For a connected reductive group $\mathsf{G}$ over $\mathbb{R}$, we study cohomological $A$-parameters, which are Arthur parameters with the infinitesimal character of a finite-dimensional representation of $\mathsf{G}(\mathbb{C})$. We prove a structure theorem for such $A$-parameters, and deduce from it that a morphism of $L$-groups which takes a regular unipotent element to a regular unipotent element respects cohomological $A$-parameters. This is used to give complete understanding of cohomological $A$-parameters for all classical groups. We review the parametrization of Adams–Johnson packets of cohomological representations of $\mathsf{G}(\mathbb{R})$ by cohomological $A$-parameters and discuss various examples. We prove that the sum of the ranks of cohomology groups in a packet on any real group (and with any infinitesimal character) is independent of the packet under consideration, and can be explicitly calculated. This result has a particularly nice form when summed over all pure inner forms.

中文翻译：

---

实还原群的上同调表示

对于连通还原群 $\mathsf{G}$ 超过 $\mathbb{电阻}$, 我们研究上同调 $\mathrm{一种}$-参数，它们是 Arthur 参数，具有有限维表示的无穷小特征 $\mathsf{G}(\mathbb{C})$. 我们证明了这样的结构定理$\mathrm{一种}$-参数，并从中推导出一个态射 $升$-将规则单能元素转换为规则单能元素的组遵守上同调 $\mathrm{一种}$-参数。这用于完全理解上同调$\mathrm{一种}$- 所有经典组的参数。我们回顾了 Adams-Johnson 包的上同调表示的参数化$\mathsf{G}(\mathbb{电阻})$ 按上同调 $\mathrm{一种}$-参数并讨论各种示例。我们证明了任何实群（和任何无穷小特征）上的包中的上同调群的秩和与所考虑的包无关，并且可以明确计算。当对所有纯内在形式求和时，这个结果具有特别好的形式。