Let G be a connected reductive linear algebraic group over a field k. Using ideas from geometric invariant theory, we study the notion of G-complete reducibility over k for a Lie subalgebra h of the Lie algebra g = Lie(G) of G and prove some results when h is solvable or char(k) = 0. We introduce the concept of a k-semisimplification h ′ of h; h ′ is a Lie subalgebra of g associated to h which is G-completely reducible over k. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that h ′ is unique up to Ad(G(k))-conjugacy in g. Moreover, we prove that the two concepts are compatible: for H a closed subgroup of G and H′ a k-semisimplification of H, the Lie algebra Lie(H′ ) is a k-semisimplification of Lie(H).
Bibliographical noteAcknowledgments: The research of this work was supported in part by the DFG (Grant #RO 1072/22-1 (project number: 498503969) to G. R¨ohrle). We thank the referees for their careful reading of the paper and their helpful comments clarifying some points. We are especially grateful to one referee for providing us with Proposition 4.4, which is a substantial improvement on a result from the original version.
Open Access funding enabled and organized by Projekt DEAL.
- G-complete reducibility
- geometric invariant theory
- cocharacter-closed orbits
- degeneration of G-orbits