$\newcommand\fg{\mathfrak g}\newcommand\gl{\mathfrak{gl}}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field, and let $G$ be a smooth, affine algebraic group over $k$. In everything below, I intend to work on the level of algebraic groups, not just of rational points. I am also happy to restrict to the reductive case, and even, if it helps, to the adjoint case. However, I am specifically interested in the case of small, positive characteristic.


There are well defined notions of a Jordan decomposition in $G$ and in the Lie algebra $\fg$ of $G$, both of which involve identifying a semisimple part and a nilpotent part that commute.


On the dual Lie algebra $\fg^\*$, I don't know any reasonable notion of when two elements commute, but one can still imagine defining at least the conjugacy class of the semisimple part, as follows: let $X^\*$ be an element of $\fg^\*$, and consider the coadjoint orbit $\mathscr O = \Ad^\*(G)X^\*$. Are the following two statements true?


1. The closure of $\mathscr O$ contains a unique closed coadjoint orbit $\mathscr S$.
2. For every closed coadjoint orbit $\mathscr S$, there are a maximal torus $T$ in $G$ and an element $S^\* \in \mathscr S$ such that $S^\*$ vanishes on all root subalgebras of $T$ in $\fg$.


If one or more is not true, then is there any other way of coming up with a reasonable notion at least of the conjugacy class of the semisimple part in the Jordan decomposition? I would prefer such a decomposition to satisfy two properties:


a. Every $G$-invariant isomorphism $\fg^\* \to \fg$ (e.g., the Killing form, when it is non-degenerate), if any, should commute with the map "taking the orbit of the semisimple part".


b. For every faithful representation $G \to \GL(V)$, we obtain $\gl(V) \cong \gl(V)^\* \to \fg^\*$, where the first isomorphism is *via* the trace pairing. For every $X \in \gl(V)$, the semisimple part of $X$ is carried into the orbit of the semisimple part of its image.



Jordan decomposition on the dual Lie algebra
With a few restrictions ($p \ne 2$ and $G$ has no components of special orthogonal type), my (1) and (2) are Theorem 4, parts (iv) and (ii), of [Kac and Weisfeiler - Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic $p$](https://doi.org/10.1016/1385-7258(76)90059-7).


