Motivic classes of stacks in finite characteristic and applications to stacks of Higgs bundles

We define a ring of motivic classes of stacks suitable for symmetric powers in finite characteristic. Let X be a smooth projective curve over a field of arbitrary characteristic. We calculate the motivic classes of the moduli stacks of semistable Higgs bundles on X. This recovers results of Fedorov, A. Soibelman and Y. Soibelman in characteristic zero, as well as those of Mozgovoy and Schiffmann for finite fields. We also obtain a simpler formula for the motivic classes of Higgs bundles in the universal $λ$-ring quotient using Mellit's results.