  Let $Q$ be an algebraic group with $q=\Lie Q$ 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 a $Q_v$-module $V/q.v$ for $v\in V$. A pair $(Q,V)$ is said to
have GIB if Vinberg's inequality turns into an equality for all $v\in V$. In
this article, we are interested in the GIB property of
$\theta$-representations, where $\theta$ is a finite order automorphism of a
simple Lie algebra $g$. An automorphism of order $m$ defines a $Z/mZ$-grading
$g=g_0+g_1+...+g_{m-1}$. If $G_0$ is the identity component of $G^\theta$, then
it acts on $\gt g_1$ and this action is called a $\theta$-representation. We
classify inner automorphisms of $gl_n$ and all finite order autmorphisms of the
exceptional Lie algebras such that $(G_0,g_1)$ has GIB and $g_1$ contains a
semisimple element.
