Relative complete reducibility and normalized subgroups
- We study a relative variant of Serre’s notion of \(\bf G\)-complete reducibility for a reductive algebraic group \(\bf G\). We let \(\bf K\) be a reductive subgroup of \(\bf G\), and consider subgroups of \(\bf G\) that normalize the identity component \(\bf K\)°. We show that such a subgroup is relatively \(\bf G\)-completely reducible with respect to \(\bf K\) if and only if its image in the automorphism group of \(\bf K\)∘ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of \(\bf G\), as well as "rational" versions over nonalgebraically closed fields.
| Author: | Maike Katharina GruchotORCiDGND, Alastair LitterickORCiDGND, Gerhard RöhrleORCiDGND |
|---|---|
| URN: | urn:nbn:de:hbz:294-78070 |
| DOI: | https://doi.org/10.1017/fms.2020.25 |
| Parent Title (English): | Forum of Mathematics, Sigma |
| Publisher: | Cambridge University Press |
| Place of publication: | Cambridge |
| Document Type: | Article |
| Language: | English |
| Date of Publication (online): | 2021/01/28 |
| Date of first Publication: | 2020/05/26 |
| Publishing Institution: | Ruhr-Universität Bochum, Universitätsbibliothek |
| Tag: | Open Access Fonds |
| Volume: | 8 |
| Issue: | Artikel e30 |
| First Page: | e30-1 |
| Last Page: | e30-32 |
| Note: | Article Processing Charge funded by the Deutsche Forschungsgemeinschaft (DFG) and the Open Access Publication Fund of Ruhr-Universität Bochum. |
| Dewey Decimal Classification: | Naturwissenschaften und Mathematik / Mathematik |
| open_access (DINI-Set): | open_access |
| faculties: | Fakultät für Mathematik |
| Licence (English): |  Creative Commons - CC BY 4.0 - Attribution 4.0 International |
