教授 小林俊行 (リー群の無限次元表現論，不連続群論) 1. 簡約リー群のユニタリ表現論
ユニタリ表現論においては，表現の誘導と制限の解明という２つの大きな テーマがある。1次元表現からの「誘導」は等質空間上の大域解析と等価であ り，半世紀以上にわたってGelfand, Harish-Chandra,大島利雄氏等により大き く発展してきた。一方，「誘導」に較べて「制限」の理論は，テータ対応など特 殊な事例を除き，未開拓のままであった。
筆者は1990年代に，ユニタリ表現の「制限」に関する種々の“悪い”現象を 調べ，逆に，“良い振舞いをする”クラスが意外にも豊富にあることを発見し，
離散的分岐則の理論を構築した([a])。その後，この理論は，新しい既約ユニタ リ表現の構成(Gross, Wallach)や組成列の研究などの表現論内部の道具として だけでなく，非対称等質空間上のL2解析やモジュラー多様体の研究など周辺 分野でも応用が見出されつつある([6, 26])。
さて，既約成分が高々一回しか現れない重複度1の表現は，既約表現を一 般化した概念である。重複度1の表現の既約分解はcanonicalであるため，道 具としての表現論が特に効力を発揮する。例えば，有用な展開定理（Taylor, Fourier, spherical, . . . )や種々の関数等式を支える代数構造には，（普通は意識 せずとも)しばしば重複度1の表現が潜んでいる。現在，筆者は，複素多様体 における可視的な作用という概念と無重複性の伝播という視点を導入し，無限 次元の場合および(組合せ論が絡む)有限次元の場合を同時に含む，重複度１ 表現の統一的な理論をめざしている([17, 27, 39])。
2.非可換調和解析
保型形式の整数論に現れるWeil表現は，表現論の立場からはメタプレクティッ ク群の最も小さいユニタリ表現(極小表現) として特徴付けられる。一般の簡 約リー群の極小表現は最高ウェイトをもたない(真空ベクトルが存在しない)た め，その構成自体がそれほど簡単ではない。一方，極小表現は，他の既約表現 の族から“孤立”しているが故に，思いがけない数学的対象に登場する可能性 も秘めている。表現が小さいということは，関数空間から見ると対称性が大き いことを意味する。そこで筆者は，極小表現を解析・幾何的な手法で構成し，
数学の異分野に現れるモデルの中で特別大きな対称性の構造を捉えようと試 みた。
その一つの試みとして，一般の擬リーマン多様体に対し，そのYamabe作用 素の大域解の空間に共形変換群の表現を構成し，特に，それが定曲率空間の場 合には不定値直交群の極小表現になることを証明した。また，そのモデルに現 れるウルトラ双曲型偏微分方程式の大域解の共形不変な保存量を発見した(こ
の保存量は，極小表現の代数的理論から内在的に存在していることが予知さ れ，ある場合には佐藤超関数のアイディアを使って構成される) (Ørstedと共 同[13, 14, 15])。さらに，Weil表現のSchr¨odingerモデルを一般化し，Fourier–
Hankel変換, Hermite半群を特別な場合として含む極小表現の「裏返し変換」
や複素半群への解析接続の公式を具体的に決定した(真野元氏との共同[46])。
3. 非リーマン等質空間における不連続群
局所から大域への研究は，20世紀の幾何学における大きな流れの一つであ り，とりわけリーマン幾何学において著しい発展をとげた。その一方で，それ 以外の基礎的な幾何構造(擬リーマン，シンプレクティック，複素, · · ·)に対し ては，局所均質性を課した場合でさえ，その大域的な性質については驚くほど 何も知られていない。
例えば計量が不定値の擬リーマン多様体においては離散群の等長な作用が必 ずしも真性不連続ではなく，リーマン多様体における古典的な不連続群論とは 著しく異なる現象(Calabi–Markus現象など)が現れる。筆者は，1980年代後 半に，リーマン幾何の枠組を超えた等質空間の不連続群論に世界で最初に本格 的に取り組み，その基盤づくりに着手した([b]など)。特に，群構造を半分忘 却することによって不連続性の概念を一般化し，不連続性の判定条件と双対定 理を得,また，高次元の既約対称空間でも「剛性」が成り立たず不連続群を連 続変形できる現象を発見した。この新しい領域の全体的な解説と未解決問題を [4]に，非リーマン対称空間におけるコンパクトなClifford–Klein形の存在問題 をA. Borelの追悼論文集(吉野太郎氏と共同[30])に著した。
Academic Service
Editor in Chief, Journal of Mathematical Society of Japan (2002–2004;
2004–2006), Editor (1998–2006).
