  Spaces of spinor-valued homogeneous polynomials, and in particular spaces of
spinor-valued spherical harmonics, are decomposed in terms of irreducible
representations of the symplectic group Sp$( p)$. These Fischer decompositions
involve spaces of homogeneous, so-called $\mathfrak{osp}(4|2)$-monogenic
polynomials, the Lie superalgebra $\mathfrak{osp}(4|2)$ being the Howe dual
partner to the symplectic group Sp$( p)$. In order to obtain Sp$(
p)$-irreducibility this new concept of $\mathfrak{osp}(4|2)$-monogenicity has
to be introduced as a refinement of quaternionic monogenicity; it is defined by
means of the four quaternionic Dirac operators, a scalar Euler operator
$\mathbb{E}$ underlying the notion of symplectic harmonicity and a
multiplicative Clifford algebra operator $P$ underlying the decomposition of
spinor space into symplectic cells. These operators $\mathbb{E}$ and $P$, and
their hermitian conjugates, arise naturally when constructing the Howe dual
pair $\mathfrak{osp}(4|2) \times$ Sp$( p)$, the action of which will make the
Fischer decomposition multiplicityfree.
