On the multiplicity spaces for branching to a spherical subgroup of minimal rank

TL;DR

This paper investigates the structure of multiplicity spaces in branching problems for complex semi-simple Lie algebras, especially when the subalgebra is spherical of minimal rank, providing a geometric description.

Contribution

It offers a new geometric description of multiplicity spaces for spherical subalgebras of minimal rank, generalizing known tensor product results.

Findings

Describes multiplicity spaces as intersections of kernels of root operators.

Recovers known tensor product decomposition results via geometric methods.

Provides explicit descriptions for spherical subalgebras of minimal rank.

Abstract

Let g be a complex semi-simple Lie algebra and g be a semisimple subalgebra of g. Consider the branching problem of decomposing the simple g-representations V as a sum of simple grepresentations V. When g = g x g, it is the tensor product decomposition. The multiplicity space Mult(V, V) satisfies V = $\oplus$ V Mult(V, V) $\otimes$ V, where the sum runs over the isomorphism classes of simple g-representations. In the case when g is spherical of minimal rank, we describe Mult(V, V) as the intersection of kernels of powers of root operators in some weight space of the dual space V * of V. When g = g x g, we recover by geometric methods a well known result.