  A semi-classical check of the Goddard-Nuyts-Olive (GNO) generalized duality
conjecture for gauge theories with adjoint Higgs fields is performed for the
case where the unbroken gauge group is non-abelian. The monopole solutions of
the theory transform under the non-abelian part of the unbroken global symmetry
and the associated component of the moduli space is a Lie group coset space.
The well-known problems in introducing collective coordinates for these
degrees-of-freedom are solved by considering suitable multi-monopole
configurations in which the long-range non-abelian fields cancel. In the
context of an $N=4$ supersymmetric gauge theory, the multiplicity of BPS
saturated states is given by the number of ground-states of a supersymmetric
quantum mechanics on the compact internal moduli space. The resulting
degeneracy is expressed as the Euler character of the coset space. In all cases
the number of states is consistent with the dimensions of the multiplets of the
unbroken dual gauge group, and hence the results provide strong support for the
GNO conjecture.
