Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups

Multiplicity-free Representations and Visible Actions on Complex Manifolds

“…If an irreducible rational representation π of G C contains a P C -invariant line, π is a pan representation in the sense of Definition 6.2.1, and its restriction to H C is multiplicity-free by Theorem 26 in Section 6. Then, it follows from Vinberg and Kimelfeld [70,Corollary 1] …”

Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs

“…⊓ ⊔ 8.10 Restriction U (p, q) ↓ U (p − 1, q) and SO(n, 2) ↓ SO(n − 1, 2) Suppose (G, H) is a reductive symmetric pair whose complexification (g C , h C ) is one of the following types: (sl(n, C), gl(n − 1, C)) (or (gl(n, C), gl(1, C) + gl(n − 1, C))), (so(n, C), so(n − 1, C)). As is classically known (see [83]), for compact (G, H) such as (U (n), U (1) × U (n − 1)) or (SO(n), SO(n − 1)), any irreducible finite dimensional representation π of G is multiplicity-free when restricted to H. For non-compact (G, H) such as (U (p, q), U (1) × U (p − 1, q)) or (SO(n, 2), SO(n − 1, 2)), an analogous theorem still holds for highest weight representations π: Theorem 8. 10.…”

Spherical orbits and representations of Uε(g)

“…Indeed, this follows from [36 consists exactly of the self-dual representations of G with highest weights in the root lattice, each occurring once (see [35, p. 229]). In this case the corresponding unipotent conjugacy class Ø in G is spherical ( [6], [44]) and, by Remark 2.31, rk(1 − z(Ø)) = rk(1 − w 0 ). It follows from the proofs of Theorems 2.11 and 2.12 that z(Ø) = w 0 (cf.…”