spring 2005
This semester we meet on Fridays from 3-4 in 406 Malott Hall.
may
13: Michel Duflo (Paris VII), Associated varieties for representations of Lie superalgebras
Location: room 253.
This is joint work with V. Serganova. The theory of finite dimensional representations of basic classical Lie superalgebras has many similarities with the theory of (infinite dimensional) Harish-Chandra modules for real semi-simple Lie algebras. We propose new tools to study these representations which play the role of the associated varieties for Harish-Chandra modules.
6: no meeting (topology festival).
april
29: no talk.
22: Reyer Sjamaar (Cornell), Varieties invariant under a Borel subgroup
Vladimir Retakh's talk has been cancelled.
This is joint work with Victor Guillemin. Atiyah proved that the moment map image of the closure of an orbit of a complex torus action is convex. Brion generalized this result to actions of a complex reductive group. We extend their results to actions of a maximal solvable subgroup.
15: Xuhua He (MIT), The G-stable pieces and parabolic character sheaves 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 arbitrary Steinberg fiber in \bar{G}. It turns out the boundary of the closure is independent of the choice of the Steinberg fibers and is a union of some G-stable pieces. Based on the these results, some conjectures about the "parabolic character sheaves" will be formulated.
13: Henrique Bursztyn (Toronto), Fusion of quasi-hamiltonian spaces
Note unusual time and date: Wednesday at 2:30. Location: room 406.
I will describe how quasi-hamiltonian spaces with Lie group valued moment maps naturally fit into the framework of Dirac geometry (much in the same way that Poisson geometry describes the ordinary theory of hamiltonian actions). As an application, I will describe the operation of "fusion" of quasi-hamiltonian G-spaces from the Dirac geometric point of view, showing how everything can be carried out for noncompact groups G.
8: no talk
1: Etienne Rassart (IAS), Polynomiality properties of type A weight and tensor product multiplicities
Kostka numbers and Littlewood-Richardson coefficients appear in the representation theory of complex semisimple Lie algebras of type A respectively as the multiplicities of weights in irreducible representations, and multiplicities of irreducible factors in tensor products of irreducibles. Despite a number of formulas for them, they are notoriously hard to compute.
Using a variety of tools from representation theory (Gelfand-Tsetlin diagrams), convex geometry (vector partition functions), symplectic geometry (Duistermaat-Heckman measure) and combinatorics (hyperplane arrangements), we show that the Kostka numbers are given by polynomials in the cells of a complex of cones. For fixed $\lambda$, the nonzero $K_{\lambda\mu}$ consist of the lattice points inside a permutahedron. By relating the complex of cones to a family of hyperplane arrangements, we provide an explanation for why the polynomials giving the Kostka numbers exhibit interesting factorization patterns in the boundary regions of the permutahedron. Some of the techniques used generalize to the case of Littlewood-Richardson coefficients.
This is joint work with Sara Billey and Victor Guillemin.
march
25: no meeting (spring break)
18: no meeting (spring break)
11: no talk this week
4: Leslie Saper (Duke), L^2-harmonic forms on locally symmetric spaces
For a Hermitian locally symmetric space, the L^2-harmonic forms have a topological interpretation: they represent the intersection cohomology of the Baily-Borel-Satake compactification (Zucker's conjecture). I will give a similar, though more delicate, topological interpretation for general locally symmetric spaces (except possibly type E_7 or E_8).
february
25: Dan Zaffran (Cornell), Families of compact complex manifolds related to Mumford's Geometric Invariant Theory
A new family of non-algebraic complex manifolds of any dimension was recently constructed by several authors. I will describe that construction, recall the basics of Mumford's GIT, and explain the link between the two.
18: Martin Kassabov (Cornell), On the Automorphism Tower of Free Nilpotent Groups
I will describe the automorphism tower of free nilpotent groups which was the topic of my PhD thesis. The main tool in studying the automorphism tower is to embed every group as a lattice in some Lie group. Using known rigidity results the automorphism group of the discrete group can be embedded into the automorphism group of the Lie group.
The main result states that the automorphism tower of the free nilpotent group Gamma(n,d) on n generators and nilpotency class d stabilizes after finitely many steps. If the nilpotency class is small compared to the number of generators, we have that the height of the automorphism tower is at most 3.
11: organizational meeting
4: Dylan Thurston (Oliver Club/Topology & Geometric Group Theory Seminar)
january
28: Sergey Arkhipov (Yale)
This semester we meet on Fridays from 3-4 in 406 Malott Hall.
december
3: Dragan Milicic (Utah), Variations on a theme of Casselman-Osborne
A number of basic results in representation theory (for example: Wigner's lemma, Casselman-Osborne's lemma) say, roughly speaking, that a certain property - obvious for a functor itself - holds also for its derived functors. The "classic" proofs in the literature are usually based on quite diverse technical tricks. We shall show how all of these statements are special cases of a simple theorem about derived functors between the corresponding derived categories and a certain "homogeneity" property with respect to their centers.
november
19: David Vogan (MIT), Comparing intertwining operators for different groups
A fundamental problem in unitary representation theory is to identify unitary representations of a (small) group with unitary representations of a (larger) group. Such identifications have an obvious value in classification problems, and they can also shed light on theoretical goals (like Langlands functoriality).
