spherical functions on real reductive groups and archimedean zeta integrals

• Project/Area Number: 24740025
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Seikei University
• Principal Investigator: Ishii Taku 成蹊大学, 理工学部, 准教授 (60406650)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 保型形式 / 保型L関数 / Whittaker関数 / アルキメデスゼータ積分 / 保型Ｌ関数 / Novodvorsky積分 / 局所ゼータ積分 / Rankin-Selberg法 / 離散系列表現

Outline of Final Research Achievements

The aim of our study is to establish analytic properties of automorphic L-functions via the integral representations method. We compute the archimedean zeta integrals which are the integral transforms of generalized spherical functions on real reductive Lie groups. As a consequence we prove the entireness, the location of possible poles and functional equations for some automorphic L-functions on GL(n) and GSp(2). Especially we show the coincidence of archimedean zeta integrals and the expected L-factors (product of Gamma functions) for the exterior square L-functions on GL(n) and the standard L-functions on GL(3)×GL(2).

Research Products

• [Int'l Joint Research] Colorado大学(米国)
• [Journal Article] The archimedean zeta integrals for GL(3)×GL(2) (2016)
  • Author(s): Miki Hirano, Taku Ishii and Tadashi Miyazaki
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci. Volume: 92 Issue: 2 Pages: 27-32
  • DOI: 10.3792/pjaa.92.27
  • NAID: 40020756994
  • Peer Reviewed / Open Access
• [Journal Article] Calculus of principal series Whittaker functions on SL(n,R). (2014)
  • Author(s): Ishii, Taku; Oda, Takayuki
  • Journal Title: J. Funct. Anal. Volume: 266 Issue: 3 Pages: 1286-1372
  • DOI: 10.1016/j.jfa.2013.11.020
  • Peer Reviewed
• [Journal Article] Archimedean zeta integrals on GL_n × GL_m and SO_{2n+1} × GL_m (2013)
  • Author(s): Taku Ishii and Eric Stade
  • Journal Title: Manuscripta Mathematica Volume: 141 Issue: 3-4 Pages: 485-536
  • DOI: 10.1007/s00229-012-0581-y
  • Peer Reviewed
• [Journal Article] Whittaker functions on orthogonal groups of odd degree (2013)
  • Author(s): Taku Ishii
  • Journal Title: Journal of Lie Theory Volume: 23 Pages: 85-112
  • Peer Reviewed
• [Presentation] Archimedean zeta integrals for GSp(2) (2015)
  • Author(s): Taku Ishiｉ
  • Organizer: Number theory seminar
  • Place of Presentation: Colorado大学
  • Year and Date: 2015-05-15
• [Presentation] Whittaker functions on Sp(2,R) and archimedean zeta integrals (2015)
  • Author(s): 石井卓
  • Organizer: 保型形式セミナー
  • Place of Presentation: 上智大学
  • Year and Date: 2015-03-14
• [Presentation] 古典群のclass one Whittaker関数の明示公式
  • Author(s): 石井卓
  • Organizer: 保型形式の整数論月例セミナー
  • Place of Presentation: 東京大学
• [Presentation] GLn(F)の既約表現のL因子とε因子
  • Author(s): 石井卓
  • Organizer: 第21回整数論サマースクール
  • Place of Presentation: 箱根高原ホテル
• [Presentation] Archimedean zeta integrals attached to non-holomorphic Siegel modular forms of degree 2
  • Author(s): 石井卓
  • Organizer: 跡公式と関連する話題について
  • Place of Presentation: 東京大学
• [Presentation] Pieri-type formulas for GL(n,R)-Whittaker functions and its application
  • Author(s): Taku Ishii
  • Organizer: 超幾何関数とその周辺
  • Place of Presentation: 東京大学玉原国際セミナーハウス
• [Presentation] Archimedean zeta integrals for the exterior square L-functions on GL(n)
  • Author(s): Taku Ishii
  • Organizer: Kyoto conference on automorphic forms
  • Place of Presentation: 京都大学
  • Invited
• [Presentation] Archimedean L-factors for standard L-functions attached to non-holomorphic Siegel modular forms of degree 2
  • Author(s): Taku Ishii
  • Organizer: 保型表現とその周辺
  • Place of Presentation: 京都大学数理解析研究所
  • Invited