  We carry out analysis and geometry on a marked configuration space
$\Omega^M_X$ over a Riemannian manifold $X$ with marks from a space $M$. We
suppose that $M$ is a homogeneous space $M$ of a Lie group $G$. As a
transformation group $\frak A$ on $\Omega_X^M$ we take the ``lifting'' to
$\Omega_X^M$ of the action on $X\times M$ of the semidirect product of the
group $\operatorname{Diff}_0(X)$ of diffeomorphisms on $X$ with compact support
and the group $G^X$ of smooth currents, i.e., all $C^\infty$ mappings of $X$
into $G$ which are equal to the identity element outside of a compact set. The
marked Poisson measure $\pi_\sigma$ on $\Omega_X^M$ with L\'evy measure
$\sigma$ on $X\times M$ is proven to be quasiinvariant under the action of
$\frak A$. Then, we derive a geometry on $\Omega_X^M$ by a natural ``lifting''
of the corresponding geometry on $X\times M$. In particular, we construct a
gradient $\nabla^\Omega$ and a divergence $\operatorname{div}^\Omega$. The
associated volume elements, i.e., all probability measures $\mu$ on
$\Omega_X^M$ with respect to which $\nabla^\Omega$ and
$\operatorname{div}^\Omega$ become dual operators on $L^2(\Omega_X^M;\mu)$, are
identified as the mixed marked Poisson measures with mean measure equal to a
multiple of $\sigma$. As a direct consequence of our results, we obtain marked
Poisson space representations of the group $\frak A$ and its Lie algebra $\frak
a$. We investigate also Dirichlet forms and Dirichlet operators connected with
(mixed) marked Poisson measures.
