International Journal of Mathematics

Kobayashi's multiplicity-free theorem asserts that irreducible unitary highest-weight representations π are multiplicity-free when restricted to every symmetric pair if π is of scalar type. The aim of this paper is to find the "classical limit" of this multiplicity-free theorem in terms of the geometry of two coadjoint orbits, for which the correspondence is predicted by the Kirillov–Kostant–Duflo orbit method.

For this, we study the Corwin–Greenleaf multiplicity function for Hermitian symmetric spaces G/K. First, we prove that for any G-coadjoint orbit and any K-coadjoint orbit if . Here, 𝔤 = 𝔨 + 𝔭 is the Cartan decomposition of the Lie algebra 𝔤 of G.

Second, we find a necessary and sufficient condition for by means of strongly orthogonal roots. This criterion may be regarded as the "classical limit" of a special case of the Hua–Kostant–Schmid–Kobayashi branching laws of holomorphic discrete series representations with respect to symmetric pairs.

In this paper, we classify all immersed hypersurfaces in the unit sphere S^m+1 with parallel para-Blaschke tensor.

We show that any Dolbeault cohomology group H^p,q(D), p ≥ 0, q ≥ 1, of an open subset D of a closed finite codimensional complex Hilbert submanifold of ℓ₂ is either zero or infinite dimensional. We also show that any continuous character of the algebra of holomorphic functions of a closed complex Hilbert submanifold M of ℓ₂ is induced by its evaluation at a point of M. Lastly, we prove that any closed split infinite dimensional complex Banach submanifold of ℓ₂ admits a nowhere critical holomorphic function.

We extend the classification of complete polynomial vector fields in two complex variables given by Brunella to cover the case of holomorphic (non-polynomial) vector fields whose underlying foliation is still polynomial.

This paper contained mathematical errors and its publication is retracted from IJM. Please contact ijm@wspc.com

We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge–Ampère equations, and investigate their regularity. These types of equations are precisely what is needed in order to construct Kähler–Einstein metrics over irreducible singular Kähler spaces with ample or trivial canonical sheaf and singular Kähler–Einstein metrics over varieties of general type.