PIRSA:22020052

Harish-Chandra bimodules in complex rank

Utiralova, A. (2022). Harish-Chandra bimodules in complex rank. Perimeter Institute. https://pirsa.org/22020052

Utiralova, Aleksandra. Harish-Chandra bimodules in complex rank. Perimeter Institute, Feb. 11, 2022, https://pirsa.org/22020052 

          @misc{ pirsa_PIRSA:22020052,
            doi = {10.48660/22020052},
            url = {https://pirsa.org/22020052},
            author = {Utiralova, Aleksandra},
            keywords = {Mathematical physics},
            language = {en},
            title = {Harish-Chandra bimodules in complex rank},
            publisher = {Perimeter Institute},
            year = {2022},
            month = {feb},
            note = {PIRSA:22020052 see, \url{https://pirsa.org}}
          }

Aleksandra Utiralova

Massachusetts Institute of Technology (MIT)

Talk number
PIRSA:22020052
Collection
Talk Type
Subject
Abstract

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 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 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 Deligne categories that 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 Deligne categories.

Zoom Link: https://pitp.zoom.us/j/93951304913?pwd=WVk1Uk54ODkyT3ZIT2ljdkwxc202Zz09