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In this paper we study the category of Harish-Chandra bimodules$HC_{\chi,\psi}$ in the Deligne category $\text{Rep}(GL_t)$. In particular, weanswer Question 3.25 posed in Pavel Etingof's paper arXiv:1407.0373 anddetermine for which central characters $\chi$ and $\psi$ this category is notzero.

Harish-Chandra bimodules in the Deligne category $\text{Rep}(GL_t)$