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Zbl 0855.58002
Slupinski, M.J.
A Hodge type decomposition for spinor valued forms. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 29, No. 1, 23-48 (1996). ISSN 0012-9593

The author defines an action of the Lie algebra $\text {sl} (2, \bbfR)$ 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 $(\text {Pin} (V), \text {sl} (2, \bbfR))$, and Howe correspondences for the pair $(\text {Spin} (V), \text {sl} (2, \bbfR))$ are also obtained. Some positivity results in this context, which are analogous to the classical infinitesimal Hodge-Riemann bilinear relations, are also proved.
[N.Papaghiuc (Iaşi)]

MSC 2000:: *58A14 Hodge theory (global analysis)
14C30 Transcendental methods

Keywords: Hodge type decomposition; spinor valued exterior forms

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