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A covariant pseudodifferential calculus on Riemann surfaces, based on theKrichever-Novikov global picture, is presented. It allows defining scalar andmatrix KP operators, together with their reductions, in higher genus. Globallydefined Miura maps are considered and the arising of polynomial or rational${\cal W}$ algebras on R.S. associated to each reduction are pointed out. Thehigher genus NLS hierarchy is analyzed in detail.Comment: 15 pages, LaTe

KP HIERARCHIES, POLYNOMIAL AND RATIONAL $\mathcal W$ ALGEBRAS ON RIEMANN SURFACES: A GLOBAL APPROACH