Researchers Database
Takano Keiji
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Last Updated :2025/04/19
Researcher Information
J-Global ID
Research Interests
Research Areas
Academic & Professional Experience
- 2005 - 2014 National Institute of Technology, Akashi College
- 2005 - 2014 Akashi National College of Technology, Associate Professor
- 2014 - 香川大学 教育学部, 准教授
- 2014 - Kagawa University, Faculty of Education, Associate Professor
- 2000 - 2005 National Institute of Technology, Akashi College
- 2000 - 2005 Akashi National College of Technology, Lecturer
Education
- 1995 - 2000 Osaka University 理学研究科 数学専攻
- - 2000 Osaka University Graduate School, Division of Natural Science Department of Mathematics
- 1993 - 1995 Tohoku University 理学研究科 数学専攻
- - 1995 Tohoku University Graduate School, Division of Natural Science Department of Mathematics
Association Memberships
Published Papers
- On some relatively cuspidal representations: Cases of galois and inner involutions on glnShin-Ichi Kato; Keiji TakanoOsaka Journal of Mathematics Osaka University 57 (3) 711 - 736 0030-6126 2020
- Relative non-cuspidality of representations induced from split parabolic subgroupsShin-ichi Kato; Keiji TakanoTokyo Journal of Mathematics 1 - 8 2020 [Refereed]
- On some relatively cuspidal representations: Cases of Galois and inner involutions on GL(n)Shin-ichi Kato; Keiji TakanoOsaka Journal of Mathematics 1 - 26 2020 [Refereed]
- On some relatively cuspidal representations of GL(n) over p-adic fieldsShin-ichi Kato; Keiji Takano数理解析研究所講究録 2013 14 - 27 2019/02
- Discrete series for symmetric spaces over p-adic fields髙野啓児; 加藤信一; Keiji Takano; Shin-ichi KatoRIMS Kokyuroku Kyoto University 1767 14-24 - 24 1880-2818 2011/10
- p-進体上の対称空間における尖点表現と離散系列表現髙野啓児日本数学会関数解析分科会特別講演アブストラクト 71-78 2010/03
- Shin-ichi Kato; Keiji TakanoJOURNAL OF FUNCTIONAL ANALYSIS ACADEMIC PRESS INC ELSEVIER SCIENCE 258 (5) 1427 - 1451 0022-1236 2010/03 [Refereed]
- Shin-ichi Kato; Keiji TakanoINTERNATIONAL MATHEMATICS RESEARCH NOTICES OXFORD UNIV PRESS 2008 1-40 1073-7928 2008 [Refereed]
- Subrepresentations for p-adic symmetric spaces髙野啓児; 加藤信一; Keiji Takano; Shin-ichi Kato京都大学数理解析研究所講究録 1523 95-108 2006/10
- Spherical functions on the symmetric variety GL(2n,F)/GL(n,E) where E/F is quadratic unramified髙野啓児RIMS Kokyuuroku Kyoto University 1321 50-61 - 61 1880-2818 2003/05
- Spherical functions in a certain distinguished model of GL(n)髙野啓児RIMS Kokyuuroku Kyoto University 1281 209-219 - 219 1880-2818 2002/08
- Spherical Functions in a Certain Distinguished Model髙野啓児博士論文（大阪大学理学研究科） 2001/03 [Refereed]
- Spherical functions in a certain distinguished model髙野啓児; Keiji TakanoJour. Math. Sci. Univ. Tokyo The University of Tokyo 7 (3) 369-400 - 400 1340-5705 2000/10 [Refereed]
- On standard L-functions for unitary groupsK TakanoPROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES JAPAN ACAD 73 (1) 5 - 9 0386-2194 1997/01 [Refereed]
- Standard L-functions for U(n,n)髙野啓児RIMS Kokyuuroku Kyoto University 909 177-189 - 189 1880-2818 1995/05
- ユニタリー群 U(n,n) のスタンダードL-関数について髙野啓児修士論文（東北大学 理学研究科） 1995/03 [Refereed]
Conference Activities & Talks
- GL(n) の内部対合に対する相対尖点表現：構造論とある種の誘導の相対非尖点性による証明の完成 [Not invited]京都大学数学教室 加藤信一教授セミナー 2018
- GL(n) の内部対合に関する相対尖点表現の構成 [Not invited]京都大学理学研究科 加藤信一教授セミナー 2017
