Meetings of the Rutgers Lie Group Seminar in Fall 1998

Friday, Sep. 18, 1998, 11:30am-12:30pm, Hill 425

Allan Knutson:
               Positivity of Littlewood-Richardson coefficients and
               the combinatorics of "honeycombs"

A central topic in representation theory and combinatorics is the
decomposition of tensor products of irreducible representations of
GLn(C). In particular, what inequalities must a triple of high weights
(\lambda,\mu,\nu) satisfy in order for the triple tensor product to have an
invariant vector? A list of inequalities derived from Schubert calculus has
recently been found, which were shown by Klyachko and Helmke-Rosenthal to be
necessary, and by Klyachko to be asymptotically sufficient.

In this talk we introduce a combinatorial model, the honeycomb, to study
invariant vectors in triple tensor products (reinterpreting work of
Berenstein and Zelevinsky). With this we (1) rederive the inequalities, (2)
determine which ones are actually relevant, and (3) show that the
inequalities are sufficient even for small weights (not just asymptotically)
-- this latter one was known as the "saturation conjecture". In particular
this implies Horn's conjecture from 1956 on the spectrum of the sum of two
Hermitian matrices.

Friday, Sep. 25, 1998, 11:30am-12:30pm, Hill 425

Yun Gao:
               Vertex operators arising from the homogeneous realization
               of \widehat{gl}n and applications to extended affine
               Lie algebras

The notion of extended affine Lie algebra (EALA, for short) is a higher
dimensional generalization of affine Kac-Moody algebras first studied by
mathematical physicists. Toroidal Lie algebras are examples of EALAs. There
are many EALAs whose coordinates allow not only the Laurent polynomial
algebra but also quantum tori, Jordan tori and the octonion torus depending
on types of EALAs. For instance, EALAs of type An-1 are tied up with
the Lie algebra gln(Cq), where Cq is a quantum torus
associated to a \nu×\nu matrix q.

In this talk, I will use the underlying Fock space for the homogeneous
vertex operator representation of the affine Lie algebra \widehat{gl}n to
construct a family of vertex operators. Consequently, we will get an
irreducible vertex operator representation for an EALA of type An-1
coordinatized by a quantum torus Cq of 2 variables.  As an
application, the character formula for the basic module of the EALA is
obtained.

Friday, Oct. 2, 1998, 11:30am-12:30pm, Hill 425

Burt Totaro:
               The Chow ring of classifying spaces

There is a natural way to approximate the classifying space of a compact Lie
group, up to homotopy, by smooth complex algebraic varieties.  This gives a
new invariant of compact Lie groups, the Chow ring of the classifying space
BG, which maps to the integral cohomology of BG.  I will describe the recent
calculations and general results about this Chow ring: in some ways, it is
better-behaved than the usual cohomology ring.

Friday, Oct. 9, 1998, 11:30am-12:30pm, Hill 425

Alan Weinstein:
               Near-homomorphisms from compact Lie groups

A near-homomorphism from a group G to a group H is a map a from G to H such
that a(gh) is "sufficiently close" to a(g)a(h) for all g and h in G. When G
and H are compact Lie groups, it was shown by Grove and Karcher in the
1970's that a near-homomorphism is near a homomorphism. At around the same
time, de la Harpe and Karoubi proved a similar theorem for the case where H
is replaced by the unitary group of a Hilbert space. In this talk, I will
discuss the case where H is a group of diffeomorphisms.

Friday, Oct. 16, 1998, 11:30am-12:30pm, Hill 425

Chris Woodward:
               Verlinde formulas as fixed point formulas

The representation ring of a loop group at a fixed level has a product
structure whose structure coefficients are certain invariants of
moduli spaces of flat connections on a two-sphere with three
punctures.  Verlinde gave formulas for these structure coefficients as
well as for the moduli spaces of higher genus Riemann surfaces.  I
will discuss a fixed point formula for loop group actions, which
specializes to the Verlinde formulas, much as the Lefschetz formula for
the spin-c index specializes to the Steinberg formula for the
structure coefficients of the representation ring of a compact group.