Managing Editor, Japanese Journal of Mathematics (2005– ).
Editor, Geometriae Dedicata (2000– ).
Editor, International Mathematics Research Notices (2002– ).
Editor, Publications RIMS (2003– ).
Editor, International Journal of Mathematics (2004– ).
Editor, International Mathematics Research Papers (2005– ).
日本数学会理事 (2003–2005; 2005– ).
博士課程学生の指導
Salma Nasrin (東京大学から数理解析研究所への指導委託),博士(数理科
学)取得, 2003
吉野太郎(東京大学から数理解析研究所への指導委託), 博士(数理科学)
取得, 2005
ポスドクの受入
JSPS reserach Fellow 2000–2001, Dr. Andreas Nilsson (Sweden) JSPS research Fellos 1999–2001, Dr. Tibor Odor (Hungary) JSPS research Fellow 2003–2005, Dr. Salma Nasrin (Bangladesch) JSPS research Fellow 2005, Dr. Andreas Nilsson (Sweden)
COE Fellow 2005–2006, 2006– , 吉野太郎
主な著作（番号は2000–2006の著作リストにおける通し番号）
[4] T. Kobayashi. Discontinuous groups for non-Riemannian homogeneous spaces. In B. Engquist and W. Schmid, editors,Mathematics Unlimited
— 2001 and Beyond, pp. 723–747. Springer-Verlag, 2001. (邦訳) “非 リーマン等質空間の不連続群論”『数学の最先端 21世紀への挑戦』第 1巻, pp. 18–73, 2002.
[6] T. Kobayashi. Branching problems of unitary representations. InProc.
of ICM 2002, Beijing, Vol. 2, pp. 615–627, 2002.
[13] T. Kobayashi and B. Ørsted. Analysis on the minimal representations of O(p, q), I. — Realization and conformal geometry. Adv. Math., Vol.
180, pp. 486–512, 2003.
[14] T. Kobayashi and B. Ørsted. Analysis on the minimal representations of O(p, q), II. — Branching laws. Adv. Math., Vol. 180, pp. 513–550, 2003.
[15] T. Kobayashi and B. Ørsted. Analysis on the minimal representations of O(p, q), III. — Ultra-hyperbolic equations on Rp−1,q−1. Adv. Math., Vol. 180, pp. 551–595, 2003.
[17] T. Kobayashi and S. Nasrin. Multiplicity one theorem in the orbit method. Amer. Math. Soc. Transl., Advances in the Mathematical Sciences, Series 2, Vol. 210, pp. 161–169, 2003. Special volume in memory of Professor F. Karpeleviˇc.
[25] 小林俊行・大島利雄. リー群と表現論. 岩波書店, 2005. 610 pp. (英訳 がSpringerから出版予定)
[26] T. Kobayashi. Restrictions of unitary representations of real reductive groups. In J.-P. Anker and B. Ørsted, editors, Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces, pp. 139–
207. Progress in Mathematics, Vol. 229, Birkh¨auser, 2005. (European
School およびハーバード大学での講義録)
[27] T. Kobayashi. Multiplicity-free representations and visible actions on complex manifolds. Publ. Res. Inst. Math. Sci., Vol. 41, pp. 497–
549, 2005. special issue commemorating the fortieth anniversary of the founding of RIMS.
[30] T. Kobayashi and T. Yoshino. Compact Clifford-Klein forms of sym- metric spaces — revisited. Pure and Appl. Math. Quarterly, Vol. 1, pp.
603–684, 2005. Special Issue: In Memory of Armand Borel.
[39] T. Kobayashi. Multiplicity-free theorems of the restrictions of uni- tary highest weight modules with respect to reductive symmetric pairs, preprint, to appear inProgress in Mathematics, Birkh¨auser.
