The index of representations associated with stabilisers

Let Q be an algebraic group with Lie algebra q and V a finite-dimensional Q-module. The index of V, denoted ind(q,V), is the minimal codimension of the Q-orbits in the dual space V∗. By Vinberg's inequality, ind(q,V∗)⩽ind(qv,∗(V/q⋅v)) for any v∈V. In this article, we study conditions that guarantee equality. In case of reductive group actions, we show that it suffices to test the nilpotent elements in V and all its slice representations. It was recently proved by J.-Y. Charbonnel that the equality for indices holds for the adjoint representation of a semisimple group. Another proof for the classical series was given by the second author. One of our goals is to understand what is going on in the case of isotropy representations of symmetric spaces.