I'm reading Milne's Introduction to Shimura varieties (http://www.jmilne.org/math/xnotes/svi.pdf) and there is something I don't get.

Let $G$ be a connected semisimple algebraic group $G$ over $\mathbb{Q}$ and let $D$ be an hermitian symmetric domain such that $G(\mathbb{R})^+$ (the $^+$ denote the identity component) acts on $D$ via a surjective homomorphism $Hol(D)^+$ with compact kernel.

Let $G^{ad}_\mathbb{R} \simeq H_1 \times \ldots \times H_r$ be the decomposition into $\mathbb{R}$-simple factors. Denote by $H_{c}$ the product of $H_i$'s such that $H_i(\mathbb{R})$ is compact. Denote by $H_{nc}$ the product of the others (which we assume is not empty !).

Do we have an isomorphism $H_{nc}(\mathbb{R})^+ \simeq Hol(D)^+$ ?