【論文解説】Langlands対応の深度保存が特異点を超える

Описание: この学術論文は、**非アルキメデス局所体上の連結簡約群**における**局所ラングランズ対応（LLC）**の理解を深めることに焦点を当てています。具体的には、**ラングランズ・パラメータの深度**という概念を修正し、その**深度保存性**をLLCの下で回復させます。これにより、**標数0の局所体**から**標数pの局所関数体**へ、**調和解析学の広範な結果を移転**するための**体系的な枠組み**が提供されます。さらに、**ℓ-近傍な体**における**平滑群スキームの切り詰められた同型**や**ヘッケ代数の同型**といった主要な定理を確立し、特に**正標数のLLCを記述する手法**について論じています。
Depth Preservation and Close-Field Transfer in the Local Langlands Correspondence
We introduce a revised notion of depth for Langlands parameters for a connected reductive $G$ defined over a nonarchimedean local field $F$ that restores depth preservation under the local Langlands correspondence (LLC) - in particular for all tori. We leverage that preservation to derive structural results that, taken together, yield a canonical transfer of broad harmonic-analytic results from characteristic $0$ to characteristic $p$. When $F$ has suitably large positive characteristic, we prove a block-by-block equivalence: each Bernstein block of $G(F)$ is equivalent to a corresponding block for some $G'(F')$ with $F'$ of characteristic $0$ $\ell$-close to $F$; using this, we show that an LLC in characteristic $0$ corresponds canonically to an LLC in characteristic $p$. For regular supercuspidals we give a direct, more structured construction via Kaletha. Along the way we recover and extend results on $\ell$-close fields -- introducing a depth-transfer function generalizing the normalized Hasse--Herbrand function, proving truncated isomorphisms for arbitrary tori and parahorics, establishing a depth and supercuspidality-preserving Kazhdan-type Hecke-algebra isomorphism for arbitrary maximal parahorics of arbitrary connected reductive groups; and a generalized Cartan decomposition for arbitrary maximal parahorics -- thereby subsuming several earlier results in the literature. Collectively, the results let one work in characteristic $0$ without loss of generality for a wide swath of harmonic analysis on $p$-adic groups.
Manish Mishra
http://arxiv.org/abs/2509.04997v1
#Langlands対応 #深度保存 #ℓ近接体 #p進群 #Bernsteinブロック #超特異表現 #Hecke環 #代数群解析