Abstract: I will present the calculation of the rational Borel-Moore homology groups for determinantal varieties, thus solving a problem of Pragacz and Ratajski. The main ingredients are Hartshorne's algebraic de Rham homology theory, the calculation of the singular cohomology of matrix orbits, and the degeneration of the Cech-de Rham spectral sequence. As a consequence, we obtain explicit formulas for the dimensions of de Rham cohomology groups of local cohomology modules, which are analogues of Lyubeznik numbers first introduced by Switala. We further determine the Hodge numbers of the singular cohomology of matrix orbits and of the Borel–Moore homology of their closures, based on Saito’s theory of mixed Hodge modules. This is joint work with Claudiu Raicu.