Generic semistability for reductive group actions

Abstract

We prove that given a reductive algebraic group G G and a rational representation ρ : G → G L ( V ) \rho : G \to \mathrm {GL}(V) defined over an algebraically closed field of characteristic 0 0 , v ∈ V v \in V is generically semistable, i.e., 0 ∉ T . v ¯ 0 \not \in \overline {T.v} for a general maximal torus T T if and only if v v is semistable with respect to the induced action of the center of G G . The proof is obtained through a detailed description of the relation between the state polytope with respect to the maximal torus T T of G G and the state polytope with respect to T ∩ [ G , G ] 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.