The index of representations associated with stabilisers

Abstract

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.