Integral expressions for derivations of multiarrangements

The construction of an explicit basis for a free multiarrangement is not easy in general. Inspired by the integral expressions for quasi-invariants of quantum Calogero-Moser systems, we present integral expressions for specific bases of certain multiarrangements. Our construction covers the cases of three lines in dimension $2$ (previously examined by Wakamiko) and free multiarrangements associated with complex reflection groups (Hoge, Mano, R\"ohrle, Stump). Furthermore, we propose a conjectural basis for the module of logarithmic vector fields of the extended Catalan arrangement of type $B_2$.