We find an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the $GL(2)$ Higgs moduli space on a Riemann surface. On one side we have the components of the Lagrangian brane of $U(1,1)$ Higgs bundles whose mirror was proposed by Nigel Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the $SL(2)$ Higgs moduli space. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin’s proposal.