| DATE |
SPEAKER |
TITLE (click to show abstract) |
| September 12th |
Alexandra Utiralova, UC Berkeley |
Harish-Chandra bimodules in complex rank
Abstract: The Deligne tensor categories are defined as an interpolation of
the categories of representations of groups GL_n, O_n, Sp_{2n} or S_n
to the complex values of the parameter n. One can extend many
classical representation-theoretic notions and constructions to this
context. These complex rank analogs of classical objects provide
insights into their stable behavior patterns as n goes to infinity.
I will talk about some of my results on Harish-Chandra bimodules in the
Deligne categories. It is known that in the classical case simple
Harish-Chandra bimodules admit a classification in terms of W-orbits
of certain pairs of weights. However, the notion of weight is not
well-defined in the setting of the Deligne categories. I will explain how
in complex rank the above-mentioned classification translates to a
condition on the corresponding (left and right) central characters.
Another interesting phenomenon arising in complex rank is that there
are two ways to define Harish-Chandra bimodules. That is, one can
either require that the center acts locally finitely on a bimodule M
or that M has a finite K-type. The two conditions are known to be
equivalent for a semi-simple Lie algebra in the classical setting,
however, in the Deligne categories, it is no longer the case. I will talk
about a way to construct examples of Harish-Chandra bimodules of
finite K-type using the ultraproduct realization of the Deligne
categories.
|
| September 19th |
Andrei Okounkov, UC Berkeley |
Constant terms from the point of Schubert calculus
Abstract:
Langlands' constant term formula is really the key to many computations with Eisenstein series.
In my talk, I will explain the geometric point of view on this formula that forms the basis
of our ongoing joint work with David Kazhdan on spectral analysis of Eisenstein series.
|
| September 26th |
Vera Serganova, UC Berkeley |
Borel-Weil-Bott theorem in the supercase.
Abstract: The classical Borel-Weil-Bott theorem computes cohomology
groups of line bundles on flag varieties associated with reductive
groups G. If the ground field has characteristic zero this cohomology
vanish in all but one degree, and in this degree we obtain an
irreducible G-module.
In positive characteristic the situation is much more complicated and
leads to the notion of the Weyl module.
The goal of this talk is to review what is known about the answer to
the similar question in the case of supergroups. In general, the
problem is still open.
I will discuss different approaches and the main difficulty. The talk
will include a crash course on supergroups and supergeometry.
|
| September 30th (Friday) Evans 736 |
Maria Gorelik , The Weizmann Institute of Science |
Root groupoid and related Lie superalgebras.
Abstract: This talk in based on a joint work with V. Serganova and V. Hinich,
arXiv:2209.06253.
We introduce a notion of a root groupoid as a replacement of the
notion of Weyl group for (Kac-Moody) Lie superalgebras. The objects of
the root
groupoid classify certain root data, the arrows are defined by
generators and relations. As an abstract groupoid the root groupoid
has many connected components and we show
that to some of them one can associate an interesting family of Lie
superalgebras which we call root superalgebras.
To each root groupoid component we associate a graph (called skeleton)
generalizing the Cayley graph of the Weyl group. The skeleton
satisfies a version of Coxeter property generalizing the fact that the
Weyl group of a Kac-Moody Lie algebra is Coxeter.
|
| Obtober 10th |
José Simental Rodríguez , Universidad Nacional Autónoma de México |
Harish-Chandra bimodules for rational Cherednik algebras
Abstract: Whenever two algebras A and A' quantize the same Poisson algebra, one can speak about a category of Harish-Chandra (A, A')-bimodules.
In this talk, we will focus on the case of rational Cherednik algebras, which quantize W-invariant functions on h \oplus h*, where W is a complex reflection group
and h its reflection representation. In this case, it is known that the category of Harish-Chandra bimodules is an artinian, abelian category with enough projectives.
However, even the number of irreducibles depends crucially on the quantization parameter and in most cases it is unknown.
In the case when W is the symmetric group we have a precise count of the number of irreducibles that we will describe, as well as a complete description of the block
containing the regular bimodule and tensor products of irreducibles in this block.
|
| October 17th |
Alex Sherman, University of Sydney |
Support varieties for Lie superalgebras
Abstract: Support theory for Lie superalgebras was introduced in the
2000s via two different approaches. One is through cohomological
support varieties, defined via the Ext algebra, which were introduced
and studied by Boe, Kujawa, and Nakano. On the other hand Duflo and
Serganova introduced associated varieties, which mimic the rank
variety construction given for finite groups. Both types of support
have been studied over the years, and have their own strengths and
difficulties. We discuss an ongoing project to understand the
connection between these two approaches. In particular we will begin
by recalling support theory for finite groups, and explore the
similarities and differences with the super case. In the end, we will
explain how in the Kac-Moody case there is a very beautiful
relationship between the two types of support, using the recently
developed notion of splitting subgroups, which take on the role of
Sylow subgroups in the super setting.
