  A galaxy merger is expected to cause the formation of a supermassive black
hole (SMBH) binary, which itself eventually coalesces through the anisotropic
emission of gravitational waves. This may result in the merged SMBH receiving a
recoil kick velocity ~100 - 1000 km/s, causing it to oscillate in the
gravitational potential of the host galaxy. The luminous quasar E1821+643,
identified as an SMBH recoil candidate via spectropolarimetry observations,
shows Doppler shifting of the broad emission lines in direct and scattered
light, consistent with a relative velocity of 2100 km/s between the quasar
nucleus and host galaxy. In this paper, we attempt to detect the expected
spatial displacement using a combination of optical spectroastrometry and
Hubble Space Telescope (HST) narrow band images. The spectroastrometry reveals
a relative spatial displacement between the quasar nucleus and the gas emitting
the [OIII]4959,5007 lines of ~130mas (~580pc) to the North-West. Our HST images
resolve the [OIII] emission on sub-arcsecond scales, showing that it is
asymmetrically distributed, extending to radial distances ~0.5 - 0.6" from the
nucleus in a wide arc running from the North-East around to the West. A
simulated spectroastrometry observation based on the HST [OIII] image indicates
that only a small fraction of the measured displacement can be attributed to
the asymmetric [OIII] emission. This displacement therefore appears to be a
real spatial offset of the quasar nucleus with respect to the narrow-line
region, presumed to be located at the host galaxy center, further supporting
the interpretation that a post-merger gravitational recoil of the SMBH has
occurred in E1821+643.
