Harish–Chandra Bimodules of Finite K-type in Deligne Categories

Abstract

Abstract We continue the study of Harish–Chandra bimodules in the setting of the Deligne categories $\operatorname {Rep}(G_{t})$ that we started in [17]. 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 $\operatorname {Rep}(G_{t})$. 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 Problem 3.23 posed in [10].