We study in this paper the restricted roots for a class of spherical homogeneous spaces of semisimple groups which includes simply connected symmetric spaces. For these spaces we give a detailed description (case by case) of the set of roots of the group associated with each restricted root of the space (the nest of the restricted root). As an application, we obtain a refinement of the Weyl dimension formula in the case of spherical representations, expressing the dimension as a product over the set of indivisible positive restricted roots.

48 pages, 26 figures. This abridged version contains the main theorems and the statements of the propositions and lemmas (without proofs), together with the root nest calculations used to obtain the main theorems. The complete details are in the paper in Journal of Lie Theory