Configuration spaces of orbits and their $S_n$-equivariant $E$-polynomials

In this paper, we study the configuration space of orbits, a generalization of the configuration space of points but for algebraic varieties that are acted by an algebraic reductive group. The main objective of this work is to study the $E$-polynomials of these spaces and their quotients by $S_n$. For this purpose, we develop a novel method for computing the $S_n$-equivariant $E$-polynomial of an algebraic variety, and we apply it to this kind of varieties.