  The linear (homogeneous and inhomogeneous) (k, N, N-k) supermultiplets of the
N-extended one-dimensional Supersymmetry Algebra induce D-module
representations for the N=2,4,8 superconformal algebras.
  For N=2, the D-module representations of the A(1,0) superalgebra are
obtained. For N=4 and scaling dimension \lambda=0, the D-module representations
of the A(1,1) superalgebra are obtained. For $\lambda\neq 0$, the D-module
representations of the D(2,1;\alpha) superalgebras are obtained, with $\alpha$
determined in terms of the scaling dimension $\lambda$ according to:
$\alpha=-2\lambda$ for k=4, i.e. the (4,4) supermultiplet, $\alpha=-\lambda$
for k=3, i.e. (3,4,1), and $\alpha=\lambda$ for k=1, i.e. (1,4,3). For
$\lambda\neq 0$ the (2,4,2) supermultiplet induces a D-module representation
for the centrally extended sl(2|2) superalgebra. For N=8, the (8,8) root
supermultiplet induces a D-module representation of the D(4,1) superalgebra at
the fixed value $\lambda=1/4$. A Lagrangian framework to construct
one-dimensional, off-shell, superconformal invariant actions from
single-particle and multi-particles D-module representations is discussed. It
is applied to explicitly construct invariant actions for the homogeneous and
inhomogeneous N=4 (1,4,3) D-module representations (in the last case for
several interacting supermultiplets of different chirality).
