  The space $D$ of Hodge structures on a fixed polarized lattice is known as
Griffiths period domain and its quotient by the isometry group of the lattice
is the moduli of polarized Hodge structures of a fixed type. When $D$ is a
Hermition symmetric domain then we have automorphic forms on $D$, which
according to Baily-Borel theorem, they give an algebraic structure to the
mentioned moduli space. In this article we slightly modify this picture by
considering the space $U$ of polarized lattices in a fixed complex vector space
with a fixed Hodge filtration and polarization. It turns out that the isometry
group of the filtration and polarization, which is an algebraic group, acts on
$U$ and the quotient is again the moduli of polarized Hodge structures. This
formulation leads us to the notion of quasi-automorphic forms which generalizes
quasi-modular forms attached to elliptic curves.
