论文标题
还原P-Adic组的内态代数和Hecke代数
Endomorphism algebras and Hecke algebras for reductive p-adic groups
论文作者
论文摘要
令G为还原的P-ADIC组，让Rep（G）^成为平滑复杂的G-eprentations类别中的Bernstein块。我们通过分析该类别的祖子π的G-尾形代数来研究REP（G）^S的结构。 我们表明，rep（g）^s与（扭曲的）杂物代数相当。该陈述以多种方式精确地制作，最重要的是，（扭曲）分级代数。就有限长度表示而言，rep（g）^s和end_g（π）-mod可以将其视为扭曲仿射Hecke代数的模块类别。 我们提出两个后果。首先，我们表明rep（g）^s和end_g（π）-mod之间的类别的等效性保留了有限长度表示的perver缩。其次，我们根据复杂的圆环和有限的组与rep（g）^s相关的有限群体提供了REP（g）^s不可约表示的分类。这证明了ABPS的猜想版本，使我们能够根据$ g $的Levi子组的超舒张表示表达不可约$ g $的代表。 我们的方法独立于类型的存在，并适用于完全普遍性。
Let G be a reductive p-adic group and let Rep(G)^s be a Bernstein block in the category of smooth complex G-representations. We investigate the structure of Rep(G)^s, by analysing the algebra of G-endomorphisms of a progenerator Πof that category. We show that Rep(G)^s is "almost" Morita equivalent with a (twisted) affine Hecke algebra. This statement is made precise in several ways, most importantly with a family of (twisted) graded algebras. It entails that, as far as finite length representations are concerned, Rep(G)^s and End_G (Π)-Mod can be treated as the module category of a twisted affine Hecke algebra. We draw two consequences. Firstly, we show that the equivalence of categories between Rep(G)^s and End_G (Π)-Mod preserves temperedness of finite length representations. Secondly, we provide a classification of the irreducible representations in Rep(G)^s, in terms of the complex torus and the finite group canonically associated to Rep(G)^s. This proves a version of the ABPS conjecture and enables us to express the set of irreducible $G$-representations in terms of the supercuspidal representations of the Levi subgroups of $G$. Our methods are independent of the existence of types, and apply in complete generality.
