发表机构
Weizmann Institute of Science; University of Maryland; Karlsruhe Institute of Technology（魏茨曼科学研究所; 马里兰大学; 卡尔斯鲁厄理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究半单李群在任意酉表示下的连续（上）同调，将可约表示情形分解为Hausdorff部分与挠部分，分别由不可约上同调表示的内含性与弱内含性刻画，并给出秩1群及SO°(n,1)的完整结果，同时证明挠上同调首阶在性质(T)下受对称空间维数平方根下界限制。
AI 中文摘要
我们研究具有任意酉表示系数的半单李群的连续同调和上同调。不可约表示的情形已由Vogan和Zuckerman确定；我们关注可约表示。（上）同调分解为其Hausdorff部分和挠部分。Hausdorff部分由不可约上同调表示的内含性所支配。我们证明挠部分由不可约上同调表示的弱内含性所支配。确切地说，我们证明一个酉表示具有非零挠当且仅当在其支撑中存在一个非孤立的上同调点。我们详细讨论了秩为1的群的例子，完全确定了群$\mathrm{SO}^\circ(n,1)$的酉上同调。对于简单李群，第一和第四署名作者表明，对于某个无不变向量的酉表示，其获得非平凡上同调的首个次数与群的秩相关。我们讨论关于挠上同调的类似问题，并证明相应的首个次数可能高得多：在具有性质(T)的情况下，它被对称空间维数的平方根所下界限制。我们使用的一个技术手段是限制到满足有限性性质的精心选择的稠密子群，我们称之为上同调见证。我们将其与满足有限性性质的群的酉上同调结果相结合。这些结果具有独立的意义。
英文摘要
We study the continuous homology and cohomology of semisimple Lie groups with coefficients in arbitrary unitary representations. The case of irreducible representations was determined by Vogan and Zuckerman; we focus on reducible representations. The (co)homology splits into its Hausdorff and torsion parts. The Hausdorff part is governed by containment of irreducible cohomological representations. We show that the torsion part is governed by weak containment of irreducible cohomological representations. Precisely, we show that a unitary representation admits non-zero torsion if and only if there is a cohomological point which is not isolated in its support. We discuss in length examples of rank-$1$ groups, completely determining unitary cohomology for the group $\mathrm{SO}^\circ(n,1)$. For a simple Lie group, the first and fourth named authors showed that the first degree in which it obtains non-trivial cohomology for some unitary representation with no invariant vectors is related to the rank of the group. We discuss the analogous question regarding torsion cohomology, and show that the corresponding first degree could be much higher: in the presence of property (T), it is bounded below by the square root of the dimension of the symmetric space. A technical device that we use is the restriction to well chosen dense subgroups which satisfy finiteness properties, which we call cohomological witnesses. We combine it with results on the unitary cohomology of groups which satisfy finiteness properties. These results are of an independent interest.
Comments61 pages, 3 figures
