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The metaplectic Howe duality and polynomial solutions for the symplectic Dirac operator. (English) Zbl 1279.30051
Summary: We study various aspects of the metaplectic Howe duality realized by the Fischer decomposition for the metaplectic representation space of polynomials on $$\mathbb R^{2n}$$ valued in the Segal-Shale-Weil representation. As a consequence, we determine symplectic monogenics, i.e. the space of polynomial solutions of the symplectic Dirac operator $$D_s$$.

##### MSC:
 30G35 Functions of hypercomplex variables and generalized variables 37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010) 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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