- （中高生にも紹介できる）初歩の暗号理論 [Not invited]香川県高等学校教育研究会数学部会春季研究大会 2016
- GL(2n,F)/GL(n,E) の相対尖点表現の構成に向けた構造論的背景 [Not invited]京都大学 加藤信一教授セミナー 2016
- p-進体上の対称空間に付随した表現論の研究 [Not invited]2015年度教育学部教員組合第1回学習会 2015
- 相対尖点表現の例：GL(2n,F)/GL(n,E) [Not invited]2015年度第3回香川セミナー（数学） 2015
- Some remarks on relatively cuspidal representations [Not invited]岡山大学ワークショップ「Structures of L2(G/H)」 2014
- GL(n,F)-distinguished models for GL(n,E) [Not invited]ワークショップ「Distinction, L2(G/H) and RLLC」 2013
- Criterion for H-relative cuspidality and discreteness [Not invited]プレ白馬ワークショップ 2013
- H-relative subrepresentation theorem [Not invited]プレ白馬ワークショップ 2013
- Relative subrepresentation theorem and some related topics [Not invited]第16回白馬整数論オータムワークショップ「球等質空間上の調和解析」 2013
- Relatively cuspidal/discrete/tempered representations for symmetric spaces [Not invited]第16回白馬整数論オータムワークショップ「球等質空間上の調和解析」 2013
- Discrete series for symmetric spaces over p-adic fields [Not invited]研究集会「保型形式と関連する跡公式、ゼータ関数の研究」 2011
- p-進体上の対称空間における尖点表現と離散系列表現 [Not invited]日本数学会年会（関数解析分科会）特別講演 2010
- Cuspidality and square integrability of representations attached to p-adic symmetric spaces [Not invited]大阪大学整数論＆保型形式セミナー 2010
MISC
- Relative Bruhat decomposition for the symmetric space GL(m)/(GL(m-q)xGL(q))髙野啓児 2018/02
- Examples of relatively cuspidal representations: Inner involutions on GL(n)髙野啓児 17 pages 2017/11
- （高校生にも紹介できる）初等整数論を用いた公開鍵暗号の理論髙野啓児 数学研究 香川県高等学校教育研究会数学部会会誌 60- 13 - 24 2017/01
- GL(2n,F)/GL(n,E) の相対尖点表現：構造論的背景について髙野啓児 11 pages 2016/08
- 相対尖点表現の例：GL(2n,F)/GL(n,E)髙野啓児 4 pages 2015/10
- p-進対称空間に付随した表現論の研究髙野啓児 7 pages 2015/07
- Rankin-Selberg Method: 基本的な例髙野啓児 第3回整数論サマースクール「等質空間と保型形式」報告集 78-89 1995/11
Research Grants & Projects
- Construction of relatively cuspidal representations attached to symmetric varieties over local fieldsJapan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2020/04 -2023/03Author : 高野 啓児本研究は局所体上の対称空間に付随した「相対尖点表現」の系統立てた構成法を探求するものである。この特殊なクラスの表現は対称空間に付随するすべての既約表現のいわば building block となるものであり、対称空間の非可換調和解析における基本構成単位とも捉えられる重要なクラスであるが、既知の実例は少ない。本研究は相対的楕円トーラスと関係した特殊な対合安定放物部分群からの誘導表現として非尖点的な相対尖点表現の新しい系列を組織的に提供しようとするものである。前年度にはこの方向で、一般線型群の内部対合およびガロア対合の場合で得られた研究成果を発表した。本年度は同様の手法で一般線型群の斜交・ 直交・ユニタリ対合の場合の構成法を研究した。対合の同値類分類が基礎体の影響で複雑となることもあり、まずはできるだけ簡潔に処理できる具体例（偶数次元の特定の形式で定まる場合など）に絞って探ったものの、既知であった極大放物部分群からの１種類の系列しか得られず、新たな成果に結びつけることができなかった。 実際には非尖点的な相対尖点表現がこれら数少ない例に限られるという可能性もある。前年度にはまた、「対合分裂な放物部分群からの誘導はジェネリックには相対尖点的にならない」という一般的な定理を発表できた。この成果を用いて相対尖点的でないと判定できる誘導表現のクラスを調べ上げる研究も行っており、現在その途中段階である。 成功していた前年度の成果と合わせ、直面しているこれらの困難についての現状報告も含めて、2021年6月の「早稲田整数論セミナー」（遠隔実施）にて研究発表を行った。
- Representation theory of homogeneous spaces over p-adic or finite fieldsJapan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2014/04 -2019/03Author : Kato Shin-ichi; TAKANO KEIJIWe studied representations and harmonic analysis of symmetric spaces associated to reductive groups with involution sigma on them over p-adic fields, as a generalization of the representation theory of these groups. Relatively cuspidal representations for symmetric spaces are the counterparts of cuspidal representations for groups and viewed as the most fundamental tools in the study of the representation theory of symmetric spaces. Under the working hypothesis that relatively cuspidal representations should correspond to anisotropic maximal sigma-split tori, we succeeded in constructing relatively cuspidal representations as induced representations of cuspidal representations from sigma-stable parabolic subgroups for certain types of symmetric spaces associated with general linear groups. We also showed that relatively cuspidal representations cannot appear as induced representations from sigma-split parabolic subgroups if the inducing representations have generic parameters.
