  The $\alpha$-modulation transform is a time-frequency transform generated by
square-integrable representations of the affine Weyl-Heisenberg group modulo
suitable subgroups. In this paper we prove new conditions that guarantee the
admissibility of a given window function. We also show that the generalized
coorbit theory can be applied to this setting, assuming specific regularity of
the windows. This then yields canonical constructions of Banach frames and
atomic decompositions in $\alpha$-modulation spaces. In particular, we prove
the existence of compactly supported (in time domain) vectors that are
admissible and satisfy all conditions within the coorbit machinery, which
considerably go beyond known results.
