  We investigate the spectral theory of the invariant Landau Hamiltonian
$\La^\nu$ acting on the space ${\mathcal{F}}^\nu_{\Gamma,\chi}$ of
$(\Gamma,\chi)$-automotphic functions on $\C^n$, for given real number $\nu>0$,
lattice $\Gamma$ of $\C^n$ and a map $\chi:\Gamma\to U(1)$ such that the
triplet $(\nu,\Gamma,\chi)$ satisfies a Riemann-Dirac quantization type
condition. More precisely, we show that the eigenspace $
{\mathcal{E}}^\nu_{\Gamma,\chi}(\lambda)=\set{f\in
{\mathcal{F}}^\nu_{\Gamma,\chi}; \La^\nu f = \nu(2\lambda+n) f}$;
$\lambda\in\C,$ is non trivial if and only if $\lambda=l=0,1,2, ...$. In such
case, ${\mathcal{E}}^\nu_{\Gamma,\chi}(l)$ is a finite dimensional vector space
whose the dimension is given explicitly. We show also that the eigenspace
${\mathcal{E}}^\nu_{\Gamma,\chi}(0)$ associated to the lowest Landau level of
$\La^\nu$ is isomorphic to the space, ${\mathcal{O}}^\nu_{\Gamma,\chi}(\C^n)$,
of holomorphic functions on $\C^n$ satisfying $$ g(z+\gamma) = \chi(\gamma)
e^{\frac \nu 2 |\gamma|^2+\nu\scal{z,\gamma}}g(z), \eqno{(*)} $$ that we can
realize also as the null space of the differential operator
$\sum\limits_{j=1}\limits^n(\frac{-\partial^2}{\partial z_j\partial \bar z_j} +
\nu \bar z_j \frac{\partial}{\partial \bar z_j})$ acting on $\mathcal C^\infty$
functions on $\C^n$ satisfying $(*)$.
