
Journal of Lie Theory 23 (2013), No. 1, 257311 Copyright Heldermann Verlag 2013 Restricted Roots and Restricted Form of the Weyl Dimension Formula for Spherical Varieties Simon Gindikin Dept. of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 088548019, U.S.A. gindikin@math.rutgers.edu Roe Goodman Dept. of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 088548019, U.S.A. goodman@math.rutgers.edu 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. Keywords: Spherical variety, symmetric space, restricted root system, Weyl dimension formula. MSC: 14M27; 17B10, 20G20, 22E46 [ Fulltextpdf (757 KB)] for subscribers only. 