Smooth representations of reductive p‐ADIC groups: structure theory via types
Smooth representations of reductive p‐ADIC groups: structure theory via types
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DOI：
10.1112/s0024611598000574
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发表时间：
1998-11
影响因子：
1.8
通讯作者：
CJ Bushnell;PC Kutzko
中科院分区：
文献类型：
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作者：
CJ Bushnell;PC Kutzko
这里，F表示非阿基米德局部域（具有有限剩余域），G是定义在F上的连通约化代数群的F点群。设R（G）表示G的光滑复表示范畴.设B（G）是对（L，σ）的集合，其中L是G的F-Levi子群，σ是L的不可约超尖点表示，取模由具有非分歧拟特征标和G-共轭的扭生成的等价关系.对于s ∈ B（G），可以附加R（G）的一个完全（交换）子范畴Rs（G）;伯恩斯坦中心理论表明R（G）是这些Rs（G）的直积。本文的目的是给出一个一般的方法来描述这些因素类别通过表示的紧开子群在一个统一的框架。固定s ∈ B（G）.设K是G的紧开子群，ρ是K的不可约光滑表示.对K，ρ是s-型的，如果它具有以下性质：G的不可约表示π包含ρ当且仅当π在Rs（G）中。设H（G，ρ）是G上紧支撑ρ-球函数的Hecke代数，若（K，ρ）是s-型，则范畴Rs（G）与H（G，ρ-模的范畴H（G，ρ）-Mod正则等价.设M是G的一个Levi子群，存在一个标准映射B（M）到B（G）.设t ∈ B（M），像s ∈ B（G）. G的一个具有Levi分支M的抛物子群的选择给出了连接Rt（M）和Rs（G）的抛物归纳函子和Jacquet限制函子.我们假设给定M中的一个t-型（KM，ρM）;本文讨论了从这个数据构造G中的一个s-型（K，ρ）的一般方法。这样就得到了这些归纳函子和限制函子用内射环同态H（M，ρM）→ H（G，ρ）的描述。该方法适用于各种各样的情况下，并包括许多以前的工作。在进一步的条件下，在某些特别有趣的情况下观察到，人们可以在一定程度上明确地描述H（G，ρ）。这使人们能够隔离的情况下，地图上的赫克代数是一个同构，这反过来又意味着强大的交织定理的类型。1991年数学学科分类：22 E50。
Here, F denotes a non‐Archimedean local field (with finite residue field) and G the group of F‐points of a connected reductive algebraic group defined over F. Let R(G) denote the category of smooth, complex representations of G. Let B(G) be the set of pairs (L, σ), where L is an F‐Levi subgroup of G and σ is an irreducible supercuspidal representation of L, taken modulo the equivalence relation generated by twisting with unramified quasi characters and G‐conjugacy. To s ∈ B(G), one can attach a full (abelian) sub‐category Rs(G) of R(G); the theory of the Bernstein centre shows that R(G) is the direct product of these Rs(G). The object of the paper is to give a general method for describing these factor categories via representations of compact open subgroups within a uniform framework. Fix s ∈ B(G). Let K be a compact open subgroup of G and ρ an irreducible smooth representation of K. The pair K, ρ is an s‐type if it has the following property: an irreducible representation π of G contains ρ if and only if π in Rs(G). Let H(G, ρ) be the Hecke algebra of compactly supported ρ‐spherical functions on G; if (K,ρ) is an s‐type, then the category Rs(G) is canonically equivalent to the category H(G, ρ)‐Mod of H(G, ρ‐modules. Let M be a Levi subgroup of G; there is a canonical map B(M) to B(G). Take t ∈ B(M) with image s ∈ B(G). The choice of a parabolic subgroup of G with Levi component M gives functors of parabolic induction and Jacquet restriction connecting Rt(M) with Rs(G). We assume given a t‐type (KM,ρM) in M; the paper concerns a general method of constructing from this data an s‐type (K,ρ) in G. One thus obtains a description of these induction and restriction functors in terms of an injective ring homomorphism H(M,ρM) → H(G,ρ). The method applies in a wide variety of cases, and subsumes much previous work. Under further conditions, observed in certain particularly interesting cases, one can go some distance to describing H(G,ρ) explicitly. This enables one to isolate cases in which the map on Hecke algebras is an isomorphism, and this in turn implies powerful intertwining theorems for the types. 1991 Mathematics Subject Classification: 22E50.