- Ramification theory of automorphic representations and arithmetic of special L-valuesJapan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2015/04 -2018/03Author : Yoshi-Hiro IshikawaNumber theory investigation usually involves quite vast area of deep mathematics,like as the Fermat Last Theolem does. The Langlands Program, which led to the settlement of FLT, has been the central strategy of arithmetic since 70s. We follow the LP to study the ramification theory of the group U(3) representations in view point of L‐/ε‐factors. Our approach is resorting to integral presentations of L‐function of automorphic forms, whose ramified factors give us arithmetic info. The point is to find nice Whittaker functions appearing in the ramified factor. We can successfully detect where/which the nice ones are only in the case of Real/unramified U(3).
- Ramified components of automorphic representations: local theory and its application to special L-valuesJapan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2012/04 -2015/03Author : ISHIKAWA YOSHI-HIRO; TSUZUKI Masao; YASUDA Seidai; TAKANO Keiji; MIYAUCHI MichitakaNumber theory investigation usually involves quite vast area of deep mathematics, like as the Fermat Last Theolem does. The Langlands Program, which led to the settlement of FLT, has been the central strategy of arithmetic since 70s. We follow the LP to study the ramification theory of the group U(3) representations in view point of L-/ε-factors. Our approach is resorting to integralpresentations of L-function of automorphic forms, whose ramified factors give us arithmetic info. The point is to find nice Whittaker functions appearing in the ramified factor. We can successfully detect where/which the nice ones are in the case of Real/unramified U(3). As an application to the global problem, we got algebraicity result for all the critical values of twisted L-function of generic cuspidal representaions on U(3).
- Representation Theory of Symmetric Spaces over Finite or Local FieldsJapan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2010/04 -2014/03Author : KATO Shin-ichi; TAKANO KeijiWe studied representations and harmonic analysis of symmetric spaces associated to reductive groups over finite or p-adic fields as a generalization of the representation theory of these groups. In the case of symmetric spaces over p-adic fields, we succeeded in constructing new examples of relatively cuspidal representations. We established criteria for tempered representations in the form analogous to the case of relatively cuspidal representations or square-integral representations. This is a natural extension of the criteria for group case. In the finite fields case, we studied a construction of relatively cuspidal representations on symmetric spaces by cohomological induction.
