spring 2004
This semester we meet on Fridays from 3:30-4:30 in 406 Malott Hall.
april
30: Oleg Chalykh (Cornell), Ideals, strongly homotopy modules, and Calogero-Moser spaces.
This is a continuation of the talk Yuri gave few weeks ago. I will provide more details on the definition and construction of A-infinity modules which serve as minimal models representing the ideals in the Weyl algebra. This gives a rather natural way to quantise the Hilbert scheme of points in C^2, leading to the Calogero-Moser spaces. Other examples will be mentioned.
23: Sam Evens (Notre Dame), Poisson structures on homogeneous spaces and compactifications
I will explain some aspects of joint work with Jiang-Hua Lu relating Poisson geometry and Lie theory. In particular, I will exhibit a Poisson structure with an open leaf on any nilpotent orbit, a Poisson structure on the DeConcini-Procesi compactification, and explain some results regarding symplectic leaves.
16: no talk
9: Eckhard Meinrenken (Toronto), The small Cartan model
This is joint work with Anton Alekseev. Let G be a compact Lie group acting smoothly on a manifold M. A classical result of Cartan states that the G-equivariant cohomology of M, with real coefficients, may be computed as the cohomology of a complex of "equivariant differential forms" on M. A few years ago, Goresky-Kottwitz-MacPherson described a "small" Cartan model for the equivariant cohomology, involving only invariant differential forms. We construct an explicit chain map from the small Cartan model into the usual Cartan model, inducing an isomorphism in cohomology. As a byproduct, we obtain a refinement of Chevalley's transgression theorem.
2: no talk
march
26: spring break
19: Werner Müller (Bonn), Weyl's law for the cuspidal spectrum of SL_n
One of the main problems in the theory of automorphic forms is the question of construction and existence of cusp forms for a given reductive group. In this talk we address the problem of existence for the group SL_n(R). We prove that for congruence subgroups of SL_n(Z) the counting function of the cuspidal spectrum satisfies Weyl's law. This settles for SL_n a conjecture of Sarnak.
12: Tara Holm (Berkeley), Equivariant cohomology of the based loop group
We describe how to exted the techniques of Goresky, Kottwitz and MacPherson to the (possibly infinite dimensional) setting of equivariant CW complexes with only even dimensional cells. We show how to compute equivariant cohomology of these spaces in a purely combinatorial way. These results give insight to many (symplectic!) examples, including certain coadjoint orbits of the loop group LG. This is joint work with M. Harada and A. Henriques.
5: Stephen Bullock (NIST), Cartan Involutions of SU(2^n) and Entanglement Dynamics
We describe a Cartan involution related to quantum computing. Associated decompositions of SU(2^n) are useful in studying the changes in entanglement of quantum data caused by applying a unitary evolution, e.g. a quantum computation.
27: Victor Protsak (Cornell), A concrete approach to primitive ideals in enveloping algebras
Primitive ideals in the universal enveloping algebra of a complex reductive Lie algebra have been completely classified. Nevertheless, the ideals themselves and the corresponding primitive quotients remain somewhat mysterious. I propose a new scheme for constructing primitive ideals by exhibiting their natural systems of generators. The construction is closely related to the theory of reductive dual pairs. By this method one obtains a family of completely prime, primitive ideals quantizing small nilpotent orbits in a classical Lie algebra.
20: Erez Lapid (Hebrew University), Classification of generic unitary representations of classical groups over local fields
13 (joint with Oliver Club): Mikhail Kogan (IAS), Mirkovic-Vilonen cycles and cluster algebras
Lusztig's construction of canonical bases in quantum groups led to many important developments in the representation theory of semi-simple Lie groups. In particular, it was applied by Berenstein-Zelevinsky to produce explicit combinatorial expressions for tensor product multiplicities, often called Littlewood-Richardson coefficients. Recently, other approaches to constructing canonical bases have been introduced. In this talk we will discuss two of them. Mirkovic and Vilonen discovered cycles inside loop Grassmannians which provide a geometric construction of canonical bases of irreducible representations of reductive Lie groups. Independently, in an attempt to provide an algebro-combinatorial framework for the study of bases dual to Lusztig's canonical bases, Fomin and Zelevinsky developed the theory of cluster algebras. In this talk, after an elementary introductions to both subjects, we discuss geometric properties of Mirkovic-Vilonen cycles and their convolutions with a view toward establishing a connection to cluster algebras. In particular, our results in type A include a parametrization of Mirkovic-Vilonen cycles by Kostant partition functions, an explicit description of these cycles in terms of the lattice model, a computation of the moment map images of these cycles, as well as a lattice description of Drinfeld's bundle used to define convolution. This is joint work with Jared Anderson.
