## 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)

### Keywords:

Hodge type decomposition; spinor valued exterior forms
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### References:

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