Meetings of the Rutgers Lie Group Seminar in Spring 1995

Tuesday, Jan. 17, 1995, 3:00pm-4:00pm, Hill 250 (special date)

Shrawan Kumar:
               The Picard group of moduli spaces of G-bundles,
               and representation theory

Friday, Jan. 27, 1995, 11:30am-12:30pm, Hill 425

Friedrich Knop:
               The image of the moment map from the differentiable
               point of view

Consider a Hamiltonian action of a compact group K on a symplectic
manifold X. We are interested in smooth functions which
Poisson-commute with all K-invariant smooth functions on X.
  All pull-back functions from the moment map have this property but
there are (slightly) more of them. We explain how this difference is
controlled by a certain subgroup W_X of the Weyl group of K.
  The group W_X is also a reflection group and is analogous to the
little Weyl group of a symmetric space. We discuss also its effect on
the automorphism group of X.

Friday, Feb. 3, 1995, 11:30am-12:30pm, Hill 425

Siddhartha Sahi:
               Euclidean Fourier transforms and unitary representations

Consider a (degenerate) principal series representation X attached to
a parabolic subgroup P=LN of a semisimple Lie group G.
  We can regard the vectors in the representation as functions on the
opposite nilradical, and then via the Euclidean Fourier transform, as
tempered distributions on N.
  If Z is a submodule of X, then the support of the distributions in Z
is a P-orbit on N, closely related to the associated variety of Z.
  It seems that (possible) unitary structures on Z can often be
described rather simply in terms of measures and distributions on
P-orbits in N.
  The talk will discuss applications of this philosophy to the case
where N is abelian.

Friday, Feb. 10, 1995, 11:30am-12:30pm, Hill 425

Roger Zierau:
               Maximal compact subvarieties in flag domains

There are many ways to realize representations as sections of a vector
bundle over a symmetric space.  A Penrose transform often does the
trick for representations associated to open orbits in a complex flag
variety.  The general construction uses a double fibration (between
the orbit D and a space of maximal compact subvarieties M(D)) along
with some standard geometric tricks.  When the representations are
highest weight representations M(D)=G/K and the general procedure
works very well and is understood.  Otherwise one needs to know more
about the space M(D).  I will discuss some recent joint work with Joe
Wolf on the structure of M(D).  I will also describe how this leads to
a realization of the representation in a completely holomorphic
setting.

Friday, Feb. 17, 1995, 11:30am-12:30pm, Hill 425

Kamal Khuri-Makdisi:
               Fourier coefficients of forms of half-integral weight

   Let f(z) = \sum_{n > 0} a_n exp(2\pi inz) be a cusp form of
half-integral weight m + ½.  If n = q^2 r with r square-free,
then by Shimura's correspondence the n-th Fourier coefficient a_n
can be expressed in terms of a_r and the Fourier coefficients of a
form g of integral weight 2m.  As for a_r, a theorem of
Waldspurger says that its square is more or less the central value of
the L-function of g twisted by the quadratic character corresponding
to Q(r^½).
  This talk will describe a method, due to Shimura, to get explicit
forms of Waldspurger's relation.  The method works well over totally
real number fields (even for nonholomorphic forms), and should generalize
straightforwardly to forms over an arbitrary number field.

Friday, Feb. 24, 1995, 11:30am-12:30pm, Hill 425

Ranee Brylinski:
               Geometric quantization and minimal unitary representations

In this talk I will give an overview of joint work with Bert Kostant
on the construction, following the precepts of geometric quantization,
of a unitary representation attached to the minimal nilpotent orbit
O. The representations we obtain are natural generalizations of the
Fock space model of the metaplectic representation. One feature of our
methods is that exceptional Lie groups are treated in exactly the same
manner as the classical ones. Roughly speaking, the main point is
that, using the geometry and invariant theory of the complexification
of O, we find new realizations of a finite-dimensional complex simple
Lie algebra g in terms of p's and q's. Then we quantize these symbols
to realize g as as a Lie algebra of pseudo-differential operators.
This leads to many results on harmonic analysis for the ensuing
unitary representation of the underlying real group.
 
Thursday, March 2, 1995, 3:00pm-4:00pm, Hill 705

Jean-Luc Brylinski:
               A dual to the McKay correspondence

Finite subgroups of SU(2) are associated to Dynkin diagrams of type A, D, E.
The McKay correspondence gives a bijection between irreducible
representations of the finite group and vertices of the extended Dynkin
diagram. The edges give the operation of tensoring with the 2-dimensional
representation.  The dual correspondence we propose is a bijection between
non-trivial conjugacy classes of the finite group and vertices of the
diagram.The ends of the diagram give 3 distinguished conjugacy classes, and
edges correspond to taking a product with an element in one of these
classes. This correspondence is based on the topology of the resolution of
singularities of the Kleinian singularity.

