Study of the Langlands functoriality via an explicit local Langlands correspondence

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14287
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Kyoto University
• Principal Investigator: Oi Masao 京都大学, 白眉センター, 特定助教 (40868171)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 局所Langlands対応 / Langlands関手性 / 正則超尖点表現 / Harish-Chandra指標 / エンドスコピー / Swan導手

Outline of Final Research Achievements

We studied the local Langlands correspondence and the Langlands functoriality for supercuspidal representations via harmonic analysis by noting the Harish-Chandra characters. In particular, we established the twisted character formula and the twisted endoscopic character relation for toral supercuspidal representations and obtained a precise description of the local Langlands correspondence for simple supercuspidal representations. We also established a simple characterization of supercuspidal representations in terms of Harish-Chandra characters. As related problems, we also studied the functorial behavior of Swan conductors of Galois representations and a new formulation of the local Langlands correspondence from a geometric perspective.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

本研究で得られた局所Langlands対応の定式化に関する成果は，Langlands関手性に関連する諸問題の根本に関わるものである．またtoral型超尖点表現や単純超尖点表現に関する明示的局所Langlands対応に関する成果は，これまで知られていなかったクラスの具体例を，きわめて精密なレベルで与えるものである．これらを踏まえると，今回の研究成果は単なる興味深い結果には留まらず，今後Langlands関手性やより一般にp進簡約群の表現論に関する研究が分野として展開されてゆく中で，実用性を発揮すると期待している．