Alastair Litterick

Let H ⊆ G be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic p > 0. In our first main theorem we show that if a closed subgroup K of H is H-completely reducible, then it is also G-completely reducible in the sense of Serre, under some restrictions on p, generalising the known case for G = GL(V). Our proof uses R. W. Richardson's notion of reductive pairs to reduce to the GL(V) case. We study Serre's notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if K is H-completely reducible, then the saturation of K in G is completely reducible in the saturation of H in G (which is again a connected reductive subgroup of G), under suitable restrictions on p, again generalising the known instance for G = GL(V). We also study saturation of finite subgroups of Lie type in G. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in the case G = GL(V).

We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral subgroups, we give an effective classification algorithm. For non-toral elementary abelian subgroups, we focus on algebraic groups of exceptional type with a view to future applications, and in this case we provide tables explicitly describing the subgroups and their local structure. We then describe how to transfer results to the corresponding finite groups of Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming work to complete the classification of elementary abelian p p -subgroups for torsion primes p p in finite groups of exceptional Lie type. Such classification results are important for determining the maximal p p -local subgroups and p p -radical subgroups, both of which play a crucial role in modular representation theory.

Let K be a reductive subgroup of a reductive group G over an algebraically closed field k. The notion of relative complete reducibility, introduced in [M. Bate, B. Martin, G. Rohde, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z. 269 (2011), no. 1, 809-832], gives a purely algebraic description of the closed K-orbits in G(n), where K acts by simultaneous conjugation on n-tuples of elements from G. This extends work of Richardson and is also a natural generalization of Serre's notion of G-complete reducibility. In this paper we revisit this idea, giving a characterization of relative G-complete reducibility, which directly generalizes equivalent formulations of G-complete reducibility. If the ambient group G is a general linear group, this characterization yields representation-theoretic criteria. Along the way, we extend and generalize several results from [M. Bate, B. Martin, G. Rohde, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z. 269 (2011), no. 1, 809-832].