  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.
