  We study a spin-orbital model for 4$d^{1}$ or 5$d^{1}$ Mott insulators in
ordered double perovskites with strong spin-orbit coupling. This model is
conveniently written in terms of pseudospin and pseudo-orbital operators
representing multipoles of the effective $j=3/2$ angular momentum. Similarities
between this model and the effective theories of Kitaev materials motivate the
proposal of a chiral spin-orbital liquid with Majorana fermion excitations. The
thermodynamic and spectroscopic properties of this quantum spin liquid are
characterized using parton mean-field theory. The heat capacity, spin-lattice
relaxation rate, and dynamic structure factor for inelastic neutron scattering
are calculated and compared with the experimental data for the spin liquid
candidate Ba$_{2}$YMoO$_{6}$. Moreover, based on a symmetry analysis, we
discuss the operators involved in resonant inelastic X-ray scattering (RIXS)
amplitudes for double perovskite compounds. In general, the RIXS cross sections
allow one to selectively probe pseudospin and pseudo-orbital degrees of
freedom. For the chiral spin-orbital liquid in particular, these cross sections
provide information about the spectrum for different flavors of Majorana
fermions.
