Date: Friday, February 25, 2022
Location: Online (1:00 PM to 2:00 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: Jenny Wilson jchw@umich.edu
