Classification of irreducible unitary modules over $\mathfrak{u}(p,q\mathrm{\mid }n)$

We classify all irreducible highest-weight unitary modules over the non-compact real form $\mathfrak{u}(p,q\mathrm{\mid }n)$ of the general linear Lie superalgebra ${\mathfrak{g}\mathfrak{l}}_{p+q\mathrm{\mid }n}$. The classification is given by explicit necessary and sufficient conditions on the highest weights, and our approach combines the Howe duality for ${\mathfrak{g}\mathfrak{l}}_{p+q\mathrm{\mid }n}$ with a quadratic invariant of the maximal compact subalgebra. Using this classification result, we also classify all irreducible lowest-weight unitary modules over $\mathfrak{u}(p,q\mathrm{\mid }n)$ via duality, and all irreducible unitary modules over $\mathfrak{u}(n\mathrm{\mid }q,p)$ via an isomorphism of Lie superalgebras.