Let $G$ be a compact connected Lie group equipped with an involution $i$. Let $H$ be the subgroup fixed by the involution. We denote by $G'$ and $H'$ the underlying reductive groups of $G$ and $H$ respectively. Now let $Z$ be a subset of the flag manifold formed of elements $x$ for which the stabilizer $S_x =\{g \in G \mid gx=x \}$ is stable under the involution. A result in
(T.MATSUKI, the orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J.Math.Soc .Japan 31 ).
tells us that the group $H$ has finitely many orbits in $Z$ and that the set of those orbits parametrizes the $H'$-orbits in the flag manifold. My question is about $H'$: is it fixed by an involution?? If Yes what is it, and how it is related with the given involution defined in $G$? Please help me to understand this.
Here is a general fact: Let $G$ be a Lie group, $G_{{\mathbb C}}$ be its complexification. Then every (Lie) automorphism $\tau$ of $G$ extends uniquely to a biholomorphic automorphism $\tau_{{\mathbb C}}$ of $G_{{\mathbb C}}$. Moreover, the automorphisms $\tau, \tau_{{\mathbb C}}$ have the same order.
This fact is an immediate consequence of the universal property of the complexification.
Thus, if $H<G$ is fixed by an involution $\tau$ of $G$ then $\tau$ extends to an involution $\tau_{{\mathbb C}}$ on $G_{{\mathbb C}}$. I claim that $$H_{{\mathbb C}}<G_{{\mathbb C}}$$ is fixed by $\tau_{{\mathbb C}}$. Indeed: Both $id$ and $\tau_{{\mathbb C}}$ extend the identity embedding $H\to G$ to an embedding $$ H_{{\mathbb C}}\to G_{{\mathbb C}}$$ By the uniqueness property of extensions (with respect to complexifications), the restrictions of $id$ and $\tau_{{\mathbb C}}$ to $H_{{\mathbb C}}$ have to coincide, i.e. $\tau_{{\mathbb C}}$ fixes $H_{{\mathbb C}}$.
With a bit more work (using the fact that $G$ and $H$ are algebraic), for compact groups $G$ one can also show that if $H$ is precisely the fixed point set of $\tau$ in $G$ then $H_{{\mathbb C}}$ is precisely the fixed point set of $\tau_{{\mathbb C}}$ in $G_{{\mathbb C}}$. Maybe this also holds for noncompact groups $G$, I am not entirely sure: It still holds if the fixed point set of $\tau_{{\mathbb C}}$ is connected.