In recent work with Schumann we have proven a conjecture of
Naito-Sagaki giving a branching rule for the decomposition of the restriction of an irreducible
representation of the special linear Lie algebra to the symplectic Lie algebra,
therein embedded as the fixed-point set of the involution obtained by the folding of
the corresponding Dyinkin diagram. This conjecture had been open for over ten years,
and provides a new approach to branching rules for non-Levi subalgebras in terms
of Littelmann paths. In this talk I will introduce the path model, explain the setting of the problem, our proof, and provide some examples of other non-Levi branching situations.
