subgroups
Consider a torus acting on a compact Kaehler manifold. Atiyah has described the momentum map image of orbits of the complexified torus. This talk concerns a nonabelian generalization, where we examine orbits of the Borel subgroup of a complex reductive group. (Work in progress, joint with Victor Guillemin.)
18: Vladimir Retakh (Rutgers), Quasideterminants and their applications
I will explain how to use noncommutative quasideterminants for understanding the structure of universal enveloping algebras, quantum algebras, Capelli identities, Yangians, etc.
11: Yi Lin (Cornell), Symplectic Hodge theory and the equivariant d\delta-lemma
J. Brylinski introduced the notion of symplectic harmonic forms. Further he conjectured that on a compact symplectic manifold every cohomology class contains a harmonic representative. Olivier Mathieu discovered for a symplectic manifold Brylinski conjecture is true if and only if it satisfies the strong Lefschetz property. Later on Victor Guillemin sharpened this result by establishing the symplectic d\delta-lemma for any compact symplectic manifold with strong Lefschetz property.
In this talk we will discuss an equivariant version of the symplecitc d\delta-lemma. We will also explain how our methods can be used to simplify Guillemin's original proof.
4: two talks by David Vogan (MIT), both in 406 Malott
1:30 pm: 431 Reasons to love unitary representations, Oliver Club talk.
4:00 pm: Cutting and pasting unitary representations
The ideas in the Oliver Club talk yesterday allow one to impose "reality" constraints on possible unitary representations of a reductive group G: certain formally self-adjoint elements of the center of the universal enveloping algebra must have real eigenvalues. When one tries to read these arguments backwards and deduce unitarity from reality conditions, it turns out that there are essential difficulties. The reality conditions follow not only from the existence of a Hilbert space structure, but from the existence of any G-invariant Hermitian form. Knapp made this precise twenty years ago, giving a simple and complete classification of irreducible representations that preserve a (possibly indefinite) Hermitian form. Classifying unitary representations amounts to understanding when a G-invariant Hermitian form must be definite. On a one-dimensional vector space, any Hermitian form is definite, because there is no room for a signature that has both a positive and a negative part. It follows that any one-dimensional Hermitian representation is unitary. I'll explain a way to apply the same idea to infinite-dimensional representations, by understanding them asglued together from indivisible pieces.
march
28 (new time: 1:30-2:30): Patrick Iglésias (Université de Provence and Cornell), Extension of the moment map to spaces which are not manifolds, and coadjoint orbits.
21: no talk because of spring break
14: no talk because of spring break
7: Karl-Hermann Neeb (Darmstadt), Period maps in infinite-dimensional Lie theory
february
28: no talk
21: Patrick Iglésias (Université de Provence and Cornell), SL(2,R)-invariant dynamical systems on the Poincaré disc: an illustration of the moment map
14: Indira Chatterji (Cornell), On the exceptional group SL_3(O)
The automorphism group of the projective plane over the octonions is an exceptional Lie group of real rank 2, mostly referred to as E_{6(-26)}. I will describe Freudental's construction of this group, which makes it look as if it was 3 by 3 matrices with coefficients in the octonions, with determinant one.
7: Julianna Tymoczko (Princeton), An Introduction to Hessenberg varieties
Hessenberg varieties form a family of subvarieties of the flag variety with important relations to representation theory, numerical analysis, quantum cohomology, and other areas. Significant examples include the Springer fiber (whose cohomology carries information about representations of the Weyl group) and the Peterson variety (which can be stratified so that the open stratum's coordinate ring gives the quantum cohomology of the flag variety).
I will discuss the geometric structure of Hessenberg varieties and will describe how several major classes of Hessenberg varieties can be paved by affines using properties of associated Lie algebras. I will show how these affines are indexed by filled Young tableaux and how their dimensions can be computed by simple combinatorial rules.
january
31: Milen Yakimov (Cornell), A Kostant type finiteness for the Kazhdan-Lusztig tensor product
Affine Kac-Moody algebras found numerous applications in topology, combinatorics, and mathematical physics. The study of certain categories of Harish-Chandra modules for them, defined by Lian and Zuckerman, is an important problem which directly relates the theory of representations of real reductive groups to the above listed areas of mathematics. Although the usual tensor product of modules for these algebras exists, it is another one, the "fusion" product of Kazhdan and Lusztig that plays a more important role but is a lot harder to study. We will describe a finiteness result for it which in a very special case implies the original result of Kazhdan and Lusztig. It is a generalization of Kostant's theorem that for any subalgebra k of a complex simple Lie algebra g which is reductive in g, the category of finite length, admissible (g,k)-modules is stable under tensoring with finite dimensional g modules. If time permits, we will explain some applications of further advanced representation theoretic techniques (Jacquet functors) for constructing block decompositions of the affine Harish-Chandra categories and studying Zuckerman-Jantzen type affine translation functors.
