Rutgers Lie Group Seminar Spring 2000

Meetings of the Rutgers Lie Group Seminar in Fall 2000


Friday, September 22, 2000, 11:30am-12:30pm, Hill 423:

Ethan Duckworth:

               Double coset problems in algebraic groups

Let G be a simple algebraic group with X and P closed subgroups. The double
coset question asks whether X\G/P is finite. Many interesting problems in
the structure theory and representation theory of groups can be viewed as
examples of the double coset question. This talk will describe some of these
problems and offer new techniques for answering the main question. The methods
include geometric reasoning about the classical groups as well as character
theory for finite groups. One theme will be to apply knowledge about algebraic
groups to finite groups, and vice versa.

Friday, September 29, 2000, 11:30am-12:30pm, Hill 423:

Alexander Kirillov:

               Family algebras

A new class of associative algebras (so-called family algebras) is
introduced and investigated. These algebras are related to an irreducible
representation of a simple complex Lie algebra. A family algebra is a sort
of a finite approximation to the enveloping algebra U(g) viewed as a
module over its center. 

It turns out that several important questions about semi-simple algebras
and their representations can be formulated, studied and sometimes solved
in terms of family algebras. 

Here we only start this program and hope that it will be continued and
developed.

Friday, October 6, 2000, 11:30am-12:30pm, Hill 423:

Daya-Nand Verma:

               On the geometry and combinatorics of the Hulsurkar Matrix 

The work of Hulsurkar (Inventiones 1974) is now a quarter of a Century old,
and yet is far from widely known in the circle of research workers of the
area concerned. His proof of a conjecture of mine (Budapest Conf. Proc.
edited by I. M. Gelfand, 1975) on the Dimension Formula of Weyl has actually
much deeper implications, that have somehow escaped wider attention.
This talk will aim at repairing the lapse.

Friday, October 13, 2000, 11:30am-12:30pm, Hill 423:

Mark Graev:

               Gelfand generalized hypergeometric functions and
               finite-dimensional representations of GL(n,C)

Friday, October 20, 2000, 11:30am-12:30pm, Hill 423:

Anatolii Vershik:

               Characters and representations of groups of infinite matrices
               over finite fields

The representation theory of the infinite symmetric group as well as the group
of infinite matrices over a finite field could be considered as asymptotic
theory of representations of finite groups. The most natural examples of the
representations are those which have generalized characters. For the case of
GL(\infty,F_q) the right approach is based on parabolic theory (Grenn, D. Faddeev,
Zelevinsky) and leads to the very important group GLB (=group of almost upper
triangular matrices). This group is locally compact continuous and has a nice
series of genralized characters. The principle subseries of those characters
will be considered.
The talk is based on a joint paper of S. Kerov and the speaker.

Friday, October 27, 2000, 11:30am-12:30pm, Hill 423:

Matthew Leingang:

               Symmetric space valued moment maps

Let G be a compact Lie group, and P a G-space with an equivariantly
closed three-form.  Then one can develop a theory of hamiltonian
G-spaces with moment maps taking values in P.  In order for such a
theory to be meaningful, the given form must obey certain nondegeneracy
assumptions.  If we assume further that P arises from a Riemannian
symmetric pair over G, the irreducible such spaces can be classified as
either g^*, G, or G_C/G.

References:
math.SG/9810064 Matthew Leingang: Symmetric pairs and moment spaces


Friday, November 3, 2000, 11:30am-12:30pm, Hill 423:

Pavel Etingof:

               The Cherednik algebra and the Calogero-Moser space

This talk is an introduction to my recent work with Victor Ginzburg.

The Cherednik algebra H_n is the algebra over C generated by the symmetric group
S_n and two sets of generators x=(x₁,...,x_n) and y=(y₁,...,y_n), where the defining
relations are the obvious commutation relations between S_n and x_i, y_i, and the
relations:

[x_i,x_j]=[y_i,y_j]=0,
[x_i,y_j]=-s_ij, i\ne j,
[x_i,y_i]=s_i₁+...+s_in.

(where s_ij is the permutation of i and j). This algebra has a filtration given
by deg(S_n)=0, deg(x_i)=deg(y_i)=1, and it is known from Cherednik's work that
gr(H_n) is the smash product C[S_n]\bullet C[x,y] (the Poincare-Birkhoff-Witt
theorem).

