Good index behaviour (based on a joint work with D. I. Panyushev)

Let $G$ be an algebraic group with Lie algebra $\mathfrak g$ and $V$ a $G$-module. The index of the representation $G:V$ is the minimal codimension of the $G$-orbits in the dual space $V^*$. Let $G_v$ be the stabiliser of $v\in V$ and $\mathfrak g\cdot v$ the tangent space to the orbit $Gv$. Say that $G:V$ has a good index behaviour (GIB) if the index of $G_v:V/(\mathfrak g\cdot v)$ equals the index of $G:V$ for each $v\in V$. In case of the (co)adjoint action of a reductive group condition of (GIB) is equivalent to the Elashvili conjecture on the index of centralises recently proved by Charbonnel. In general, the index of $G_v:V/(\mathfrak g\cdot v)$ is greater or equal than the index of $G:V$.
In this talk, we give several conditions, which are sufficient for (GIB). Then the isotropy representations of symmetric pairs are studied in details. It turns out, that they do not always have (GIB), so Elashvili conjecture cannnot be generalised to all symmetric spaces of reductive groups.