Unitary Howe dualities for fermionic and bosonic algebras and related Dirac operators

In this paper we use the canonical complex structure $\mathbb{J}$ on ${\mathbb{R}}^{2n}$ to introduce a twist of the symplectic Dirac operator. As a matter of fact, these operators can be interpreted as the bosonic analogues of the Dirac operators on a Hermitian manifold. Moreover, we prove that the algebra of these symplectic Dirac operators is isomorphic to the Lie algebra $\mathfrak{s}\mathfrak{u}(1,2)$ which leads to the Howe dual pair $(\mathsf{U}(n),\mathfrak{s}\mathfrak{u}(1,2))$.