On the Signature of Certain Spherical Representations

Let G be a real semisimple Lie group. Let g0 be the Lie algebra of G. We will denote the complexification of any vector space V0 by V and its dual by V ∗. Let g0 = k0 + p0 be a fixed Cartan decomposition corresponding to the Cartan involution θ of g0. Let K be the corresponding maximal compact subgroup of G. Let us fix T ⊂ K a maximal torus. Define S ⊂ t∗ as the set of weights of T that are sums of distinct non-compact roots. We say that (μ, Vμ) an irreducible representation of K is unitarily small if the weights of μ lie in the convex hull of S. Let us state the following: