  Unimodular gravity (UG) is an important theory which may explain the
smallness of the cosmological constant. To get insight into the covariant
quantization of UG, we discuss the BRST quantization of General Relativity (GR)
with a cosmological constant in the unimodular gauge. We develop a novel way to
gauge fix the transverse diffeomorphism (TDiff) and then further to fulfill the
unimodular gauge. This process requires the introduction of an additional pair
of BRST doublets which decouple from the physical sector together with the
other three pairs of BRST doublets for the TDiff. We show that the physical
spectrum is the same as GR in the usual covariant gauge fixing. We then study a
theory derived by making "Fourier transform" of GR in the unimodular gauge with
respect to the cosmological constant as a candidate of "quantum UG." We clarify
the difference from GR and point out problems in this theory.
