Embedding types and canonical affine maps between Bruhat-Tits buildings of classical groups
Embedding types and canonical affine maps between Bruhat-Tits buildings of classical groups
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经典群的 Bruhat-Tits 建筑物之间的嵌入类型和规范仿射映射
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发表时间：
2010
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通讯作者：
Daniel Skodlerack
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作者：
Daniel Skodlerack
P. Broussous 和 S. Stevens 研究了扩大的 Bruhat-Tits 建筑物之间的地图，以构建 p 进酉群的类型。他们需要尊重 Moy-Prasad 过滤的地图。该属性称为 (CLF)，即与李代数过滤的兼容性。在本文的第一部分中，我们将他们在此类映射上的结果推广到四元数代数情况。令 k0 为留数特征的 p 进场，而不是二。我们考虑在 k0 上定义的酉群 G:=U(h) 的半单 k0 有理李代数元素 beta，其具有符号厄米形式 h。设 H 为 G 中 beta 的中心化器。我们证明了从放大的 Bruhat-Tits 建筑物 B^1(H,k0) 到 B^1(G,k0) 的仿射 H(k0) 等变 CLF 映射 j 的存在性。正如 Broussous 所推测的，如果 H 的所有因子均不与 k0 阶 1 的各向同性正交群 k0 同构，并且所有因子均为酉群，则 CLF 性质决定 j。在较弱的假设下，仿射 CLF 映射 j 仅在 H^0(k0) 中心下等变，它在 B^1(H,k0) 的平移之前唯一确定。第二部分致力于通过 CLF 映射的几何形状对嵌入类型进行解码。 Broussous 和 M. Grabitz 对嵌入类型进行了研究。我们考虑具有 p 进中心 F 的有限指数除法代数 D。Budhnell-Kutzko 框架中 GLn(D) 的简单类型的构造需要对必须满足刚性属性的地层进行研究。给出一个层特别意味着固定一个由 Mn(D) 中的域扩展 E|F 和一个在 E^x 共轭下稳定的遗传阶 a 组成的对 (E,a)，换句话说，我们将 E^x 的嵌入固定到 a 的归一化器中。 Broussous 和 Grabitz 对这些对进行了不变量分类。我们描述并证明了一种使用 CLF 映射的几何结构来解码这些不变量的方法。
P. Broussous and S. Stevens studied maps between enlarged Bruhat-Tits buildings to construct types for p-adic unitary groups. They needed maps which respect the Moy-Prasad filtrations. That property is called (CLF), i.e. compatibility with the Lie algebra filtrations. In the first part of this thesis we generalise their results on such maps to the Quaternion-algebra case. Let k0 be a p-adic field of residue characteristic not two. We consider a semisimple k0-rational Lie algebra element beta of a unitary group G:=U(h) defined over k0 with a signed hermitian form h. Let H be the centraliser of beta in G. We prove the existence of an affine H(k0)-equivariant CLF-map j from the enlarged Bruhat-Tits building B^1(H,k0) to B^1(G,k0). As conjectured by Broussous the CLF-property determines j, if none of the factors of H is k0-isomorphic to the isotropic orthogonal group of k0-rank one and all factors are unitary groups. Under the weaker assumption that the affine CLF-map j is only equivariant under the center of H^0(k0) it is uniquely determined up to a translation of B^1(H,k0). The second part is devoted to the decoding of embedding types by the geometry of a CLF-map. Embedding types have been studied by Broussous and M. Grabitz. We consider a division algebra D of finite index with a p-adic center F. The construction of simple types for GLn(D) in the Budhnell-Kutzko framework required an investigation of strata which had to fulfil a rigidity property. Giving a stratum especially means to fix a pair (E,a) consisting of a field extension E|F in Mn(D) and a hereditary order a which is stable under conjugation by E^x, in other words we fix an embedding of E^x into the normalizer of a. Broussous and Grabitz classified these pairs with invariants. We describe and prove a way to decode these invariants using the geometry of a CLF-map.