Nikolaus Conference 2022

The notions of \(G\)-complete reducibility and \(G\)-irreducibility for an algebraic group \(H\) inside a reductive group \(G\) were introduced by J.P. Serre. They generalize the concepts of complete reducibility, respectively irreducibility, from representation theory. For example, in representation theory an \(H\)-module \(V\) is said to be irreducible if \(V\) does not contain any non-trivial \(H\)-stable subspace. In the more general setting, we pass from homomorphisms \(H\to GL(V)\) to homomorphisms \(H\to G\) for an arbitrary reductive group \(G\). A subgroup \(H\) of \(G\) is called \(G\)-irreducible if it is not contained in any proper parabolic subgroup of \(G\). For \(G=GL(V)\) we have that \(H\) is \(GL(V)\)-irreducible if and only if \(V\) is an irreducible \(H\)-module. Moreover, there is also a group intrinsic view of complete reducibility generalizing to the definition of \(G\)-complete reducibility. Therefore, a subgroup \(H\) in \(GL(V)\) is \(GL(V)\)-complete reducibility if and only if \(V\) is a completely reducible \(H\)-module. This short talk states the main definitions and facts from the general structure theory of algebraic groups needed here and introduces Serre's concepts.