Derived invariants arising from the Albanese map

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