# Harish-Chandra bimodules in complex rank

February 16, 2022

Zoom seminar **Quadratic forms, Linear algebraic groups and Beyond**, 8:30-9:30am

https://uottawa-ca.zoom.us/j/99397061432?pwd=LzNBS1UrSVNJQ1ZiU28yWHAyNlV...

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 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.