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 \pm 13 km s ^ { -1 } .
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 } < 2.1 \times 10 ^ { 10 } M _ { \odot } at 1 \sigma level ( < 3.1 \times 10 ^ { 10 } M _ { \odot } at 3 \sigma ) , which is consistent with black hole masses expected from the local and redshift-evolved M _ { BH } - \sigma _ { * } relations , M _ { BH } \sim 10 ^ { 9 } -10 ^ { 10 } M _ { \odot } .