Invariant measures on moduli spaces of twisted holomorphic 1-forms and strata of dilation surfaces

The moduli space of twisted holomorphic 1-forms on Riemann surfaces, equivalently dilation surfaces with scaling, admits a stratification and GL(2,R)-action as in the case of moduli spaces of translation surfaces. We produce an analogue of Masur-Veech measure, i.e. an SL(2,R)-invariant Lebesgue class measure on strata or explicit covers thereof. This relies on a novel computation of cohomology with coefficients for the mapping class group. The computation produces a framed mapping class group invariant measure on representation varieties that naturally appear as the codomains of the periods maps that coordinatize strata.