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A Hodge type decomposition for spinor valued forms. (English) Zbl 0855.58002

The author defines an action of the Lie algebra \(\text{sl} (2, \mathbb{R})\) on the space of spinor valued exterior forms \(\Lambda \otimes S\) associated to an Euclidean vector space \((V, g)\). This action commutes with the natural action of \(\text{Pin} (V, g)\), and the author obtains a decomposition of \(\Lambda \otimes S\) in terms of primitive elements analogous to the classical Hodge-Lefschetz pointwise decomposition of the exterior algebra of a Kähler manifold. This gives rise to Howe correspondences for the pair \((\text{Pin} (V), \text{sl} (2, \mathbb{R}))\), and Howe correspondences for the pair \((\text{Spin} (V), \text{sl} (2, \mathbb{R}))\) are also obtained. Some positivity results in this context, which are analogous to the classical infinitesimal Hodge-Riemann bilinear relations, are also proved.

MSC:

58A14 Hodge theory in global analysis
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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References:

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