Complete reducibility: variations on a theme of Serre

manuscripta mathematica, vol. 168, pp. 439–451

Abstract

Abstract In this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and $$\sigma $$ σ -complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

Authors 3

1. Maike Gruchot Aachen Lehrstuhl für Algebra und Zahlentheorie

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen University, Pontdriesch 14/16, 52062, Aachen, Germany

   Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen University, Aachen, Germany

2. Alastair Litterick

   University of Essex

   Affiliation as printed

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

   Department of Mathematical Sciences, University of Essex, Colchester, Essex, UK

3. Gerhard Röhrle corresponding

   Ruhr University Bochum

   Affiliation as printed

   Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780, Bochum, Germany

   FakultÄt für Mathematik, Ruhr UniversitÄt Bochum, Bochum, Germany