Generic semistability for reductive group actions

Abstract:

We prove that given a reductive algebraic group $G$ and a rational representation $\rho : G \to \mathrm {GL}(V)$ defined over an algebraically closed field of characteristic $0$, $v \in V$ is generically semistable, i.e., $0 \not \in \overline {T.v}$ for a general maximal torus $T$ if and only if $v$ is semistable with respect to the induced action of the center of $G$. The proof is obtained through a detailed description of the relation between the state polytope with respect to the maximal torus $T$ of $G$ and the state polytope with respect to $T \cap [G,G]$. We also consider the case of solvable groups and prove that the generic semistability implies the center semistability but not the other way around.