On the multiplicity spaces for branching to a spherical subgroup 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.