论文标题
古典P-ADIC组的主载体代数
Affine Hecke algebras for classical p-adic groups
论文作者
论文摘要
我们考虑了四类的经典群体在非架构的本地领域F上：符号，（特殊）正交，通用（S）销钉和统一。这些群体不必在F上进行准整体。本文的主要目标是通过Hecke代数获得此类G的本地Langlands通信。 对于光滑的复杂G代表类别中的每个Bernstein Block Rep（g）s，（扩展的）仿射Hecke代数H（S）可以与Heiermann方法相关联。另一方面，对于g的每个伯恩斯坦组件$φ_e（g）^{s^\ vee} $ for G的增强l-parameters的空间φ_e（g），一个人也可以将（扩展的）offine hecke hecke代数相关联，说$ h（s^\ vee）$。对于基于REP（G）^的超级矛盾表示，由于Moeglin和Arthur，可以通过内窥镜检查局部兰兰对应。使用它，我们将每个rep（g）^s分配一个唯一的$φ_e（g）^{s^\ vee} $。 我们的主要新结果是代数同构$ h（s）^{op} \ to H（s^\ vee）$，直至内部自动形态。结合早期的工作，它提供了满足Borel的Desiderata的注入性局部Langlands对应关系IRR（G） - >φ_e（G）。该参数化图也可能是溢出的，但在所有情况下我们都不能证明这一点。 我们的框架适合（重新）证明有关平滑的G-代理（不一定可降低）的许多结果，并将其与L参数空间的几何形状相关联。特别是，我们的兰兰斯参数化产生了一种独立的方法，可以根据约旦块和Levi亚组的Jordan块和超矛盾表示分类。我们表明，它与Moeglin和Tadić二十年前获得的离散系列的分类相吻合。
We consider four classes of classical groups over a non-archimedean local field F: symplectic, (special) orthogonal, general (s)pin and unitary. These groups need not be quasi-split over F. The main goal of the paper is to obtain a local Langlands correspondence for any group G of this kind, via Hecke algebras. To each Bernstein block Rep(G)^s in the category of smooth complex G-representations, an (extended) affine Hecke algebra H(s) can be associated with the method of Heiermann. On the other hand, to each Bernstein component $Φ_e (G)^{s^\vee}$ of the space Φ_e (G) of enhanced L-parameters for G, one can also associate an (extended) affine Hecke algebra, say $H (s^\vee)$. For the supercuspidal representations underlying Rep(G)^s, a local Langlands correspondence is available via endoscopy, due to Moeglin and Arthur. Using that we assign to each Rep(G)^s a unique $Φ_e (G)^{s^\vee}$. Our main new result is an algebra isomorphism $H(s)^{op} \to H (s^\vee)$, canonical up to inner automorphisms. In combination with earlier work, that provides an injective local Langlands correspondence Irr(G) -> Φ_e (G) which satisfies Borel's desiderata. This parametrization map is probably surjective as well, but we could not show that in all cases. Our framework is suitable to (re)prove many results about smooth G-representations (not necessarily reducible), and to relate them to the geometry of a space of L-parameters. In particular our Langlands parametrization yields an independent way to classify discrete series G-representations in terms of Jordan blocks and supercuspidal representations of Levi subgroups. We show that it coincides with the classification of the discrete series obtained twenty years ago by Moeglin and Tadić.
