Seminar Event Detail

Representation Stability

Date:  Friday, February 25, 2022
Location:  Online (1:00 PM to 2:00 PM)

Title:  Harish-Chandra 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 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


Speaker:  Alexandra Utiralova
Institution:  MIT

Event Organizer:   Jenny Wilson


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