A HODGE TYPE DECOMPOSITION FOR SPINOR VALUED FORMS

In this paper we define an action of the Lie algebra sl(2, R) on the s pace of spinor valued exterior forms Lambda X S associated to a euclid ean vector space (V, g). This action commutes with the natural action of Pin(V, g) and we obtain a decomposition of Lambda X S in terms of p rimitive elements analogous to the classical Hodge-Lefschetz pointwise decomposition of the exterior algebra of a Kahler manifold. This give s rise to Howe correspondences for the pair (Pin(V), sl(2, R)) and How e correspondences for the pair (Spin(V), sl(2, R)) are also obtained. We prove some positivity results in this context, which are analogous to the classical, infinitesimal Hodge-Riemann bilinear relations.