On the CR-structure of compact group orbits associated with bounded symmetric domains

Abstract

Let E be a complex vector space of finite dimension and let K ⊂ GL(E) be a compact connected subgroup. Then for fixed a ∈ E the orbit K := K(a) is a real-analytic submanifold of E that inherits various structures from E. For instance, choosing a K-invariant positive definite inner product (x|y) on E makes K a Riemannian manifold on which K acts transitively by isometries. On the other hand, K inherits from E a Cauchy–Riemann structure (CR-structure), that is given by the distribution of the maximal complex subspaces Hx K := Tx K ∩ iTx K of the real tangent spaces Tx K ⊂ E, x ∈ K , together with the complex structure on every Hx K (multiplication by i). The subspace Hx K is called the holomorphic tangent space to K at x (see [9] and [4] as general references for CR-manifolds). Of interest for the geometry of the orbit K = K(a) with respect to its CR-structure is the study of the CR-functions (or more generally CR-mappings) on K , i.e. of smooth functions f : K → C that satisfy the tangential Cauchy–Riemann differential equations in the sense that the restriction of the differential df to every holomorphic tangent space is complex linear. For instance, all holomorphic functions defined in an open neighbourhood of K ⊂ E give by restriction real-analytic CR-functions on K . Actually, we deal with the more general continuous CR-functions on K (which satisfy by definition the tangential CR-equations in the distribution sense, or equivalently, which are locally uniform limits of sequences of smooth CR-functions due to the approximation theorem of BaouendiTreves [6]). In this context it is of interest to determine the space of all