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Iwahori–Hecke algebra and unramified local L-functions

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Abstract

In this paper, we compute the Hecke action of a certain test function on the space of an unramified principal series of a connected reductive group over a non-archimedean local field. The key of our computation is the theory of Iwahori–Hecke algebra. As an application, we obtain a new expression of the local L-functions of unramified representations.

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Acknowledgements

The authors are grateful to Miyu Suzuki for encouragement and constructive advice on a draft of this paper. The authors also thank Hiraku Atobe, Yoichi Mieda, and Lei Zhang for their helpful comments. The authors would like to express their sincere gratitude to Thomas Haines for his detailed explanation about how to prove our result via Iwahori–Hecke algebras. He also kindly answered a lot of questions from the authors and encouraged them. Finally, the authors are also grateful to the referee for giving them constructive comments for improving this paper. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan and JSPS KAKENHI Grant Number 17J05451 and 20K14287 (Oi), 17J02456 (Sakamoto), and 17J01075 and 20J00024 (Tamori). R.S. was also supported by RIKEN Center for Advanced Intelligence Project (AIP).

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Authors and Affiliations

  1. Department of Mathematics (Hakubi Center), Kyoto University, Kitashirakawa, Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan

    Masao Oi

  2. Department of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-851, Japan

    Ryotaro Sakamoto

  3. Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan

    Hiroyoshi Tamori

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  1. Masao Oi
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  2. Ryotaro Sakamoto
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Correspondence to Masao Oi.

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Oi, M., Sakamoto, R. & Tamori, H. Iwahori–Hecke algebra and unramified local L-functions. Math. Z. 303, 59 (2023). https://doi.org/10.1007/s00209-023-03214-9

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