L2L2 dimensions of spaces of braid-invariant harmonic forms

Let XX be a Riemannian manifold endowed with a co-compact isometric action of an infinite discrete group. We consider L2L2 spaces of harmonic vector-valued forms on the product manifold XNXN that are invariant with respect to an action of the braid group BNBN, and compute their von Neumann dimensions (the braided   L2L2-Betti numbers).