Some of the least well-understood unitary representations for reductive groups are the complementary series, whose unitarity is a consequence of positivity of some intertwining operator. In order to compare complementary series for different groups, one needs a way to compare intertwining operators for these groups. I'll talk about a method for doing that: specifically, for relating intertwining operators for spherical principal series in a small group to those for non-spherical principal series in a larger group. In interesting examples the small group need not even be a subgroup of the larger one.
12: Farkhod Eshmatov (Cornell), An Example of an A-infinity Functor
The notion of an A-infinity functor was introduced by M.Kontsevich in connection with his Mirror Symmetry Conjecture. In this talk we will explain Kontsevich's definition in the setting of DG categories and discuss an example of an A-infinity functor arising in deformation theory.
5: Toby Stafford (Michigan), Cherednik algebras and Hilbert schemes of points.
The work discussed in this lecture is joint with Iain Gordon. The Cherednik algebra H_c of type A_{n-1} is a particular deformation of the twisted group ring of a Weyl algebra by the symmetric group S_n. In their short history, Cherednik algebras have been influential in a surprising range of subjects: for example they have been used to answer questions in integrable systems, combinatorics and symplectic quotient singularities.
In many ways Cherednik algebras behave rather like primitive factor rings of enveloping algebras. In this talk we will show, in a manner reminiscent of the Beilinson-Bernstein equivalence of categories, that H_c can also be regarded as a noncommutative deformation of the Hilbert scheme of points in the plane. This can be used to show the close connection between the representation theory of H_c and the geometry of Hilbert schemes. In turn this allows one to use the rich theory of Hilbert schemes, notably Haiman's work on the n! conjecture, to understand the representation theory of Cherednik algebras.
october
29: Leticia Barchini (Oklahoma State, Stillwater), Stein extensions of real symmetric spaces
Let \Omega be an open convex domain in \R^n and let H(\Omega) be the space of harmonic functions on \Omega. It is known that there exists a maximal domain D\subset \C^n$ so that all functions in H(\omega) admit a holomorphic continuation to D.
When G/K is a Riemannian symmetric space of noncompact type there exists a domain \omega_{AG} in G_C/K_C with the remarkable property that all eigenfunctions of the invariant differential operators have a holomorphic extension to \omega_{AG}, and this domain is maximal for this property.
The domain \omega_{AG} has been the focus of intense study in recent years. The goal of my talk is to present a survey of results on \omega_{AG}, its relation to the geometry of flag manifolds and its relevance in representation theory.
22: Arvind Nair (Tata and UMichigan), A Lefschetz property for arithmetic ball quotients
Arithmetic subgroups of unitary groups of signature (1,n) act on the complex unit ball, with the quotients being finite-volume locally symmetric varieties. These ball quotients contain certain totally geodesic subvarieties of complex codimension one, coming from subgroups which are unitary groups of signature (1,n-1). I shall discuss the geometry of these ball quotients, and an injectivity result for restriction of cohomology to these subvarieties, generalizing a result of Oda.
15: Peter Trapa (Utah), Two bases of Weyl group representations
Suppose W is a Weyl group. The purpose of this talk is to recall two bases of each (special) representation of W. One is purely algebraic (originating in the study of primitive ideals in enveloping algebras); the other is essentially geometric (originating in the study of the Springer fiber). Tanisaki conjectured that these two bases should be related by an upper-triangular matrix. McGovern proved this for classical cases. I'll outline a completely different approach to the conjecture which (perhaps surprisingly) uses real reductive Lie groups in an essential way.
8: no talk (Fall break)
1: Kevin Wortman (Cornell), Finiteness properties of arithmetic groups over function fields
In joint work with Kai-Uwe Bux, we verify a conjecture that an arithmetic subgroup of a reductive group defined over a global function field is of type FP_\infty if and only if the semisimple rank of the reductive group over the global field equals 0.
This conjecture had its roots in investigations of Serre and Stuhler into cohomology of arithmetic groups.
september
24: Alessandra Pantano (Cornell), Signatures of Intertwining Operators and Weyl Group Representations II
17: Alessandra Pantano (Cornell), Signatures of Intertwining Operators and Weyl Group Representations I
For split groups, it is possible to discuss the non-unitarity of a spherical principal series by means of Weyl group computations. Indeed, the Hermitian form on some K-types (that we call "PETITE") only depends on a certain representation of the Weyl group. In the first part of the talk we describe this setting. In the second part, we give a method for constructing petite K-types.
10: no talk
3: organizational meeting