6: Yuri Berest (Cornell), A_{\infty}-Modules and Noncommutative Hilbert Schemes
january
30: Jeff Adams (University of Maryland), Non-linear covers of real groups
Non-linear groups, such as the metaplectic two-fold cover of Sp(2n), play an important role in number theory. In this talk I will give simple proofs of some facts about non-linear groups which were known previously from the classification of real groups. In particular I will give a necessary and sufficient condition for a real group to have such a cover (they are surprisingly common).
This semester we meet on Fridays from 3:30-4:30 in 406 Malott Hall.
december
11: Siddharta Sahi (Rutgers University), Triple groups and double Hecke algebras
Note unusual time and location: Thursday 11 December at 3 pm in 206 Malott Hall.
We define a new class of objects which we call triple groups and relate them with Cherednik's double affine Hecke algebras. As an immediate consequences we obtain new descriptions of double affine Weyl and Artin groups, double affine Hecke algebras, and also the corresponding elliptic objects. From the new descriptions we recover results of Cherednik on automorphisms of double affine Hecke algebras. This is joint work with Bogdan Ion at Michigan.
november
21: Nicolas Guay (University of Chicago), Category O for rational Cherednik algebras
Rational Cherednik algebras have appeared recently as examples of symplectic reflection algebras and as degenerate forms of double affine Hecke algebras. For a Cherednik algebra A associated to a Weyl group W, one can define the notion of the category O of modules over A. I will explain how it is similar to the category O for a semisimple Lie algebra. It is possible to define a functor, denoted KZ, from O to the category of modules over a certain finite Hecke algebra H. I will explain the double-centralizer property of KZ and how this functor allows us to view the category of finitely generated modules over H as a quotient of O. Finally, I will mention the connection between cell modules for H and standard modules in O.
14: no talk
7: Thomas Nevins (University of Michigan), Noncommutative deformations and Hilbert schemes of points
Recent work by several people has revealed a fascinating role for noncommutative algebra in the study of Hilbert schemes of points. I will review some aspects of this story and describe joint work with J. T. Stafford that constructs deformations of Hilbert schemes from a large class of noncommutative algebras. In particular, a choice of a smooth plane cubic curve E gives a family of symplectic deformations of the Hilbert scheme of points on P^2 - E.
october
31: Bertram Kostant (MIT), Geometric Quantization and the Symplectic Emergence of Exceptional Lie groups
This is joint work with Ranee Brylinski. Let $(X,\omega)$ be a symplectic manifold so that the space of smooth functions $C^{\infty}(X)$ is a Poisson algebra. Assume $[\omega]\in H^2(X,\Bbb R)$ is integral so that there exists a (quantum) line bundle with connection $(L,\nabla)$ over $X$ such that $$\omega=curv\,(L,\nabla)$$. Prequantization defines a representation $\pi: C^{\infty}(X) \to End\Gamma(L)$. Using $L$ we may a construct a symplectic manifold $(X^e,\omega^e)$ where $\dim X^e=\dim X+2$ and where $\pi$ is realized as ordinary Hamiltonian action on functions. In a word
Prequantization on $X$ is classical mechanics in a manifold of dimension $\dim X+2$.
Since $(X,\omega)$ arises from $(X^e,\omega^e)$ by symplectic reduction we may refer to the construction of $(X^e,\omega^e)$ as symplectic induction. When $(X,\omega)$ is a suitable coadjoint orbit of certain classical groups we show that a double cover of $(X^e,\omega^e)$ is open and dense in a coadjoint orbit of certain exceptional Lie groups. In particular starting from classical groups this gives rise to a symplectic construction of the Lie groups $E_6$, $E_7$ and $E_8$.
24: Ivan Penkov (Riverside), Generalized Harish Chandra modules
A generalized Harish Chandra module is a module which is of finite type over a not necessarily symmetric subalgebra. Zuckerman, Serganova and myself are currently working on a possible classification of irreducible Harish Chandra modules over a reductive Lie algebra. In this talk I will outline the program and some of the results proved so far. I will discuss in some detail generalized Harish Chandra modules which are of finite type over the principal three dimensional subalgebra.
17: Megumi Harada (University of Toronto), The Gel'fand-Cetlin-Molev integrable system on coadjoint orbits of U(n,\H)
In the 1950s, Gel'fand and Cetlin constructed a canonical basis for finite-dimensional irreducible representations of the unitary group U(n,\C). Using the theory of geometric quantization, which relates representation theory to symplectic (and Poisson) geometry, Guillemin and Sternberg constructed in 1983 a large group of commuting symmetries -- the "Gel'fand-Cetlin torus action" -- on the coadjoint orbits of U(n,\C), thus making the orbits into integrable systems. In this talk, I will review the history and context of this Gel'fand-Cetlin story, and then describe how a similar theory works for the case of the quaternionic unitary group U(n,\H).
10: Fall break
3: Victor Protsak (Cornell), Howe duality for enveloping algebras
september
26: Yuri Berest (Cornell), Morita Equivalence of Cherednik Algebras
19: Reyer Sjamaar (Cornell), Imploding the double of a Lie group
12: cancelled
5: organizational meeting