  Let $Q$ be an algebraic group and $V$ a $Q$-module. The index of $V$ is the
minimal codimension of the $Q$-orbits in the dual space $V^*$. There is a
general inequality, due to Vinberg, relating the index of $V$ and the index of
$Q_v$-module $V/q.v$ for any $v\in V$. In this article, we study conditions
that guarantee us the equality. It was recently proved by Charbonnel (Bull.
Soc. Math. France. v.132, 2004) that such an equality holds for the adjoint
representation of a semisimple group. Another proof for the classical series
was given by the second author, see math.RT/0407065. One of our goals, which is
almost achieved, is to understand what is going on in the case of isotropy
representations of symmetric spaces.
