Semisimplifications and representations of the General Linear Supergroup

Thorsten Heidersdorf; Rainer Weissauer

paper

Semisimplifications and representations of the General Linear Supergroup

arXiv:2501.05298

Abstract

We study the semisimplification of the full karoubian subcategory generated by the irreducible finite dimensional representations of the algebraic supergroup $G L (m ∣ n)$ over an algebraically closed field of characteristic zero. This semisimplification is equivalent to the representations of a pro-reductive group $H_{m ∣ n}$ . We show that there is a canonical decomposition $H_{m ∣ n} ≅ G L (m - n) \times H_{n ∣ n}$ , thereby reducing the determination of $H_{m ∣ n}$ to the equal rank case $m = n$ which was treated in a previous paper.

v4: Minor changes in Section 6 (reordering); v3: Added an appendix ("A variant of the $I^\cdot$ functor"); v2: very minor changes (typos etc)