Cohomology of buildings and of their automorphism groups
Cohomology of buildings and of their automorphism groups
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建筑物及其自同构群的上同调
DOI：
10.1007/s00222-002-0242-y
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发表时间：
2002
影响因子：
3.1
通讯作者：
T. Januszkiewicz
中科院分区：
文献类型：
--
作者：
J. Dymara;T. Januszkiewicz
建筑物是由山雀建造的。它们是理解真实的半单群及其p-adic类似物的工具。这无疑是两种最有趣的情况，但现有的结构提供了许多更有趣的例子，其中一些是非常对称的。虽然经典（球形或欧几里得）建筑物对应于球形或欧几里得反射群，但我们主要关注与其他（无限）反射群相关的建筑物。这类例子，本文地址是卡茨-穆迪建筑，如描述的山雀[Ti]。这是一个非常大的家庭，我们使用的正是这些建筑的对称性。我们最初对建筑物上同调的兴趣是我们怀疑建筑物提供了Kazhdan群的丰富来源。回想一下，局部紧群G是Kazhdan群（或具有性质（T）），如果G的平凡表示在G的所有不可约酉表示的空间中是孤立的。等价地，G是Kazhdan群，如果H1 ct（G，ρ）= 0：即，它的第一个连续上同调群的系数在任何酉表示为零。在本文中，我们使用后一种定义（为了它们的等价性，参见。[HV]）。事实证明，我们不仅可以给出一个几乎确定的声明，当一个Kac-Moody的自同构群
Buildings were created by Tits. They serve as a vehicle for understanding real semi-simple groups and their p-adic analogs. These are undoubtedly the two most interesting cases, but existing constructions provide many more interesting examples, some of which are remarkably symmetric. While classical (spherical or Euclidean) buildings correspond to spherical or Euclidean reflection groups, we are mainly concerned with buildings related to other (infinite) reflection groups. The class of examples which this paper addresses is that of Kac–Moody buildings, as described by Tits [Ti]. It is a very large family, and it is precisely the great symmetry of these buildings that we are using. Our initial interest in the cohomology of buildings was that we suspected that buildings provided a rich source of Kazhdan groups. Recall that a locally compact group G is a Kazhdan group (or has Property (T)) if the trivial representation of G is isolated in the space of all irreducible unitary representations of G. Equivalently, G is a Kazhdan group if H1 ct(G, ρ) = 0: i.e., its first continuous cohomology group with coefficients in any unitary representation vanishes. In this paper we use the latter definition (for their equivalence, cf. [HV]). It turns out that not only can we give an almost definitive statement about when the automorphism group of a Kac–Moody