G-complete reducibility in non-connected groups

Abstract

In this paper we present an algorithm for determining whether a subgroup $H$ of a non-connected reductive group $G$ is $G$-completely reducible. The algorithm consists of a series of reductions; at each step, we perform operations involving connected groups, such as checking whether a certain subgroup of $G^0$ is $G^0$-cr. This essentially reduces the problem of determining $G$-complete reducibility to the connected case.