[46] T. Kobayashi and G. Mano. The inversion formula and holomorphic extension of the minimal representation of the conformal group, preprint, 57 pp. (Roger Howe還暦記念号に投稿中)
2000年以前の引用
[a] T. Kobayashi. Discrete decomposability of the restriction ofAq(λ) with respect to reductive subgroups, Part I,Invent. Math., Vol. 117, pp. 181–
205, 1994; Part II — micro-local analysis and asymptotic K-support, Ann. of Math., Vol. 147, pp. 709–729, 1998; Part III — restric- tion of Harish-Chandra modules and associated varieties,Invent. Math., Vol. 131, pp. 229–256, 1998.
[b] T. Kobayashi. Proper action on a homogeneous space of reductive type.
Math. Ann., Vol. 285, pp. 249–263, 1989.
主な講演（番号は2000–2006の講演リストにおける通し番号）
[14] Conformal Geometry and Analysis on Minimal Representations, Work- shop on Integral Geometry in Representation Theory, MSRI, Berkeley, USA, 8–12 October 2001.
[25] Branching Problems of Unitary Representations, (invited lecture), Inter- national Congress of Mathematicians (ICM 2002), Beijing, China, 20–28 August 2002.
[51] Visible Actions on Complex Manifolds and Multiplicity-free Representa- tions, Conference on Lie Groups and Representation Theory, Odense, Denmark, 10–13 August 2004.
[52] Schr¨odinger Model of the Minimal Representation of the Lorentz Group O(p, q), Workshop on Harmonic Analysis and Homogeneous Spaces (in honor of Professor G. van Dijk), Lorentz Center, Leiden University, the Netherlands, 23–26 August 2004.
[53] Restriction of Unitary Representations — Discrete and continuous spec- trum, (plenary lecture), Sixth Pan-African Congress of Mathematicians (PACOM2004), Institut National des Sciences Appliqu´ees et de la Tech- nologie (INSAT), Tunis, Tunisia, September 2004.
[62] Analysis on Homogeneous Spaces Revisited — From Viewpoint of Branch- ing Laws of Unitary Representations, Harmonic Analysis on Lie Groups and Symmetric Spaces in honor of Jacques Faraut, Joint meeting of Sem- inar Sophus Lie, Nancy, France, 10–11 June 2005.
[65] Multiplicity-free Theorem and Visible Actions on Complex Manifolds, The Asian Mathematical Conference (AMC2005), Singapore, 20–23 July 2005.
[69] Restrictions of Unitary Representations of Real Reductive Groups, In- ternational Conference on Representations of Real Reductive Groups, in honor of R. Parthasarathy, Mumbai, India, 2–5 January 2006.
[70] Multiplicity-free Representations and Visible Actions on Complex Mani- folds, (opening lecture), International Conference on Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory (on the Occasion of Professor Roger Howe’s 60th Birthday), Singapore, 9–11 January 2006.
Summer School・大学院講義などによる連続講義・連続講演
（番号は2000–2006の講演リストにおける通し番号）
国外
[6] Branching Problems and Unitary Representations, (8 lectures), Euro- pean School on Group Theory, SDU-Odense University, Denmark, 14–26 August 2000.
[12] Restriction of Representations to Reductive Subgroups, (24 lectures), Har- vard University, USA, April–May 2001.
[19] Conformal Geometry and Analysis on Minimal Representations, (3 ple- nary lectures), The 22th Winter School on Geometry and Physics, Srni, Czech Republic, 12–19 January 2002.
[80] Unitary Representations, Restrictions, and their Applications, (course lectures), Summer School 2006 on Microlocal and Geometric Methods in Representation Theory, G¨unzburg, Germany, 17–28 July 2006. promised.
[81] TBA, (course lectures), Summer School on Geometry Topology, Analysis of Locally Symmetric Spaces and Discrete Groups, Beijing, Republic of China, 17 July–4 August 2006. promised.
国内
[16] 分岐則の重複度1定理, (3回講演), 東京大学数理科学研究科, 5, 12, 19 December 2001.
[21] 無限次元表現の分岐則,集中講義(5回),東京大学数理科学研究科, 17–21 June 2002.
[43] 不定値計量をもつ等質空間への群作用, 集中講義, 東京工業大学, 8–12 December 2003.
[45] 複素多様体における可視的な作用と表現論への応用について, 集中講義 (5回), 東京大学, 19–23 January 2004.
[58] Multiplicity-free Representations and Visible Actions on Complex Mani- folds, 集中講義(4回),大阪市立大学, 31 January–4 February 2005.
[73] 群作用とリー群の表現論,集中講義(5回),広島大学, 24–28 April 2006.
