Overlap lattice fermion in a gravitational field

We construct a lattice Dirac operator of the overlap-type which describes the propagation of the Dirac fermion in an external gravitational field. The local Lorentz symmetry is manifestly realized as a lattice gauge symmetry, while the general coordinate invariance is expected to be restored only in the continuum limit. Our doubler-free Dirac operator obeys the conventional Ginsparg-Wilson relation and possesses a gamma_5 hermiticity with respect to the inner product which is suggested from the general coordinate invariance. The lattice index theorem in the presence of a gravitational field holds and the classical continuum limit of the index density reproduces the Dirac genus. Reduction to a single Majorana fermion is possible for $8k+2$ and $8k+4$ dimensions, but not for $8k$ dimensions as being consistent with the existence of the global gravitational/gauge anomalies in the latter dimensions. Other Lorentz representations, such as the spinor-vector and the bi-spinor representations, can also be treated. Matter fields with a definite chirality (with respect to the lattice-modified chiral matrix) is briefly considered