Harish-Chandra Bimodules in Complex Rank

Abstract

In this work we study Harish-Chandra bimodules in the setting of the Deligne categories Rep(Gₜ). We classify all pairs (χ, ψ) of central characters of U(gₜ) for which there exists a non-zero Harish-Chandra bimodule over gₜ, such that the two copies of the center Z(U(gₜ)) act via characters χ and ψ. Thus, we answer Question 3.25 posed in Pavel Etingof’s paper Representation Theory in Complex Rank II.

We also construct a family of Harish-Chandra bimodules that interpolate 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 Rep(Gₜ). The full classification of (simple) finite K-type bimodules is yet unknown.

This construction also yields some examples of central characters χ of the universal enveloping algebra U(gₜ) for which the quotient Uₓ is not simple, and, thereby, it allows us to partially solve Problem 3.23 posed in Representation Theory in Complex Rank II.