  We prove quantitative equidistribution results for actions of Abelian
subgroups of the $2g+1$ dimensional Heisenberg group acting on compact
$2g+1$-dimensional homogeneous nilmanifolds. The results are based on the study
of the $C^\infty$-cohomology of the action of such groups, on tame estimates of
the associated cohomological equations and on a renormalisation method
initially applied by Forni to surface flows and by Forni and the second author
to other parabolic flows. As an application we obtain bounds for finite Theta
sums defined by real quadratic forms in $g$ variables, generalizing the
classical results of Hardy and Littlewood \cite{MR1555099, MR1555214} and the
optimal result of Fiedler, Jurkat and K\"orner \cite{MR0563894} to higher
dimension.
