ON EXTENDED WEIGHT MONOIDS OF SPHERICAL HOMOGENEOUS SPACES

Given a connected reductive complex algebraic group G and a spherical subgroup H ⊂ G , the extended weight monoid Λ ̂ G + G / H encodes the G -module structures on spaces of global sections of all G -linearized line bundles on G / H . Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P ⊂ G , in this paper we obtain a description of Λ ̂ G + G / H via the set of simple spherical roots of G / H together with certain combinatorial data explicitly computed from the pair ( P ; H ). As an application, we deduce a new proof of a result of Avdeev and Gorfinkel describing Λ ̂ G + G / H in the case where H is strongly solvable.