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..


If I understood your question correctly: it is not always that $F({\mathbb R})/K_F$ inherits a Hermitian structure.

For example, take $G:=\text{PU}(n,1)$, $K:=\text{U}(n)$, and $F:=\text{PO}(n,1)$. Then $G/K$ is the holomorphic $n$-ball in ${\mathbb C}^n$ equipped with its "complex hyperbolic" Hermitian structure. In its turn, $F/K_F$ is the usual $n$-dimensional real hyperbolic space (a space form). If $n$ is odd, it possesses no Hermitian structure.


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