  We present Keck spectroscopic observations of z>6 Lyman-break galaxy (LBG)
candidates in the Subaru Deep Field (SDF). The candidates were selected as
i'-dropout objects down to z'=27 AB magnitudes from an ultra-deep SDF z'-band
image. With the Keck spectroscopy we identified 19 LBGs with prominent Ly_alpha
emission lines at 6< z < 6.4. The median value of the Ly_alpha rest-frame
equivalent widths (EWs) is ~50 A, with four EWs >100 A. This well-defined
spectroscopic sample spans a UV-continuum luminosity range of -21.8<
M_{UV}<-19.5 (0.6~5 L*_{UV}) and a Ly_alpha luminosity range of (0.3~3) x
10^{43} erg s^{-1} (0.3~3 L*_ {Ly_alpha}). We derive the UV and Ly_alpha
luminosity functions (LFs) from our sample at <z>~6.2 after we correct for
sample incompleteness. We find that our measurement of the UV LF is consistent
with the results of previous studies based on photometric LBG samples at 5<z<7.
Our Ly_alpha LF is also generally in agreement with the results of
Ly_alpha-emitter surveys at z~5.7 and 6.6. This study shows that deep
spectroscopic observations of LBGs can provide unique constraints on both the
UV and Ly_alpha LFs at z>6.
