  We investigate a possibility of odd-parity multipole orderings in a locally
noncentrosymmetric tetragonal compound CeCoSi. By performing symmetrical and
microscopic mean-field analyses on a two-orbital tight-binding model, we
propose potential odd-parity multipoles hidden in staggered antiferromagnetic
and antiferroquadrupole orderings in CeCoSi. We show that $3z^2-r^2$ type of
the magnetic quadrupole is induced by the staggered magnetic dipole ordering
for a large crystal-field splitting between two orbitals, while $xy$ type of
the electric toroidal quadrupole is emergent by the staggered electric
quadrupole ordering for a small crystal-field splitting. Furthermore, we
discuss a magneto-electric effect and elastic-electric effect due to the
odd-parity multipoles, which will be useful to identify order parameters in
CeCoSi.