Friday, March 3, 1995, 11:30am-12:30pm, Hill 425

Sam Evens:
               Characteristic cycles for the loop Graßmannian

We explain how to compute characteristic cycles of D-modules in several
settings relevant to representation theory. The loop Graßmannian is a
homogeneous space for a loop group LG with the property that its equivariant
D-modules correspond to finite dimensional representations of the Langlands
dual of G. In this setting, we compute the characteristic cycles.
   If time permits, we will discuss some examples related to Springer
representations and affine Hecke algebra representations. This talk is based
on joint work with Ivan Mirkovic.

Friday, March 10, 1995, 10:30am-11:30am, Hill 425

Yael Karshon:
               Hamiltonian torus actions

We will define a Hamiltonian torus action on a symplectic manifold M. The
best understood examples are smooth compact toric varieties.  These are in
one to one correspondence with a set of polytopes in R^n where n=½dim M.
The symplectic quotients of these spaces are single points.
   Compact spaces whose symplectic quotients have dimension 2 are currently
studied; those with dim M = 4 are completely understood. In the "generic"
case, the action extends to form a toric variety (where the word "generic"
is used in a somewhat loose way).

Friday, March 10, 1995, 12:00pm-1:00pm, Hill 425

Wolfgang Soergel:
               n-cohomology of limits of discrete series

We drag with heavy machinery this problem into the realm of combinatorics
of Weyl groups. Some known cases can be recovered this way, and some up to
now unknown cases can still be treated in a satisfactory way.

Friday, March 17, 1995, 11:30am-12:30pm, Hill 425

               Spring break. No meeting.

Friday, March 24, 1995, 10:30am-11:30am, Hill 425

Peter Heinzner:
               Equivariant holomorphic vector bundles
                
The "Oka Principle" can be vaguely stated as follows

   On a reduced Stein space X, problems which can be cohomologically
   formulated have only topological obstructions. In other words, such
   problems are holomorphically solvable if and only if they are
   continuously solvable.

For example, Grauert has proved that the categories of topological or
complex analytic principal bundles over a Stein space X with a complex Lie
group as structure group are the same.
    In this talk we explain the equivariant version of Grauert's Oka
Principle for a compact Lie group of holomorphic transformations on X.

Theorem. (Join work with Frank Kutzschebauch) Let K be a compact Lie group
of holomorphic transformations of a reduced Stein space X. Then every
topological complex K-vector bundle over X is K-isomorphic to a holomorphic
K-vector bundle over X. Moreover, two holomorphic K-vector bundles over X
are holomorphically K-isomorphic if and only if they are topologically
K-isomorphic.

Friday, March 24, 1995, 12:00pm-1:00pm, Hill 425

Joseph Wolf:
               Flag domains and realization of representations
               by the double fibration construction

The representation theory of semisimple Lie groups makes some fascinating
connections between harmonic analysis, differential geometry and analytic
number theory. For example one can study representations by constructing
them in a variety of geometric settings. I'll survey some of those
constructions and describe some recent progress. Level: graduate students
should be able to follow most of the talk.

Friday, March 31, 1995, 11:30am-12:30pm, Hill 425

Soji Kaneyuki:
               Signatures of roots and a new characterization
               of causal symmetric spaces

A signature function on a root system of a real semisimple Lie algebra g is
a {±1}-valued multiplicative character of the root lattice. A causal
symmetric space is a symmetric space admitting an invariant smooth cone
field in tangent spaces. Simple causal symmetric spaces were classified by
G.I. Ol'shansky (1982).

(1) We show that every signature function arises from a Z-gradation of g.
From this point of view, semisimple symmetric spaces of type K_e,
introduced by Oshima-Sekiguchi, break up into two classes: K_eI and K_eII.

(2) We give a characterization of simple causal symmetric spaces in terms
of K_eI.

Friday, April 7, 1995, 11:30am-12:30pm, Hill 425

Toshiyuki Kobayashi:
               Discrete decomposability of A_q(l) with respect to
               reductive subgroups and its applications

Friday, April 14, 1995, 11:30am-12:30pm, Hill 425

Robert Shalla:
               The universal cover of SU(2,1) and algebraic
               D-module techniques in representation theory

Some recent work of Hecht, Milicic, Schmid, and Wolf uses the theory of
algebraic D-modules to study the representations of a semisimple Lie group,
G, with finite center. A straightforward generalization of their
constructions can also produce useful realizations of the irreducible
representations of G when the center is infinite.

In this talk G will be the universal cover of SU(2,1) and we shall discuss
some techniques of algebraic D-modules in this setting.  These techniques
give a simple description of the irreducible tempered representations and of
the composition series of standard modules.  We can also understand, in
geometric terms, the "new" phenomena which arise in the infinite center case.

Friday, April 21, 1995, 11:30am-12:30pm, Hill 425

Kailash Misra:
               Application of crystal basis to affine
               Lie algebra representations 

The crystal basis associated with integrable highest weight representations
of affine Lie algebras provide useful combinatorial tool to study the
corresponding representations. In this talk we will discuss how this tool can
be used to show that any integrable highest representation of classical
affine Lie algebras can be embedded in the tensor product of level one
representations of the corresponding affine Lie algebras.

Friday, April 28, 1995, 11:30am-12:30pm, Hill 425

Leon Ehrenpreis:
               Differential equations and representation theory