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 $H\_{nc}(\mathbb{R})^+ \simeq Hol(D)^+$ ?



surjective homomorphism with compact kernel (Milne's note on Shimura varieties)
You will need to add a condition that $D$ has noncompact type.


Since we have a surjective homomorphism $G(\mathbb{R})^+ \to Hol(D)^+$ with compact kernel, it suffices to show that $Hol(D)^+$ has trivial centre (so this homomorphism factors through $H\_{nc}(\mathbb{R})^+$) and that $Hol(D)^+$ has no compact factors. The second part is essentially the definition of $D$ having noncompact type.


I could not find the fact that $Hol(D)^+$ has trivial centre explicitly stated in Helgason (but I might not have looked hard enough). It is part of Theorem 8.7.9 in Wolf, with the following proof:


Let $K \subset Hol(D)^+$ be the stabiliser of a point $p \in D$ and let $u\_p \colon U\_1 \to Hol(D)^+$ be the homomorphism from Theorem 1.9 of Milne. The first paragraph of the proof of Milne's Theorem 1.21 shows that $Lie(K)$ is the subspace of $Lie(Hol(D)^+)$ on which $u\_p(-1)$ acts trivially.


Let $L \subset Hol(D)^+$ be the centraliser of $u\_p$.
The uniqueness of $u\_p$ implies that $K \subset L$.
Since $u\_p(-1)$ acts trivially on $Lie(L)$, the previous paragraph implies that $L^+ = K^+$.
Since $L$ is the centraliser of a torus, it is connected, and so $L = K$.


The centre of $Hol(D)^+$ is contained in $L$, and so in $K$. But the intersection of $K$ with the centre of $Hol(D)^+$ is trivial by Helgason, chapter IV, Theorem 3.3(ii). So the centre of $Hol(D)^+$ is trivial.


