  We present the results of deep spectroscopy for the central region of the
cluster lens SDSS J1004+4112 with the Subaru telescope. A secure detection of
an emission line of the faint blue stellar object (component E) near the center
of the brightest cluster galaxy (G1) confirms that it is the central fifth
image of the lensed quasar system. In addition, we measure the stellar velocity
dispersion of G1 to be sigma_* = 352+-13 km/s. We combine these results to
obtain constraints on the mass M_BH of the putative black hole (BH) at the
center of the inactive galaxy G1, and hence on the M_BH-sigma_* relation at the
lens redshift z_l=0.68. From detailed mass modeling, we place an upper limit on
the black hole mass, M_BH < 2.1x10^{10}M_sun at 1-sigma level
(<3.1x10^{10}M_sun at 3-sigma), which is consistent with black hole masses
expected from the local and redshift-evolved M_BH-sigma_* relations,
M_BH~10^{9}-10^{10}M_sun.
