The UC Berkeley Representation theory and tensor categories seminar |
---|

DATE |
SPEAKER |
TITLE (click to show abstract) |

September 12th | Alexandra Utiralova, UC Berkeley |
## Harish-Chandra bimodules in complex rankAbstract: 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 calculusAbstract: 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 algebrasAbstract: 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 superalgebrasAbstract: 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 categoryAbstract: 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 categoriesAbstract:``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 theoryAbstract: 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 productFeigin-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 GroupsAbstract: 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 orbitsAbstract: 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 localizationAbstract: 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. |