Derived invariants arising from the Albanese map

Let $a_X:X\to \mathrm{A}\mathrm{l}\mathrm{b}X$ be the Albanese map of a smooth complex projective variety. Roughly speaking in this note we prove that for all $i\ge 0$ and $\alpha \in {\mathrm{P}\mathrm{i}\mathrm{c}}^{0}X$, the cohomology ranks $h^i(\mathrm{A}\mathrm{l}\mathrm{b}X,{a_X}_{\ast }{\omega }_X\otimes P_{\alpha })$ 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.