[75] 複素解析的手法によるユニタリ表現の話題から, 集中講義(5回),東京大 学, 8–12 May 2006.
著作（2000–2006）
[1] T. Kobayashi. Discretely decomposable restrictions of unitary repre- sentations of reductive Lie groups — examples and conjectures. In T. Kobayashi, M. Kashiwara, T. Matsuki, K. Nishiyama, and T. Os- hima, editors,Advanced Study in Pure Mathematics, Analysis on Homoge- neous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto, Vol. 26, pp. 98–126, 2000.
[2] T. Kobayashi. Branching laws of unitary highest weight modules with respect to semisimple symmetric pairs. Tangunsbericht, Representation Theory and Complex Analysis, Vol. 18, pp. 15–16, 2000.
[3] T. Kobayashi, M. Kashiwara, T. Matsuki, K. Nishiyama, T. Oshima (eds.). Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto. Adv. Stud. Pure Math. 2000. ISBN 4-314- 10138-5.
[4] T. Kobayashi. Discontinuous groups for non-Riemannian homogeneous spaces. In B. Engquist and W. Schmid, editors, Mathematics Unlimited
— 2001 and Beyond, pp. 723–747. Springer-Verlag, 2001.
[5] 小林俊行. 名著発掘 L. S.ポントリャーギン著『連続群論』. 数学のたの しみ,第23巻, pp. 110–119. 2001.
[6] T. Kobayashi. Branching problems of unitary representations. In Proc.
of ICM 2002, Beijing, Vol. 2, pp. 615–627, 2002.
[7] 小林俊行. 非リーマン等質空間の不連続群論(英語からの翻訳). 数学の最 先端 21世紀への挑戦, 第1巻, pp. 18–73. シュプリンガー・フェアラーク 東京, 2002.
[8] T. Kobayashi. Introduction to actions of discrete groups on pseudo- Riemannian homogeneous manifolds. Acta Appl. Math., Vol. 73, pp. 115–
131, 2002.
[9] 小林俊行. きっかけはいろんなこと 数学まなびはじめ. 数学のたのしみ, 第29巻, pp. 9–19. 日本評論社, 2002. 『数学まなびはじめ』第2集，日本 評論社，2006に再録.
[10] T. Kobayashi. On the canonical inner product on the space of global solutions of the Yamabe operator. InProceedings of Differential Geometry Symposium on “Various Geometric Structures”, pp. 4–5, 2002.
[11] 小林俊行. 山辺作用素の解空間における不変な内積について. Surikaiseki Kokyuroku, RIMS, Vol. 1294, pp. 76–86, 2002. 「可換代数系の表現と調 和解析」研究集会報告集(太田琢也 編).
[12] T. Kobayashi. Branching problems of unitary representations. In S. Sano, editor, Proceedings of Symposium of Representation Theory, held at Fuji Haitsu, November 2002, pp. 39–52, 2002.
[13] T. Kobayashi and B. Ørsted. Analysis on the minimal representations of O(p, q), I. — Realization and conformal geometry. Adv. Math., Vol. 180, pp. 486–512, 2003.
[14] T. Kobayashi and B. Ørsted. Analysis on the minimal representations of O(p, q), II. — Branching laws. Adv. Math., Vol. 180, pp. 513–550, 2003.
[15] T. Kobayashi and B. Ørsted. Analysis on the minimal representations of O(p, q), III. — Ultra-hyperbolic equations on Rp−1,q−1. Adv. Math., Vol.
180, pp. 551–595, 2003.
[16] T. Kobayashi. Conformal geometry and global solutions to the Yam- abe equations on classical pseudo-Riemannian manifolds. Supplemento di Rendiconti del Circolo Matematico di Palermo, Serie II, Vol. 71, pp.
15–40, 2003. Lecture Notes of the 22th Winter School 2002 on Geometry and Physics, Czech Republic.
[17] T. Kobayashi and S. Nasrin. Multiplicity one theorem in the orbit method.
Amer. Math. Soc. Transl., Advances in the Mathematical Sciences, Series 2, Vol. 210, pp. 161–169, 2003. Special volume in memory of Professor F.
Karpeleviˇc.
[18] 小林俊行. O(p, q) の極小ユニタリ表現のシュレディンガーモデル.
Surikaiseki Kokyuroku, RIMS, Vol. 1342, pp. 107–116, 2003. 「IV 型 対称領域上の保型形式の研究」短期共同研究報告集(織田孝幸 編).
