I'm interested in understanding the following lemma, which Vogan states (Lemma 4.8) in his paper on the Local Langlands Conjectures (omitting the "well-known" proof).
Suppose $G$ is a complex semisimple algebraic group, and $s$ is a semisimple automorphism of $G$. Let $H$ be the fixed points of $s$ (a reductive subgroup of $G$). For each complex number $t$ that is not a root of unity, define $$\mathfrak{g}_t = \{X\in\mathfrak{g} | s(X) = tX\}.$$ Then $H$ acts on the vector space $\mathfrak{g}_t$ with finitely many orbits.
Why is this true? The proof is not well-known to me or the handful of people I asked.
