Classification of irreducible representations of affine group superschemes and the division superalgebras of their endomorphisms

TL;DR

This paper classifies irreducible representations of affine group superschemes over fields of characteristic not two, computes their endomorphism division superalgebras, and provides numerical results for quasi-reductive algebraic supergroups.

Contribution

It introduces a classification method for irreducible representations over separable closures and Galois twists, and computes associated division superalgebras, advancing understanding of supergroup representations.

Findings

Classification of irreducible representations over separable closures

Computation of division superalgebras of endomorphisms

Numerical results for quasi-reductive algebraic supergroups

Abstract

In this paper, we classify irreducible representations of affine group superschemes over fields $F$ of characteristic not two in terms of those over a separable closure $F^{\mathrm{sep}}$ and their Galois twists. We also compute the division superalgebras of their endomorphisms. Finally, we give numerical conclusions for quasi-reductive algebraic supergroups under certain conditions, based on Shibata's Borel--Weil theory for split quasi-reductive algebraic supergroups.