Let $G$ be a semisimple group over $Q$ and $K$ a maximal compact subgroup of $G(R)^+$. I am assuming that $G(R)^+/K$ has a structure of a non-compact Hermitian symmetric domain. Let $g= p + k$ be corresponding Cartan decomposition of $g = Lie(G(R)^+)$.
Now let $F$ be a semisimple subgroup of $G$ (hence defined over $Q$ !) and let $K_F = K \cap F(R)^+$. Let $f = p_F + k_F$ be the Cartan decomposition of $Lie(F(R)^+)$ with $k_F = Lie(K_F)$.
I would like to know if it is always the case that $F(R)^+/K_F$ inherits a Hermitian symmetric structure. I understand that this comes down to showing that $p_F$ inherits a complex structure from that of $p$...
This may be naive or trivial but I am not familiar with this theory and literature..