Abstract Let G be a connected complex simple Lie group, and let G ^ d {\widehat{G}^{\mathrm{d}}} be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that G ^ d {\widehat{G}^{\mathrm{d}}} consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of G ^ d {\widehat{G}^{\mathrm{d}}} come from L ^ d {\widehat{L}^{\mathrm{d}}} via cohomological induction and they are all in the good range. Here L runs over the Levi factors of proper θ-stable parabolic subgroups of G. It follows that figuring out G ^ d {\widehat{G}^{\mathrm{d}}} requires a finite calculation in total. As an application, we report a complete description of F ^ 4 d {\widehat{F}_{4}^{\mathrm{d}}} .

中文翻译：

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具有狄拉克上同调的酉表示：复李群的有限性结果

摘要 令 G 为连通复单李群，令 G ^ d {\widehat{G}^{\mathrm{d}}} 为具有非零狄拉克上同调的不可约酉表示的所有等价类的集合。我们证明 G ^ d {\widehat{G}^{\mathrm{d}}} 由两部分组成：有限多的分散表示和有限多的表示串。此外，G ^ d {\widehat{G}^{\mathrm{d}}} 的串通过上同调归纳来自 L ^ d {\widehat{L}^{\mathrm{d}}}在良好的范围内。这里 L 运行在 G 的适当 θ 稳定抛物线子群的 Levi 因子上。因此，计算 G ^ d {\widehat{G}^{\mathrm{d}}} 总共需要一个有限的计算。作为一个应用程序，我们报告了 F ^ 4 d {\widehat{F}_{4}^{\mathrm{d}}} 的完整描述。