Complete reducibility for Lie subalgebras and semisimplification

This is the authors' abstract. We don't add key points for this paper.

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 h of the Lie algebra g = Lie (G) g = Lie ( G ) of G and prove some results when h h is solvable or char0.55542pt(k)= 0 char ( k ) = 0 . We introduce the concept of a k - semisimplification h' h ′ of h h ; h' h ′ is a Lie subalgebra of g g associated to h 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' h ′ is unique up to Ad0.55542pt(G(k)) Ad ( G ( k ) ) -conjugacy in g g . Moreover, we prove that the two concepts are compatible: for H a closed subgroup of G and $$H'$$ H ′ a k -semisimplification of H , the Lie algebra Lie0.55542pt(H') Lie ( H ′ ) is a k -semisimplification of Lie0.55542pt(H) Lie ( H ) .