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

In this paper we use the canonical complex structure \mathbb{J}𝕁 on \mathbb{R}^{2n}ℝ2n to introduce a twist of the symplectic Dirac operator. This can be interpreted as the bosonic analogue of the Dirac operators on a Hermitian manifold. Moreover, we prove that the algebra of these Dirac operators is isomorphic to the Lie algebra \mathfrak{su}(1,2)𝔰𝔲(1,2) which leads to the Howe dual pair (U(n),\mathfrak{su}(1,2))(U(n),𝔰𝔲(1,2)).