  Spectral measures provide invariants for braided subfactors via fusion
modules. In this paper we study joint spectral measures associated to the
compact connected rank two Lie group $SO(5)$ and its double cover the compact
connected, simply-connected rank two Lie group $Sp(2)$, including the McKay
graphs for the irreducible representations of $Sp(2)$ and $SO(5)$ and their
maximal tori, and fusion modules associated to the $Sp(2)$ modular invariants.
