Complete reducibility for Lie subalgebras and semisimplification

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 $\mathfrak{h}$ of the Lie algebra $\mathfrak{g}=Lie(G)$ of $G$ and prove some results when $\mathfrak{h}$ is solvable or $char(k)=0$. We introduce the concept of a $k$-semisimplification ${\mathfrak{h}}^{\mathrm{\prime }}$ of $\mathfrak{h}$; ${\mathfrak{h}}^{\mathrm{\prime }}$ is a Lie subalgebra of $\mathfrak{g}$ associated to $\mathfrak{h}$ 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 ${\mathfrak{h}}^{\mathrm{\prime }}$ is unique up to $Ad(G(k))$-conjugacy in $\mathfrak{g}$. Moreover, we prove that the two concepts are compatible: for $H$ a closed subgroup of $G$ and $H^{\mathrm{\prime }}$ a $k$-semisimplification of $H$, the Lie algebra $Lie(H^{\mathrm{\prime }})$ is a $k$-semisimplification of $Lie(H)$.