论文标题
$ p $ adiC groups的Hecke代数的有限性
Finiteness for Hecke algebras of $p$-adic groups
论文作者
论文摘要
让$ g $成为一个还原的集团，这是残留特征$ p $的非Archimedean本地场$ f $ f $。我们证明，$ g（f）$的Hecke代数，其系数为$ {\ Mathbb z} _ {\ ell} $ - $ \ ell $不等于$ p $的Algebra $ r $是有限生成的模块，而这些中心是这些中心的$ r $ r $ r $ -Alge-r $ -Algebras。遵循伯恩斯坦（Bernstein）的原始策略，我们随后推断出“第二个伴随性”为$ g（f）$的平滑表示，并在任何环$ r $中都具有$ p $可逆的$ r $中的系数。这些结果已经猜想了很长时间。解锁问题的关键新工具是在兰兰兹参数方面定义的某个“偏移代数”与$ g（f）$的伯恩斯坦中心之间的特定“游览代数”之间的fargues-scholze形态。使用这座桥，我们的主要结果是代表理论上的对应物是当地兰兰兹参数的粗模量空间之间某些形态的有限性，我们在这里也证明了这一点，这可能具有独立的兴趣
Let $G$ be a reductive group over a non-archimedean local field $F$ of residue characteristic $p$. We prove that the Hecke algebras of $G(F)$ with coefficients in a ${\mathbb Z}_{\ell}$-algebra $R$ for $\ell$ not equal to $p$ are finitely generated modules over their centers, and that these centers are finitely generated $R$-algebras. Following Bernstein's original strategy, we then deduce that "second adjointness" holds for smooth representations of $G(F)$ with coefficients in any ring $R$ in which $p$ is invertible. These results had been conjectured for a long time. The crucial new tool that unlocks the problem is the Fargues-Scholze morphism between a certain "excursion algebra" defined on the Langlands parameters side and the Bernstein center of $G(F)$. Using this bridge, our main results are representation theoretic counterparts of the finiteness of certain morphisms between coarse moduli spaces of local Langlands parameters that we also prove here, which may be of independent interest