- Study onε-factor of automorphic representations and conductor of remified componentsJapan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2009 -2011Author : ISHIKAWA Yoshihiro; MORIYAMA Tomonori; YASUDA Seidai; MIYAUCHI Michitaka; TAKANO KeijiNumber theory investigation usually involves quite vast area of deep mathematics, like as the Fermat Last Theolem does. The Langlands Program, which led to the settlement of FLT, has been the central strategy of arithmetic since 70s. We follow the LP to study the ramification theory of the group U(3) representations in view point of L-/ε-factors. Our approach is resorting to integral presentations of L-function of automorphic forms, whose ramified factors give us arithmetic info. The point is to find nice Whittaker functions appearing in the ramified factor. We can successfully detect where/which the nice ones are in the case of Real/unramified U(3).
- Representation Theory of Symmetric Spaces over Finite or Local FieldsJapan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2006 -2009Author : KATO Shin-ichi; MATSUKI Toshihiko; NISHIYAMA Kyo; TAKANO KeijiAs a natural generalization of the representation theory of reductive groups, we studied representations of symmetric spaces attached to these groups. In the case of groups over p-adic fields, we established criteria for (relatively) cuspidal representations and square-integral representations in the form analogous to the group case. Moreover we proved the symmetric space version of subrepresentation theorem. In the case of groups over finite fields, we studied a construction of cuspidal representations on symmetric spaces by cohomological induction.
- Towards ramification theory of automorphic representations : Ramified representations and their L-factorsJapan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2007 -2008Author : ISHIKAWA Yoshihiro; MORIYAMA Tomonori; YASUDA Seidai; YOSHINO Yuji; TAKANO Keiji; WAKATSUKI Satoshiフェルマ予想(FLT)の様な数論の問題は, 非常に広範で深い理論を駆使して研究される。FLTの証明をも含み, 70年代より数論研究の支柱たり続けているLanglandsプログラムに沿って, 比較的小さい群U(3), GSp(4)の場合に, その分岐表現と付随するL-関数を研究した。方針は, L-関数を上の群を対称性にもつ保型形式という"関数"の積分変換で表示し, その積分の分岐因子を(一般化)ホイタッカー関数を通じて明示的に研究する。表現の分岐が激しくない簡易な場合に, L-因子を計算した。分岐が激しい場合にも, 部分群からのアプローチが有効で有ることが判った。
- Representation theoretic research of spherical functions on p-adic homogeneous spacesJapan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2003 -2005Author : KATO Shin-ichi; SAITO Hiroshi; MATSUKI Toshihiko; NISHIYAMA Kyo; MURASE Atsushi; TAKANO KeijiS.Kato, the Head investigator, studied the spherical functions on symmetric spaces over p-adic fields, together with K.Takano. By using orbit decomposition of symmetric spaces under maximal compact subgroups (Cartan decomposition, general formula of which is still in conjectural form), we obtained a Macdonald-type formula for spherical functions which expressed the value on tori by a sum over the Weyl groups of symmetric spaces (the little Weyl groups). The problem to have explicit formulas for spherical functions in general remained. However, for several examples including quadratic base change of symplectic groups, we had such formulas. As a byproduct of our study of symmetric spaces, we obtained a representation theoretical result about the representations of symmetric spaces (more precisely, about distinguished admissible representations for symmetric subgroups of reductive groups) : We succeeded in establishing a relative version (=symmetric space version) of Jacquet's subrepresentation theorem which asserts that for any irreducible admissible representation V of a p-adic reductive group G, there exists at least one parabolic P and one irreducible cuspidal W such that V may be embedded into the induced representation of W from P under the assumption of the Cartan decomposions. Namely, by defining the notion of relative cuspidality, we showed that any irreducible representation of a symmetric space can be embedded in a induced representation associated with a pair consisting of a sigma-split parabolic subgroup and an irreducible distinguished representation of its Levi subgroup. This result can be viewed as a first step to generalize the harmonic analysis on p-adic groups to that on symmetric spaces. It is interesting to build representation theory of symmetric spaces on p-adic groups and/or other groups over various fields by using the notion of "relative cuspidality". Other investigators also obtained several results on automorphic representations and automorphic forms (H.Saito and A.Murase ), and on structure theory and representation theory of real Lie groups (T.Matsuki and K.Nishiyama).
- 局所体上の代数群、対称空間の表現論Date (from‐to) : 2005
- Representation Theory of algebraic groups and symmetric spaces over local fieldsDate (from‐to) : 2005
- 教育学