Good index behaviour of $\theta$-representations, I

Let $Q$ be an algebraic group with $q=\text{Lie}Q$ and $V$ a $Q$-module. The index of $V$ is the minimal codimension of the $Q$-orbits in the dual space $V^{\ast }$. There is a general inequality, due to Vinberg, relating the index of $V$ and the index of a $Q_v$-module $V\mathrm{/}q\mathrm{.}v$ 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 $Z\mathrm{/}mZ$-grading $g=g_0+g_1+\mathrm{.}\mathrm{.}\mathrm{.}+g_{m-1}$. If $G_0$ is the identity component of $G^{\theta }$, then it acts on $>g_1$ and this action is called a $\theta$-representation. We classify inner automorphisms of $g{l}_n$ 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.