Unitary Branching for $\mathfrak{sl}(m\vert n), \mathfrak{osp}(m\vert 2n)$ and $F(4)$

Abstract

We determine the branching of irreducible unitary representations of the basic classical Lie superalgebras $\mathfrak{sl}(m|n)$, $\mathfrak{osp}(m|2n)$, and $F(4)$ under restriction to their even subalgebras. The proof combines Dirac inequalities for the Huang-Pand\v{z}i\'{c}-Dirac operator $\operatorname{D}_{\mathfrak{g},\mathfrak{g}_{\bar{0}}}$ with a super analogue of Freudenthal's recursion derived from the Dirac formalism. We extend the Dirac method to positive systems that are not adapted to the relevant real form by transporting the Dirac inequality through odd reflections and determining the resulting correction terms. This yields finite recursive procedures that determine the occurring even constituents and their branching multiplicities, together with closed formulas for the latter, uniformly in the degree of atypicality. For $\mathfrak{sl}(m|n)$, the corrected Dirac inequalities also give a characterization of unitarity for all positive systems relevant to unitary representations.