|
| October 24th |
Eugene Gorsky, UC Davis |
The Trace of the affine Hecke category
Abstract: Elias and Mackaay-Thiel defined and studied the category of
affine Soergel bimodules categorifying the affine Hecke algebra. I will
talk about the "categorical trace" of this category, and its relations
with the elliptic Hall algebra and the skein algebra of the torus. This
is a joint work with Andrei Negut.
|
| October 31st |
Colleen Delaney, UC Berkeley |
Tinkering with tensor categories
Abstract:``Zesting” is a fun way to build a new tensor category out of an old one that has a particularly combinatorial and diagrammatic flavor. In the presence of additional structures like braidings, ribbon twists, group actions, and module categories, zesting takes on additional meaning that can be interpreted both physically and quantum topologically. I will review the zesting construction with an emphasis on those of its applications that may be interesting to algebraists, for example the categorification of fusion rings and the classification of fusion categories. I will probably mine the audience for knowledge about how one should think about zesting from the point of view of representation theory of (weak) Hopf algebras.
This talk will feature joint work with Cesar Galindo, Julia Plavnik, Eric Rowell, and Qing Zhang as well as Sung Kim and also work in progress with Calvin McPhail-Snyder.
|
| Cancelled |
Ben Webster, University of Waterloo, Perimeter Institute |
Heisenberg and Kac-Moody categorical actions in representation theory
Abstract: One of the central concepts in the representation theory of the symmetric group is that you can use induction and restriction functors to compare the symmetric
groups of different ranks; in the representation theory of Lie algebras (of type A), using projective functors to compare different blocks plays a similarly key role.
In fact, these two structures (and many generalizations) can be unified into the action of a single monoidal category, the Heisenberg category.
I'll discuss examples of these actions, and how they can be turned into a more refined and powerful object, a categorical Kac-Moody action.
|
| November 7th |
Ilya Dumanskiy, MIT |
A geometric approach to Feigin-Loktev fusion product
Feigin-Loktev fusion product between cyclic graded modules over the current algebra was introduced in 1998. Although its definition is elementary, very little is known about its properties. We introduce a way to study it geometrically. This approach also gives Borel-Weil type theorems for the Beilinson-Drinfeld Grassmannian and the global convolution diagram.
The talk is based on joint works with Evgeny Feigin and Michael Finkelberg.
|
| November 14th |
Christopher Ryba , UC Berkeley |
Stable Centres of Finite General Linear Groups
Abstract:
The Jucys-Murphy elements provide a powerful tool for understanding the representation theory of symmetric groups.
Using a classical result of Farahat and Higman, one can obtain particular character formulae for symmetric groups.
An analogous version of the Farahat-Higman algebra exists for general linear groups over a finite field
(joint work with Arun Kannan). Leveraging q-Schur-Weyl duality, one obtains character formulae just like in
the case of symmetric groups (this also elucidates the structure of the Faraht-Higman algebra).
Time permitting, we will discuss the existence of Jucys-Murphy elements for general linear groups over finite fields.
|
| Novermber 21st |
Ivan Loseu , Yale University |
Harish-Chandra modules over quantizations of nilpotent orbits
Abstract: Let O be a nilpotent orbit in a semisimple Lie algebra over
the complex numbers. Then it makes sense to talk about filtered
quantizations of O, these are certain associative algebras that
necessarily come with a preferred homomorphism from the universal
enveloping algebra. Assume that the codimension of the boundary of O
is at least 4, this is the case for all birationally rigid orbits (but
six in the exceptional type), for example. In my talk I will explain a
geometric classification of faithful irreducible Harish-Chandra
modules over quantizations of O, concentrating on the case of
canonical quantizations -- this gives rise to modules that could be
called unipotent. The talk is based on a joint paper with Shilin Yu
(in preparation).
|
| November 28th |
No seminar. |
|
| December 5th |
Dmitry Vaintrob, IHES |
The derived Duflo-Serganova functor in supergeometry and
homological localization
Abstract: I will talk about ongoing work with Vera Serganova and Alex
Sherman. I will define a derived version of the Duflo-Serganova
functor, which implies new invariants of certain equivariant sheaves
on non-affine algebraic varieties. Using this functor and the theory
of super-blowups, one obtains a surprisingly general class of
"supersymmetric localization" results, reducing computations on large
equivariant supervarieties to smaller closed subvarieties, extending a
result of Serganova and Sherman.
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