I'm trying to understand one example of a symmetric space in Postnikov's Riemannian Geometry book but I'm unsure if I'm messing up one of the identifications, or just the algebra. Here's the setup:
A symmetric space is a manifold $M$ equipped with a smooth family $(s_x)_{x \in M}$ of smooth involutions such that each $x$ is an isolated fixed point for $s_x$, and the relation $s_x\circ s_y \circ s_x = s_{s_xy}$ holds for all $x,y \in M$.
Fact: every Lie group $G$ is a symmetric space in the above sense, with $(s_a^G)_{a \in G}$ given by $s_a(x) = ax^{-1}a$.
Ok, great, now let $\sigma$ be an involutive automorphism of $G$, so that ${\rm Fix}(\sigma)$ is a closed subgroup of $G$ and the quotient $G/{\rm Fix}(\sigma)$ makes sense (as a smooth manifold, not a Lie group --- ${\rm Fix}(\sigma)$ is not normal unless $\sigma$ is trivial). Also consider $G_\sigma = \{g\sigma(g)^{-1} \mid g \in G\}$, and note that there is a bijection $$G/{\rm Fix}(\sigma) \ni g\,{\rm Fix}(\sigma) \mapsto g\sigma(g)^{-1} \in G_\sigma. \tag{$\ast$}$$He claims that $G_\sigma$ is a symmetric space with $$s_{a\sigma(a)^{-1}}^{G_\sigma}(x\sigma(x)^{-1}) = a\sigma(a^{-1}x) \sigma(a\sigma(a^{-1}x))^{-1}.\tag{$\dagger$}$$I was trying to figure out how to obtain such formula (instead of just checking it works) as follows:
- First, I have checked that if $a,x \in G$ and $b,y \in {\rm Fix}(\sigma)$, then $\sigma(s^G_a(x)) = \sigma(s^G_{ab}(xy))$.
- By the above, I can define $s^{G/{\rm Fix}(\sigma)}_{a\,{\rm Fix}(\sigma)}(x\,{\rm Fix}(\sigma)) = s^G_a(x)\,{\rm Fix}(\sigma)$.
- Transfer these involutions to $G_\sigma$ using the bijection $(\ast)$.
This recipe would give me $$s^{G_\sigma}_{a\sigma(a)^{-1}}(x\sigma(x)^{-1}) = s_a^G(x) \sigma(s_a^G(x))^{-1} \tag{$\ddagger$} .$$It is not clear to me that $(\dagger)$ and $(\ddagger)$ agree. Completely unfolding everything, we get that $(\dagger)$ becomes $$a\sigma(a)^{-1}\sigma(x)x^{-1}a\sigma(a)^{-1},$$while $(\ddagger)$ becomes $$ax^{-1}a\sigma(a)^{-1}\sigma(x) \sigma(a)^{-1}.$$
How are the two proposed symmetric structures related? Or if not related, what was my mistake here? The author says that the $s^{G_\sigma}$'s are just the restrictions of the $s^G$'s to $G_\sigma$, but passing the $s^G$'s to the quotient and the bijection $(\ast)$ are all natural procedures, so the results should have been equal.