Invariants of almost complex and almost K\"ahler manifolds

Let $(M^{2n},J)$ be a compact almost complex manifold. The almost complex invariant $h^{p,q}_J$ is defined as the complex dimension of the cohomology space $\left\{\left[\alpha\right]\in H^{p+q}_{dR}(M^{2n};\mathbb{C}) \,\vert\,\alpha\in A^{p,q}(M^{2n}),\, d\alpha = 0 \right\}$. When $2n=4$, it has many interesting properties. Endow $(M^{2n},J)$ with an almost Hermitian metric $g$. The number $h^{p,q}_d$, i.e., the complex dimension of the space of Hodge-de Rham harmonic $(p,q)$-forms, is almost K\"ahler invariant when $2n=4$. In this paper we study the relationship between $h^{p,q}_J$ and $h^{p,q}_d$ in dimension $2n\ge4$. We prove $h^{n,0}_J=0$ if $J$ is non integrable and show that $h^{p,0}_d$ is almost K\"ahler invariant. If $M^{2n}$ is a compact quotient of a completely solvable Lie group and $(J,g,\omega)$ is left invariant, we find information also on $h^{1,1}_d$. Finally we study the $\mathcal{C}^\infty$-pure and $\mathcal{C}^\infty$-full properties of $J$ on $n$-forms for the special dimension $2n=4m$.