24: Alexei Oblomkov (MIT), Symmetric spaces and orthogonal polynomials in the classical and quantum cases
From classical works on harmonic analysis we know that K-invariant functions on G/K (coming from K-spherical representations L_\lambda of G) restricted to the maximal torus T give us a basis {P_\lambda} in the space of Weyl group invariant polynomials on T. The polynomials P_\lambda are proportional to the Jacobi polynomials with some special values of the parameters (which are essentially linear expressions in the ranks of K and G).
We will explain how one can generalize this construction to get a three-parameter family of Jacobi polynomials for the root system BC_n. In second part of the talk we will give the quantum version of the results from the first part of the talk. The quantum version of the construction yields a five-parameter (i.e. the most general) family of Macdonald-Koornwinder polynomials. The second part of the talk is based on joint work with Jasper Stokman.
fall 2002
november
22: Farkhod Eshmatov (Cornell) Regular algebras and noncommutative spaces
In the late 80s Artin, Tate, Shelter, and Van den Berg introduced the notion of regular algebras. They noticed that the special quotients of these algebras are isomorphic to 'quantum polynomial rings' (the quantum polynomial rings are noncommutative rings with the same projective scheme as polynomial rings). This allows us to study regular algebras by means of classial algebraic geometry. In this expository talk I will explain the Classification Theorem for regular algebras. If time permits I will try to explain how these algebras give rise to 'noncommutative projective schemes'.
15: Joshua Lansky (Bucknell University), K-types and Base Change for U(3)
It is possible to make base change fairly explicit in certain cases given some knowledge of characters and of Bruhat-Tits theory. We will look at an example involving base change from U(3) to GL(3). This is joint work in progress with Jeff Adler.
8: Dan Ciubotaru (Cornell), Unitary spherical dual for p-adic groups
For G a split reductive p-adic group, D. Barbasch and A. Moy showed that the question of determining the unitary representations of G with Iwahori fixed vectors can be translated into an algebraic problem concerning the associated Iwahori-Hecke algebra. Furthermore, in this algebraic setting, they classified the spherical unitary dual of the classical groups. I hope to present the main steps of this reduction and exemplify how the classification works.
1: Jonathan Weitsman (UCSC), Lattice points in convex polytopes, signature operators, and multidimensional Euler-MacLaurin formulas (joint with Yael Karshon and Shlomo Sternberg)
Special time and venue: 10 am in 224 Malott.
october
25: Oleg Chalykh, Cherednik algebras and differential operators on singular algebraic varieties II
18: Oleg Chalykh (Loughborough, visiting Cornell), Cherednik algebras and differential operators on singular algebraic varieties I
Cherednik algebras were introduced by Etingof and Ginzburg as an important subclass in the family of symplectic reflection algebras. From an algebraic point of view, they are universal deformations of the cross-product of a Coxeter group with the Weyl algebra. In the first part I will talk about their representation theory due to Opdam-Rouquier and Berest-Etingof-Ginzburg, which has several amusing (dis)similarities with the much-studied case of the universal enveloping algebras. The second talk will be devoted to an important application of Cherednik algebras: constructing examples of singular affine algebraic varieties on which the ring of differential operators has a particularly nice structure. This is based on the results of Berest-Etingof-Ginzburg and the joint results of Yuri Berest and the speaker.
11: no seminar because of Fall break
4: Victor Kac (MIT), Quantum reduction
september
27: Bent Ørsted (Odense, visiting Cornell), The Gromov norm of the Kaehler class and the Maslov index
The Maslov index is an integer-valued invariant associated with triples of Lagrangian subspaces in a symplectic vector space. It appears for example in the metaplectic representation, and in the study of asymptotics of oscillatory integrals. In this talk we give a geometric interpretation and generalization of the Maslov index in terms of an integral of the Kaehler form of a Hermitian symmetric space over geodesic triangles; this may also be thought of as a bounded cohomology class, and we find its Gromov norm on compact Hermitian locally symmetric spaces. This is a report on joint work with J.-L. Clerc.
20: cancelled by speaker
13: Nolan Wallach (UCSD), Holomorphic continuation of Jacquet integrals
In this lecture are discussed some generalizations of the speaker's earlier work on holomorphic continuation of Jacquet integrals that are general enough to apply to the work that he did on quaternionic representations. In this generalization one must replace standard Bruhat theory with an extension due to Kolk and Varadarajan.