TYPES IN SL n
TYPES IN SL n
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SL n 中的类型
DOI：
10.1112/s002461150201359x
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发表时间：
2002
影响因子：
1.8
通讯作者：
Alan Roche
中科院分区：
文献类型：
--
作者：
David Goldberg;Alan Roche
设 F 为非阿基米德局部域，G 为在 F 上定义的连通还原代数群的 F 点群。将 RG 写为 G 的平滑复表示类别。根据 [1]，RG 具有规范分解，作为某些不可分解子类别（称为分量）的乘积。为了通过限制紧开子群的方法来阐明 RG 的研究，Bushnell 和 Kutzko 引入了组件类型的概念。这是一对 J, r，其中 J 是 G 的紧开子群，r 是 J 的不可约平滑表示，使得给定分量的对象恰好是由它们的 r 同型向量生成的 RG 的对象。根据[6, 4.3]，承认类型 J, r 的分量等价于 Hecke 代数 HG, r（G 上紧支持 r 球函数的卷积代数）上的右模范畴。Bushnell 和 Kutzko 在 GLn F 上的工作为这些群提供了一组完整的类型，即当 G GLn F 时 RG 的每个分量的类型。它还描述 相应的赫克代数的结构。在这里，我们使用这项工作的一部分，以及 [5] 和 [6] 的通用机制，在 SLn F 中构建某些类型族。我们的结果与 [5] 的结果相结合，为这些组生成了一套完整的类型。我们将在续集中讨论相应的赫克代数的结构。为了更仔细地讨论本文的内容，我们现在假设对[6]的符号和术语有一定的了解。特别是，如果 L 是 G 的列维子群，并且 j 是 L 的不可约上尖点表示，我们将 L, j G 写为由 j 确定的 G 惯性类，将 BG 写为所有此类的集合（如 [6，§ 1] 中）。设G GLn F 和GH SLn F。一般来说，如果H 是G 的子群，我们写成Hh H∩ GH。修复 G 的 Levi 子群 L。在第 1 节和第 2 节中，我们为 LTG 构造了一组完整的上尖点 LH 类型（即 BLh 的每个元素的类型），其形式为 LH，jH LH）。该结构与[5]中研究的案例LG相似，并且关键依赖于此。 Bushnell 和 Kutzko 根据 [3, 6.2] 的 G 中的最大简单类型定义了 GH 中的最大简单类型族。然后他们证明 GH 物体具有与其同名的 G 物体类似的属性。更准确地说，他们表明 GH 的每个不可约超尖端表示 pH 都包含 GH 中的最大简单类型，该类型在 GH 共轭范围内是唯一的，并且每个这样的简单类型都是 GH，pH Gh 型。
Let F be a non-Archimedean local field and G the group of F-points of a connected reductive algebraic group defined over F. Write RG for the category of smooth complex representations of G. By [1], RG has a canonical decomposition as a product of certain indecomposable subcategories, called components. To clarify the study of RG by the method of restriction to compact open subgroups, Bushnell and Kutzko have introduced the concept of a type for a component. This is a pair J, r with J a compact open subgroup of G and r an irreducible smooth representation of J such that the objects of the given component are precisely the objects of RG that are generated by their r-isotypic vectors. A component that admits a type J, r is equivalent, by [6, 4.3], to the category of right modules over the Hecke algebra HG, r, the convolution algebra of compactly supported r-spherical functions on G.Bushnell and Kutzko's work on GLn F provides a complete set of types for these groups, that is, a type of each component of RG when G GLn F. It also describes the structure of the corresponding Hecke algebras. Here we use a part of this work, along with [5] and the general machinery of [6], to construct certain families of types in SLn F. Our results, in combination with those of [5], yield a complete set of types for these groups. We will address the structure of the corresponding Hecke algebras in a sequel. To discuss the contents of the paper more closely, we now assume some familiarity with the notation and terminology of [6]. In particular, if L is a Levi subgroup of G and j is an irreducible supercuspidal representation of L, we write L, j G for the G-inertial class determined by j and BG for the set of all such classes (as in [6, § 1]). Let G GLn F and GH SLn F. In general, if H is a subgroup of G, we write Hh H∩ GH. Fix a Levi subgroup L of G. In §§ 1 and 2 we construct, for LTG, a complete set of supercuspidal LH-types (that is, a type for each element of BLh) of the form LH, jH LH). The construction parallels, and relies crucially upon, the case LG which is studied in [5]. There Bushnell and Kutzko define the family of maximal simple types in GH in terms of the maximal simple types in G of [3, 6.2]. They then prove that the GH-objects have analogous properties to their G-namesakes. More precisely, they show that each irreducible supercuspidal representation pH of GH contains a maximal simple type in GH, which is unique up to GH-conjugation, and that each such simple type is a GH, pH Gh-type.