zbMATH — the first resource for mathematics

Fixed points of quasi-semi-simple automorphisms. (Points fixes des automorphismes quasi-semi-simples.) (French. Abridged English version) Zbl 1001.20043
Summary: Let \(\mathbf G\) be a connected algebraic reductive group over an algebraically closed field. An algebraic automorphism \(\sigma\) of \(\mathbf G\) is quasi-semi-simple if it stabilises a pair of a maximal torus of \(\mathbf G\) and a Borel subgroup of \(\mathbf G\) containing it; then the connected component \(({\mathbf G}^\sigma)^0\) of the fixed-point group \({\mathbf G}^\sigma\) is a reductive group. We give an explicit description of its root system which allows us, as promised [in F. Digne, J. Michel, Ann. Sci. Éc. Norm. Supér., IV. Sér. 27, No. 3, 345-406 (1994; Zbl 0846.20040), 1.15] to (belatedly) complete the proof which was left incomplete there.

20G15 Linear algebraic groups over arbitrary fields
20E36 Automorphisms of infinite groups
Full Text: DOI
[1] Bourbaki, N., Groupes et algèbres de Lie, chapters 4, 5 et 6, (1981), Masson
[2] Carter, R., Simple groups of Lie type, (1972), Wiley · Zbl 0248.20015
[3] Digne, F.; Michel, J., Groupes réductifs non connexes, Ann. E.N.S., 27, 345-406, (1994) · Zbl 0846.20040
[4] Springer, T.A., Linear algebraic groups, Prog. math., (1998), Birkhäuser · Zbl 0927.20024
[5] Steinberg, R., Endomorphisms of linear algebraic groups, Mem. amer. math. soc., 80, (1968) · Zbl 0164.02902
[6] Tits, J., Constantes de structure des algèbres de Lie semi-simples, Publ. math. IHéS, 31, 21-58, (1966)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.