In this chapter we treat Mumford–Tate domains in an axiomatic fashion. Historically, the concept of “Shimura domain” arose first. Such domains have a representation as bounded symmetric domains. The tautological variation of Hodge flag over such a domain satisfies Griffiths’ transversality, i.e.,we have a variation of Hodge structure over it. This characterizes them among Mumford–Tate domains and makes them easier to study directly from a Hodge theoretic perspective, which we do in Section 16.1. To make the transition to the axiomatic treatment, one views a Mumford–Tate domain as an entire conjugacy class of a given Hodge structure. To get a polarizable Hodge structure, the connected group of automorphisms of the domain must be a reductive group of Hodge type and, conversely, these are the groups that act transitively on Mumford–Tate domains. We explain this in Section 16.2. In Section 16.3 we give the promised axiomatic treatment of Mumford–Tate varieties parallel to Deligne's axiomatic treatment of Shimura varieties. The chapter ends with Section 16.4 where we give examples of Hodge structures given by representations of the classical simple groups.

Shimura Domains

Basic Properties and Classification

Definition 16.1.1 A Shimura domain is a Mumford–Tate subdomain D of some period domain such that the tautological Hodge flag restricts to a variation of Hodge structure on D.

Remark It may very well happen that the same Shimura domain can be embedded in different period domains and so the tautological variation can a priori be different. However, there is a way to give a more abstract treatment of Shimura domains which starts with the algebraic group under which the domain is homogeneous. See Proposition 16.3.3.

Our main result concerning Shimura domains is Proposition 16.1.9. Before we state it, let us first give some examples.