Notations: • G : \underline{complex} s.s. \underline{algebraic} group;

• H ⊂ G : a \underline{spherical} subgroup (see below);

• B ⊂ G : a Borel subgroup s.t. BH ⊂ G dense;

• 𝔅_{H} ⊃ 𝔅_{H}(1) : set of B-orbits in G/H (resp. codim. 1-orbits);

• T ⊂ B : a maximal torus which we will fix;

• R ⊃ R^{+} : root system of (G, T) and +-ve system;

• X^{∗}(T) = Hom(T, 𝔾_{m}) and X_{∗}(T) = Hom(𝔾_{m}, T);

• α^{∨} ∈ X_{∗}(T) & 𝔾_{m}^{α} : coroot corr. to α ∈ R and its image in T;

• U_{α} : one-parameter unipotent subgroup corr. to α;

• W := N_{G}(T)/T : the Weyl group of G;

• P_{α} ⊂ G : min’l parabolic subgroup corr. to α ∈ R.