Harish-Chandra bimodules in complex rank

Abstract: The 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 cateogories. 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 the Deligne categories, it 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 the Deligne categories.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry, Physics, and Representation Theory Seminar

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