Symplectic Dirac operators on homogeneous spaces
Abstract
27 pages We define symplectic Dirac operators on homogeneous spaces and study their representation-theoretic role. For an invariant polarization, the symplectic Dirac operator de- composes into two symplectic Dolbeault operators. We compute their commutator as the natural symplectic analogue of the square of the classical Dirac operator. Our first main result gives a necessary and sufficient condition for this commutator to satisfy a Parthasarathy-type formula. We further prove that, whenever this condition fails, no cubic perturbation of the symplectic Dol- beault operators can yield such a formula, in contrast with Kostant’s cubic Dirac operator in the orthogonal setting. As applications, 27 pageswe establish an sl2-structure generated by the symplectic Dolbeault operators and derive Dirac-type inequalities for unitary representations of Hermitian symmetric spaces labelled by the levels of the symmetric powers of the antiholomorphic tangent space at the identity. The level-zero inequality recovers the standard Parthasarathy–Dirac in- equality, while the higher levels inequalities yield new constraints. For SU (1, n), we show that, for representations with a specific Kraljevi´c corner, the level-one inequality strengthens all basic Parthasarathy inequalities of the first kind for particular K-types, precisely those satisfying an explicit highest-weight condition.
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BibTeX
@misc{ciubotaru2026symplectic,
title = {{Symplectic Dirac operators on homogeneous spaces}},
author = {Dan Ciubotaru and Gang Liu and Salah Mehdi and Nicolas Prudhon},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09111},
eprint = {2609.09111},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05781976v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symplectic Dirac operators on homogeneous spaces
AU - Dan Ciubotaru
AU - Gang Liu
AU - Salah Mehdi
AU - Nicolas Prudhon
PY - 2026
DA - 2026/09/09
PB - HAL
DO - 10.48550/arxiv.2609.09111
UR - https://hal.science/hal-05781976v1
AB - 27 pages We define symplectic Dirac operators on homogeneous spaces and study their representation-theoretic role. For an invariant polarization, the symplectic Dirac operator de- composes into two symplectic Dolbeault operators. We compute their commutator as the natural symplectic analogue of the square of the classical Dirac operator. Our first main result gives a necessary and sufficient condition for this commutator to satisfy a Parthasarathy-type formula. We further prove that, whenever this condition fails, no cubic perturbation of the symplectic Dol- beault operators can yield such a formula, in contrast with Kostant’s cubic Dirac operator in the orthogonal setting. As applications, 27 pageswe establish an sl2-structure generated by the symplectic Dolbeault operators and derive Dirac-type inequalities for unitary representations of Hermitian symmetric spaces labelled by the levels of the symmetric powers of the antiholomorphic tangent space at the identity. The level-zero inequality recovers the standard Parthasarathy–Dirac in- equality, while the higher levels inequalities yield new constraints. For SU (1, n), we show that, for representations with a specific Kraljevi´c corner, the level-one inequality strengthens all basic Parthasarathy inequalities of the first kind for particular K-types, precisely those satisfying an explicit highest-weight condition.
ER -