Unitary Branching for sl(m| n), osp(m| 2n) and F(4)
Abstract
We determine the branching of irreducible unitary representations of the basic classical Lie superalgebras sl(m|n), osp(m|2n), and F(4) under restriction to their even subalgebras. The proof combines Dirac inequalities for the Huang-Pandžić-Dirac operator Dg,g0̅ 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 sl(m|n), the corrected Dirac inequalities also give a characterization of unitarity for all positive systems relevant to unitary representations.
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BibTeX
@misc{schmidt2026unitary,
title = {{Unitary Branching for $\mathfraksl(m\vert n), \mathfrakosp(m\vert 2n)$ and $F(4)$}},
author = {Steffen Schmidt},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20332},
eprint = {2609.20332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20332},
}
RIS
TY - UNPB
TI - Unitary Branching for sl(m| n), osp(m| 2n) and F(4)
AU - Steffen Schmidt
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20332
UR - https://arxiv.org/abs/2609.20332
AB - We determine the branching of irreducible unitary representations of the basic classical Lie superalgebras sl(m|n), osp(m|2n), and F(4) under restriction to their even subalgebras. The proof combines Dirac inequalities for the Huang-Pandžić-Dirac operator D_g,g_0̅ 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 sl(m|n), the corrected Dirac inequalities also give a characterization of unitarity for all positive systems relevant to unitary representations.
ER -