  In an effort to measure the Rossiter-McLaughlin effect for the TRAPPIST-1
system, we performed high-resolution spectroscopy during transits of planets e,
f, and b. The spectra were obtained with the InfraRed Doppler spectrograph on
the Subaru 8.2-m telescope, and were supplemented with simultaneous photometry
obtained with a 1-m telescope of the Las Cumbres Observatory Global Telescope.
By analyzing the anomalous radial velocities, we found the projected stellar
obliquity to be $\lambda=1\pm 28$ degrees under the assumption that the three
planets have coplanar orbits, although we caution that the radial-velocity data
show correlated noise of unknown origin. We also sought evidence for the
expected deformations of the stellar absorption lines, and thereby detected the
"Doppler shadow" of planet b with a false alarm probability of $1.7\,\%$. The
joint analysis of the observed residual cross-correlation map including the
three transits gave $\lambda=19_{-15}^{+13}$ degrees. These results indicate
that the the TRAPPIST-1 star is not strongly misaligned with the common orbital
plane of the planets, although further observations are encouraged to verify
this conclusion.
