Symmetric subgroups of rational groups of hermitian type

A rational group of hermitian type is a Q-simple algebraic group G such that the symmetric space D of maximal compact subgroups of the real Lie group G(R) is a hermitian symmetric space of the non-compact type. There are two major classes of subgroups of G of importance to the geometry of D and to arithmetic quotients XΓ = Γ\D of D: parabolic subgroups and reductive subgroups. The former are connected with the boundary of the domain D, while the latter are connected with submanifolds D′ of the interior of D, a priori just symmetric spaces. Under certain assumptions on the reductive subgroup, D′ ⊂ D is a holomorphic symmetric embedding, displaying D′ as a sub-hermitian symmetric space. It is these latter groups we study in this paper, and we call them quite generally “symmetric subgroups”; these are the subgroups occuring in the title. The purpose of this paper is to prove an existence result of the following kind: given a maximal Q-parabolic P ⊂ G, which is the stabilizer of a boundary component F , meaning that P (R) = N(F ), there exists a Q-subgroup N ⊂ G, which is a symmetric subgroup defining a subdomain DN ⊂ D such that: F is a boundary component of DN . To formulate this precisely, we make the following assumptions on G which are to be in effect throughout the paper: G is isotropic, and G(R) is not a product of groups of type SL2(R). The basis of our formulation is the notion of incident parabolic and symmetric subgroups. We first define this for the real groups. Fix a maximal R-parabolic P ⊂ G(R), and let as above F ⊂ D∗ denote the corresponding boundary component stabilized by P . First assume that F is positive-dimensional. Let N ⊂ G(R) be a symmetric subgroup; we shall say N and P are incident, if the following conditions are satisfied.