  Let $a_X:X\rightarrow \mathrm{Alb}\, X$ be the Albanese map of a smooth
complex projective variety. Roughly speaking in this note we prove that for all
$i \geq 0$ and $\alpha\in \mathrm{Pic}^0\, X$, the cohomology ranks
$h^i(\mathrm{Alb}\, X, \,{a_X}_* \omega_X\otimes P_\alpha)$ are derived
invariants. In the case of varieties of maximal Albanese dimension this proves
conjectures of Popa and Lombardi-Popa -including the derived invariance of the
Hodge numbers $h^{0,j}$ -- and a weaker version of them for arbitrary
varieties. Finally we provide an application to derived invariance of certain
irregular fibrations.
