The cofree representation characterization conjecture for connected reductive groups

Let $V$ be a stable representation of a connected reductive group $G$. Recall that $V$ is pure when its strictly semistable locus $V^{\operatorname{sss}}$ has pure codimension one, npure when it is pure and every irreducible component of $V^{\operatorname{sss}}$ maps to a divisor under the quotient map, and cnpure when it is npure and every irreducible component maps to a principal divisor. A representation is cofree when its invariant ring is polynomial and the quotient map has the expected properties.

The cofree representation characterization conjecture. \begin{enumerate} \item If $V$ is cofree, then it is cnpure. \item If $G$ is semisimple and $V$ is irreducible, then purity of $V$ implies that $V$ is cofree. \end{enumerate}

The conjecture characterizes cofree representations through the geometry of the strictly semistable locus. The paper proves it for irreducible representations of simple Lie groups and establishes the corresponding equivalence for stable torus representations; the general connected reductive case remains open.