Let $X = \text{SL}(n,\mathbb{R})/\text{SO}(n)$ be the symmetric space associated to $G = \text{SL}(n,\mathbb{R})$.
Every element $\phi \in \text{SL}(n,\mathbb{R})$ acts as an isometry on $X$ and hence we can consider its displacement function $$ d_\phi \colon X \to \mathbb{R}, \quad x \mapsto d(\phi \cdot x,x) $$ and the possibly empty subspace $\text{Min}(\phi)$ of $X$ where the minimum of $d_\phi$ is attained. This is a symmetric subspaces of $X$ via the inclusion $$ G(\phi)\Big/\Bigl(G(\phi) \cap \text{SO}(n)\Bigr) \to \text{SL}(n,\mathbb{R})\big/\text{SO}(n), $$ where $G(\phi) = \{ g \in \text{SL}(n,\mathbb{R}) : g \text{Min}(\phi) = \text{Min}(\phi) \}$.
Question: I'm trying to understand $G(\phi)$ in the case that $\phi$ has $n$ distinct real eigenvalues. It is clear to me that the centralizer $Z(\phi)$ of $\phi$ in $\text{SL}(n,\mathbb{R})$ is contained in $G(\phi)$. I read that $$ G(\phi) \cong Z(\phi) \rtimes \text{Sym}(n), $$ where the symmetric group permutes the eigenspaces of $\phi$. However, I do not understand what the isomorphism should be and how an element of $G(\phi)$ is related to (coming from) an element of $Z(\phi)$.
The first thing to prove is that the min-set of such $\phi$ is a maximal flat in $X$. (A proof takes some effort, the key fact is that if a geodesic in $X$ is parallel to a regular geodesic in a maximal flat, then it is contained in this flat.) If you think of $X$ as the space of positive definite matrices with unit determinant and $\phi$ a diagonal matrix, then the minset of $\phi$ is the intersection of $X$ with the set of diagonal matrices. Can you finish given this information?
I suggest you take a look in Eberlein's book "Manifolds of nonpositive curvature", Chicago University Press, 1996.