Harish-Chandra bimodules of finite $K$-type in Deligne categories

TL;DR

This paper extends the theory of Harish-Chandra bimodules within Deligne categories, constructing new bimodules with finite $K$-type and providing insights into their classification and associated algebraic structures.

Contribution

It introduces a family of Harish-Chandra bimodules with finite $K$-type in Deligne categories, generalizing classical simple bimodules and addressing questions about universal enveloping algebra quotients.

Findings

Constructed new Harish-Chandra bimodules with finite $K$-type.

Provided examples of non-simple quotients of universal enveloping algebras.

Partially answered a question by Pavel Etingof regarding algebraic structures.

Abstract

We continue the study of Harish-Chandra bimodules in the setting of the Deligne categories $\mathrm{Rep}(G_t)$, that was started in the previous work of the first author (arXiv:2002.01555). In this work we construct a family of Harish-Chandra bimodules that generalize simple finite dimensional bimodules in the classical case. It turns out that they have finite $K$-type, which is a non-vacuous condition for the Harish-Chandra bimodules in $\mathrm{Rep}(G_t)$. The full classification of (simple) finite $K$-type bimodules is yet unknown. This construction also yields some examples of central characters $\chi$ of the universal enveloping algebra $U(\mathfrak{g}_t)$ for which the quotient $U_\chi$ is not simple, and, thereby, it allows us to partially solve a question posed by Pavel Etingof in one of his works.