Ginzburg and I proved the following:

Theorem.
1. Let Z_n be the center of H_n. Then gr(Z_n)=C[x,y]^S_n
2. Let M_n=Spec(Z_n). Then M_n is a smooth, affine, symplectic algebraic variety
of dimension 2n.
3. There exists an algebraic vector bundle V on M_n of dimension n! such that
H_n=End(V). The group S_n\subset H_n acts on fibers of this bundle as in the
regular representation. In particular, all irreduicble representations of H_n
are of dimension n! and are parametrized by points of M_n. 
4. The variety M_n is isomorphic to the Kazhdan-Kostant-Sternberg-Wilson
"Calogero-Moser space", which is the set of pairs of n by n matrices X, Y
such that [X,Y]+1 has rank 1, modulo conjugation.

Some, but not all, of these results can be generalized to the case when S_n
is replaced by any Coxeter group and even any group generated by symplectic
reflections: some as theorems, some as conjectures.

I will try to describe some of these results and also their quantum analogs.

References:
math.AG/0011114 Pavel Etingof, Victor Ginzburg: Symplectic reflection algebras,
Calogero-Moser system, and deformed Harish-Chandra homomorphism

Friday, November 10, 2000, 11:30am-12:30pm, Hill 423:

Alexander Klyachko:

               Random walks on symmetric spaces and matrix spectral problems

The classical spectral problems of linear algebra, such as
        1) spectrum of  product of unitary matrices, 
        2) spectrum of sum of Hermitian matrices, 
        3) singular spectrum of product of complex matrices
may be treated as questions about support of a probability measure coming
from a random walk on symmetric spaces, associated with compact semisimple
group G, namely
        1) group G itself,
        2) its Lie algebra L_G,
        3) the dual symmetric space X_G=G(C)/G, 
where G(C) is the complexification of G. By  evaluating  the corresponding
probability distributions, we reduce the singular spectral problem to the
Hermitian one, recently solved by the author. This proves the Thompson
conjecture. An elegant geometric solution of the remaining unitary problem
is given by Angihotry and Woodward. 

Friday, November 17, 2000, 11:30am-12:30pm, Hill 423:

Donald Richards:

               Total positivity, finite reflection groups, and a
               formula of Harish-Chandra

We define the concept of "total positivity" with respect to finite reflection
groups W. For the case in which W is the group of permutations on n symbols,
this notion reduces to the classical formulation of total positivity. We prove
a basic composition formula for this generalization of total positivity;
in the case in which W is the Weyl group for a compact connected Lie group
we apply an integral formula of Harish-Chandra to construct examples of
totally positive functions. In particular, we deduce that the function
K(x,y) = exp(xy), where x and y are reals, is totally positive with respect to
any Weyl group W. As an application of these results, we derive an FKG-type
correlation inequality in the case in which W is the Weyl group of SO(5); and
we derive some new positivity inequalities for the classical Bessel functions.

Friday, November 24, 2000, 11:30am-12:30pm, Hill 423:

               Thanksgiving. No meeting

Friday, December 1, 2000, 11:30am-12:30pm, Hill 423:

Genkai Zhang:

               Berezin transform on real bounded symmetric domains

We consider a weighted Bergman space on a complex bounded symmetric domain D_c
and its restriction to a real bounded symmetric domain D=G/K. We find the
spectral decomposition of the Bergman space as representation of G and the
spectral measure. This amounts to calculating the symbol of the Berezin
transform as a function of invariant differential operators on D.

Friday, December 8, 2000, 11:30am-12:30pm, Hill 423:

Jeb Willenbring:

               q-multiplicity and the Kostant-Rallis theorem

In this talk, we will describe a stable range in the graded space of harmonic
polynomials associated to the (GL(n),O(n)) case of the Kostant-Rallis theorem.
In the stable range, the q-multiplicity of an arbitrary O(n) representation is
deduced from certain symmetric function identities and classical branching laws.
By q-multiplicity we mean the generalization of a multiplicity formula for an
irreducible representation in a graded space to a generating function for the
multiplicity in the graded components. The q-multiplicity refines the
(non-graded) multiplicity.

Also presented will be joint work with N. Wallach in which a q-analog of the
Kostant-Rallis theorem is given for the symmetric pair (SL(4),SO(4)). One
significant aspect of this example is that it has implications in the study of
quantum computation.

Further motivation is provided by a problem from classical invariant theory.
Specifically, the complex orthogonal group acts on the n x n matrices by
restricting the adjoint action of GL(n,C). This provides us with an action on
the ring of complex valued polynomial functions on the matrices. A combinatorial
description of an initial segment of the Hilbert series for the invariant
polynomials under this action will be given.