Four types of (super)conformal mechanics: D-module reps and invariant actions

(Super)conformal mechanics in one dimension is induced by parabolic or hyperbolic/trigonometric transformations, either homogeneous (for a scaling dimension $\lambda$) or inhomogeneous (at $\lambda=0$, with $\rho$ an inhomogeneity parameter). Four types of (super)conformal actions are thus obtained. With the exclusion of the homogeneous parabolic case, dimensional constants are present. Both the inhomogeneity and the insertion of $\lambda$ generalize the construction of Papadopoulos [CQG 30 (2013) 075018; arXiv:1210.1719]. Inhomogeneous $D$-module reps are presented for the $d=1$ superconformal algebras $osp(1|2)$, $sl(2|1)$, $B(1,1)$ and $A(1,1)$. For centerless superVirasoro algebras $D$-module reps are presented (in the homogeneous case for ${\cal N}=1,2,3,4$; in the inhomogeneous case for ${\cal N}=1,2,3$). The four types of $d=1$ superconformal actions are derived for ${\cal N}=1,2,4$ systems. When ${\cal N}=4$, the homogeneously-induced actions are $D(2,1;\alpha)$-invariant ($\alpha$ is critically linked to $\lambda$); the inhomogeneously-induced actions are $A(1,1)$-invariant.