发表机构
University of Oxford; Université de Lorraine（牛津大学; 洛林大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在齐性空间上定义辛Dirac算子，分解为辛Dolbeault算子，给出换位子满足Parthasarathy型公式的充要条件，并应用于推导Dirac型不等式，其中一级不等式加强了特定K-类型的Parthasarathy不等式。
AI 中文摘要
我们在齐性空间上定义了辛Dirac算子，并研究了它们在表示论中的作用。对于不变极化，辛Dirac算子分解为两个辛Dolbeault算子。我们计算了它们的换位子，作为经典Dirac算子平方的自然辛类比。我们的第一个主要结果给出了该换位子满足Parthasarathy型公式的充分必要条件。我们进一步证明，当该条件不成立时，辛Dolbeault算子的任何三次扰动都不能产生这样的公式，这与正交情形下Kostant的三次Dirac算子形成对比。作为应用，我们建立了由辛Dolbeault算子生成的${\mathfrak s}{\mathfrak l}_2$-结构，并推导了厄米对称空间上酉表示的Dirac型不等式，这些不等式由恒等元处反全切空间的对称幂的级别标记。零级不等式恢复了标准的Parthasarathy-Dirac不等式，而更高级别的不等式产生了新的约束。对于$SU(1,n)$，我们证明，对于具有特定Kraljević角的表示，一级不等式加强了特定$K$-类型的第一类所有基本Parthasarathy不等式，这些$K$-类型恰好满足一个显式的最高权条件。
英文摘要
We define symplectic Dirac operators on homogeneous spaces and study their representation-theoretic role. For an invariant polarization, the symplectic Dirac operator decomposes 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 Dolbeault operators can yield such a formula, in contrast with Kostant's cubic Dirac operator in the orthogonal setting. As applications, we establish an ${\mathfrak s}{\mathfrak l}_2$-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 inequality, while the higher levels inequalities yield new constraints. For $SU(1,n)$, we show that, for representations with a specific Kraljević 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.
Comments27 pages
