Date: Monday, January 10, 2022
Location: Virtual (4:00 PM to 5:00 PM)
Title: HarishChandra bimodules in complex rank
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 representationtheoretic 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 HarishChandra bimodules in Deligne categories. It is known that in the classical case simple HarishChandra bimodules admit a classification in terms of Worbits of certain pairs of weights. However, the notion of weight is not welldefined in the setting of Deligne categories. I will explain how in complex rank the abovementioned 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 HarishChandra bimodules. That is, one can either require that the center acts locally finitely on a bimodule M or that M has a finite Ktype. The two conditions are known to be equivalent for a semisimple 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 HarishChandra bimodules of finite Ktype using the ultraproduct realization of Deligne categories.
Files:
Speaker: Alexandra Utiralova
Institution: MIT
Event Organizer: Kartik Prasanna
