Fischer decomposition for \(\mathfrak{osp}(4|2)\)-monogenics in quaternionic Clifford analysis (Q2831033)

Continuing in the philosophy of their earlier work [Adv. Appl. Clifford Algebr. 24, No. 4, 955--980 (2014; Zbl 1364.15016); Ann. Global Anal. Geom. 46, No. 4, 409--430 (2014; Zbl 1337.15021)], the authors establish a Fischer decomposition of spaces of spinor-valued spherical harmonics in terms of \(\operatorname{Sp}(p)\)-irreducibles (where \(\operatorname {Sp}(p)\) is the symplectic group). To do so, the spinor space \(\mathbb S\) is split (as described in [Zbl 1364.15016], and briefly recalled in the manuscript) into symplectic cells which are fundamental representations of \(\operatorname{Sp}(p)\). Since spaces of quaternionic monogenic homogeneous polynomials with values in a symplectic cell are not \(\operatorname {Sp}(p)\)-irreducible, in order to obtain \(\operatorname {Sp}(p)\)-irreducibility the authors introduce \(\mathfrak{osp}(4|2)\)-monogenicity as a refinement of quaternionic monogenicity. Needed to define \(\mathfrak{osp}(4|2)\)-monogenicity are four quaternionic Dirac operators, a scalar Euler differential operator \({\mathcal E}\) and a Clifford multiplication operator \(P\) in a manner dictated by the Lie superalgebra \(\mathfrak{osp}(4|2)\). The operators \({\mathcal E}\) and \(P\) arise when constructing the Howe dual pair \(\mathfrak{osp}(4|2)\times \operatorname{Sp}(p)\), the action of which leads to a multiplicity free Fischer decomposition.NEWLINENEWLINE The manuscript includes brief sections on the basics of Clifford algebra and Euclidean and Hermitian Clifford analysis.