Fall, 2015
- Speaker Vladimir Retakh, Rutgers University
- Speaker Baiying Liu, Institute for Advanced Study
- Title On cuspidality of Arthur packets of
quasi-split classical groups
- Time/place 9/25/2015, Friday, 12:00 in Hill 705
- Abstract Based on the theory of endoscopy,
Arthur classified the automorphic discrete spectrum of
quasi-split classical groups up to global Arthur
packets parametrized by Arthur parameters. Towards studying
representations in each Arthur packet, a natural question one
may ask is that whether a given Arthur packet has cuspidal
representations or not. In this talk, I will introduce some
recent progress on this aspect, which is based on relations
between the structure of Fourier coefficients of automorphic
forms in an Arthur packet and the structure of the corresponding
Arthur parameter. This work is joint with Dihua Jiang.
- Speaker Philippe Di Francesco, University of
Illinois at Urbana-Champaign
- Title Whittaker functions, cluster algebras and
Macdonald difference operators
- Time/place 10/2/2015, Friday, 12:00 in Hill 705
- Abstract Whittaker functions are solutions of
Toda-type differential/difference
equations, originally built out of Whittaker vectors, playing
a central role in representation theory of Lie algebras. We give
a new statistical weighted path formulation for these vectors,
valid for simple and affine Lie algebras as well as the quantum
algebra $U_q(sl_n)$.
We then consider graded tensor products of current algebra
$g[t]$-modules, and show that their characters obey difference
equations, generalizing the difference Toda equation, allowing
for viewing graded characters as generalized Whittaker
functions. This uses a constant term expression
for the characters involving a solution of the quantum Q-system,
a set of non-commuting integrable recursion relations that are
particular mutations of a quantum cluster algebra attached to
the Lie algebra $g$.
Finally, we obtain a new compact expression for graded $sl_n$
characters by constructing a representation of the quantum
Q-system via generalized Macdonald difference operators.
(Based on joint works with R. Kedem and B. Turmunkh and with R. Kedem.)
- Speaker Raul Gomez, Cornell University
- Title Invariant trilinear forms on induced
representations of real rank one groups
- Time/place 10/9/2015, Friday, 12:00 in Hill 705
- Abstract Bernstein and Reznikov introduced a
triple integral formula to describe a family of invariant
trilinear forms for induced representations of
PGL(2, R). However, they left open the question of
computing the full space of invariants. Using the definition
of the Schwartz space of a Nash manifold, together with some
homological algebra, we will show how to describe the
remaining trilinear forms. We will then show how these results
can be extended to induced representations of real rank one
groups, refining in this way some results previously obtained by
Clerc, Kobayashi, Ørsted and Pevzner. This is ongoing joint
work with Birgit Speh.
- Speaker Nolan Wallach, University of California at
San Diego
- Title Some applications of geometric invariant theory
- Time/place 10/16/2015, Friday, 12:00 in Hill 705
- Abstract
The lecture will begin with an introduction to the notion of quantum entanglement that is aimed at mathematicians. The rest of the lecture will be devoted to specific examples of applications of GIT that have appeared in the physics literature over the past ten years. Time permitting, open problems will be discussed.
- Speaker Jinwei Yang, University of Notre Dame
- Title Differential equations for logarithmic intertwining
operators for strongly graded vertex algebras
- Time/place 10/23/2015, Friday, 12:00 in Hill 705
- Abstract We introduce an A-pattern associated to each
element of a strongly graded module with respect to an abelian group A.
Under certain assumptions involving A-pattern of elements in a strongly
graded module and C_1-cofiniteness condition, we derive differential
equations for matrix elements of products and iterates of logarithmic
intertwining operators among strongly graded modules. I will also show that
the assumptions hold for the main examples of strongly graded vertex
algebras and their modules.
- Speaker Anton M. Zeitlin, Columbia University
- Title Decorated Super-Teichmueller Space
- Time/place 11/6/2015, Friday, 12:00 in Hill 705
- AbstractIn this talk I will describe the construction of the analogue of Penner's
coordinates on the decorated super-Teichmueller space of a surface with
$s\ge 1$ punctures, which is a principal bundle over the super-Teichmueller
space. We will discuss all necessary ingredients e.g. super-version of the
Ptolemy transformations, combinatorial approach to the description of the
spin structures on punctured surfaces as well as the even Ptolemy-invariant
2-form, which is the generalization of the Weil-Petersson 2-form. Based on
the preprint arXiv:1509.06302.
- Speaker David Vogan, MIT
- Speaker Apoorva Khare, Stanford University
- Title Standard parabolic subsets of highest weight modules
- Time/place 11/20/2015, Friday, 12:00 in Hill 705
- Abstract In 1991, Vinberg initiated the study of faces of weight polytopes, corresponding to finite-dimensional modules over a complex semisimple Lie algebra $\mathfrak{g}$. He showed that to each subset of simple roots, there corresponds a dominant face of the weight polytope. This defines a ``face map" on the power set of the simple roots. Recently, Cellini-Marietti [IMRN, 2015] classified the fibers of the face map for the adjoint representation for simple $\mathfrak{g}$, i.e., the redundancies between the faces. These redundancies are not known for any other finite- or infinite-dimensional highest weight module (except Verma modules).
We present a complete resolution of the above problem, for arbitrary highest weight modules $\mathbb{V}^\lambda$ over any semisimple $\mathfrak{g}$. We show that the fibers of the aforementioned face map are always intervals. We also compute the f-polynomials, affine hulls, vertex sets, and Weyl group stabilizers of these faces. Remarkably, all of these formulas are type-free and depend only on the highest weight, the set of simple roots, and the "integrability data" of the module $\mathbb{V}^\lambda$.
- Speaker Haisheng Li, Rutgers University at Camden
- Title Vertex super-algebras associated to $Z$-algebras of certain levels
- Time/place 12/4/2015, Friday, 12:00 in Hill 705
- Abstract In this talk, I will report on recent work about associating the
Lepowsky-Wilson $Z$-algebra of certain levels with vertex (super)algebras.
More specifically, we associate C((t))-module vertex super-algebras
and equivariant quasi modules to the Z-algebra of any nonzero level
\ell such that 2/\ell is an integer.
|
is a subgroup of index two in 
. Clifford theory describes very simple and powerful
relationships between the representation theory of 
for
the set of irreducible representations of 
defines a permutation 
of order two of 
of order two of 
.
and
determines the other in a very simple way.