Friday, Oct. 23, 1998, 11:30am-12:30pm, Hill 425

Gail Letzter:
              Quantum Symmetric Pairs

Let \theta be an involution of a semisimple Lie algebra g, let g\theta denote
the fixed Lie subalgebra, and assume that the Cartan subalgebra has been chosen
in a suitable way. We construct a quantum analog of U(g\theta) which can be
characterized as the unique maximal right coideal in the quantized enveloping
algebra of g which specializes to U(g\theta).

Friday, Oct. 30, 1998, 11:30am-12:30pm, Hill 425

Konstanze Rietsch:
               Total positivity and equivariant cohomology of flag varieties

We give a brief introduction to Lusztig's theory of total positivity for
reductive algebraic groups. Then we study total positivity in Dale
Peterson's geometric model for the equivariant cohomology of a flag variety,
and describe a connection between Lusztig's canonical basis of the
enveloping algebra of U-, and the Schubert basis of HT(G/B), in which
positivity properties are preserved.

Friday, Nov. 6, 1998, 11:30am-12:30pm, Hill 425

Dan Sage:
               Fixed point varieties on affine flag manifolds and affine
               Springer representation

Let G be a semisimple, simply connected complex Lie group with Lie algebra
g, and let F be the field of Laurent series in one variable over C. The
affine flag manifolds of G are the spaces of parahoric subalgebras (a local
field analogue of parabolic subalgebras) of a given type in the Lie algebra
with scalars extended to F.  Fixed point varieties on affine flag manifolds
are the spaces of parahoric subalgebras containing a given nil-elliptic
element; they have important applications to representation theory. We
define representations of the affine Weyl group in the homology of these
varieties, generalizing Kazhdan and Lusztig's topological construction of
Springer's representations to the affine context. We also show how to
define actions of the multiplicative group on certain of these fixed point
varieties. We use these actions to find simple formulas for their Euler
characteristics and in favorable cases, to obtain cell decompositions for
them.

Friday, Nov. 13, 1998, 11:30am-12:30pm, Hill 425

Dragan Milicic:
               Bruhat filtrations of smooth principal series

Smooth principal series representations of real reductive Lie groups have a
natural filtration which is implicit in the classical work of Bruhat on
irreducibility of unitary principal series. This filtration can be used to
study some cohomological properties of representations. As an illustration,
among other things, we are going to describe how to deduce some results of
Kostant and Wallach on Whittaker vectors. This is a joint work with Bill
Casselman and Henryk Hecht.

Friday, Nov. 20, 1998, 11:30am-12:30pm, Hill 425

Alexander Kirillov:
               The affine analogue of Macdonald inner product identities

We give a conjectural analogue of the formula for the norms of the Macdonald's
polynomials for affine root system (of type A). This generalization is highly
non-trivial; even  defining these norms is a difficult task. Our approach
is based on the representation theory of affine Lie algebras. We show that
these norms are closely related with the norms of conformal blocks in
Wess-Zumino-Witten conformal field theory, and give a proof of the inner
product formula for the root system Â1. 

Friday, Nov. 27, 1998, 11:30am-12:30pm, Hill 425

               Thanksgiving

Friday, Dec. 4, 1998, 11:30am-12:30pm, Hill 425

Gopal Prasad:
               Towards the classification of irreducible admissible
               representations of reductive p-adic groups

I will describe my joint work with Allen Moy in which we have introduced the
notion of an "unramified type" and proved that any admissible irreducible
representation contains one and any two contained in the same irreducible
representation are "associates" of each other. This allows us to define a
nonnegative rational valued invariant, called the "depth", of an irreducible
representation. Extending the work of Armand Borel on representations with
vectors fixed under an Iwahori subgroup, we can give a complete description
of the representations of depth zero. We are currently looking at how to
classify representations of positive depth. We have some geometric ideas
which may be of use. If time permits, I will describe these ideas.