Resumen:
We study the cohomology of the action of an abelian subgroup
\(P \approx R^d\) of the \((2g+1)\) -dimensional Heisenberg group \(H^g\)
on the algebra of smooth functions on a homogeneous manifold \(\Gamma
\backslash H^g\).
We show that these actions are cohomologically tamely stable in the sense
of Katok,
and deduce, using renormalization on an appropriate moduli space
(a method used by Flaminio and Forni in the \(g=1\) case),
a quantitative equidistribution result depending on some diophantine
conditions on \(P\).
As an application, we obtain bounds on certain exponential sums,
higher dimensional generalizations of the classical bounds obtained by
Hardy and Littlewood for theta sums.
This is joint work with Livio Flaminio.
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