Restricted Roots and Restricted Form of Weyl Dimension Formula for Spherical Varieties

Abstract

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.