ON RELATIVE COMPLETE REDUCIBILITY

The Quarterly Journal of Mathematics, vol. 71, pp. 321–334

Abstract

Abstract 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. Röhrle, 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. Röhrle, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z.269 (2011), no. 1, 809–832].

Authors 5

1. Christopher Attenborough

   University of York

   Affiliation as printed

   Department of Mathematics, University of York, York YO10 5DD, UK

2. Michael Bate

   University of York

   Affiliation as printed

   Department of Mathematics, University of York, York YO10 5DD, UK

3. Maike Gruchot Aachen Lehrstuhl D für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl D für Mathematik, RWTH Aachen University, D-52062 Aachen, Germany

4. Alastair Litterick

   University of Essex

   Affiliation as printed

   Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, UK

5. Gerhard Röhrle corresponding

   Ruhr University Bochum

   Affiliation as printed

   Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany