Why is hermitian symmetric domain simply connected?

Question

In the proof of thereom 1.9 in Milne's note of Shimura varieties, he uses prop 1.14 but does not give a reference for the proof that any hermitian symmetric domain is simply connected, where could I find one?

Answer 1

In fact, hermitian symmetric spaces (simple, of non-compact type) have Harish-Chandra models which are convex open subsets of some $\mathbb C^n$.

These can be made explicit and elementary (as Siegel did, also see Piatetski-Shapiro's book) for the classical groups and domains.

For compact type, I do not know any comparably simple blanket assertion, although in the classical cases one can do it case-by-case.