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We find an agreement of equivariant indices of semi-classical homomorphismsbetween pairwise mirror branes in the GL(2) Higgs moduli space on a Riemannsurface. 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 evenexterior powers of the hyperholomorphic Dirac bundle on the SL(2) Higgs modulispace. The agreement arises from a mysterious functional equation. This givesstrong computational evidence for Hitchin's proposal.

Mirror symmetry with branes by equivariant Verlinde formulae