Burnside rings for Real 2-representation theory: The linear theory

Abstract

This paper is a fundamental study of the Real [Formula: see text]-representation theory of [Formula: see text]-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a [Formula: see text]-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real [Formula: see text]-representations as a Real variant of the Burnside ring of the fundamental group of the [Formula: see text]-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to [Formula: see text]-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real [Formula: see text]-representation theory of [Formula: see text]-groups, as studied by the second author.