[19] 小林俊行. Multiplicity one theorem on branching laws and geometry of complex manifolds. Surikaiseki Kokyuroku, RIMS, Vol. 1348, pp. 1–9,
2003. 「Lie Theory のひろがりと新たな進展」研究集会報告集(有木進 編).
[20] T. Kobayashi and A. Nilsson. Characterizing multipliers by relative in- variance. Surikaiseki Kokyuroku, RIMS, Vol. 1348, pp. 10–22, 2003. Ex- pansion of Lie Theory and New Advances (organized by S. Ariki).
[21] 小林俊行. 非リーマン等質空間の不連続群について. 日本数学会秋季総合 分科会(2003年9月)幾何学分科会特別講演要旨, pp. 75–89, 2003.
[22] T. Kobayashi. Geometry of multiplicity-free representations of GL(n), visible actions on flag varieties, and triunity. Acta Appl. Math., Vol. 81, pp. 129–146, 2004.
[23] T. Kobayashi, H. Ochiai, and H. Tagawa, editors. Symposium on Repre- sentation Theory 2004, Awajishima, 2004. 164 pp., ISBN 4-9902328-0-1.
[24] T. Kobayashi and G. Mano. Integral formulas for the minimal represen- tations for O(p,2). Acta Appl. Math., Vol. 86, pp. 103–113, 2005.
[25] 小林俊行・大島利雄. リー群と表現論. 岩波書店, 2005. 610 pp.
[26] T. Kobayashi. Restrictions of unitary representations of real reductive groups. In J.-P. Anker and B. Ørsted, editors, Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces, pp. 139–207.
Progress in Mathematics, Vol. 229, Birkh¨auser, 2005.
[27] T. Kobayashi. Multiplicity-free representations and visible actions on complex manifolds.Publ. Res. Inst. Math. Sci., Vol. 41, pp. 497–549, 2005.
special issue commemorating the fortieth anniversary of the founding of RIMS.
[28] T. Kobayashi. Theory of discrete decomposable branching laws of unitary representations of semisimple Lie groups and some applications. Sugaku Expositions, Vol. 18, Amer. Math. Soc., pp. 1–37, 2005. 日本語からの 翻訳.
[29] 小林俊行. 非リーマン等質空間の不連続群について(論説). 数学, Vol. 57, pp. 267–281, 2005. 英訳がアメリカ数学会より出版予定.
[30] T. Kobayashi and T. Yoshino. Compact Clifford-Klein forms of symmetric spaces — revisited. Pure and Appl. Math. Quarterly, Vol. 1, pp. 603–684, 2005. Special Issue: In Memory of Armand Borel.
[31] 小林俊行・真野元. O(p,2)の極小表現と反転の積分表示. Surikaiseki Kokyuroku, RIMS, Vol. 1410, pp. 173–187, 2005. 「表現論および等質 空間上の調和解析」研究集会報告集(井上順子 編).
[32] 小林俊行. Fourier transform of a minimal K-type vector in the minimal representation ofO(p+ 1, q+ 1). Surikaiseki Kokyuroku, RIMS, Vol. 1421, pp. 1–11, 2005. Sp(2,R)とSU(2,2)上の保型形式, III」研究集会報告集 (織田孝幸編).
[33] T. Kobayashi and A. Nilsson. Invariant multipliers and O(p, q)-action.
In S. Aoki, S. Kato, and H. Oda, editors, Proceedings of Symposium on Representation Theory 2005, held at Kakegawa, November 15–18, 2005, pp. 10–21.
[34] T. Kobayashi. Multiplicity-free representations and visible actions on complex manifolds. In S. Aoki, S. Kato, and H. Oda, editors, Proceed- ings of Symposium on Representation Theory 2005, held at Kakegawa, November 15–18, 2005, pp. 33–66.
[35] 小林俊行・真野元. O(p, q) の極小表現の反転を与える積分作用素.
Surikaiseki Kokyuroku, RIMS, Vol. 1467, pp. 51–61, 2006. 「群の表現 と調和解析の広がり」研究集会報告集(河添健 編).
[36] T. Kobayashi and S. Nasrin. Deformation space of discontinuous groups Zk for a nilmanifold Rk+1. Surikaiseki Kokyuroku, RIMS, Vol. 1467, pp.
101–111, 2006. Representation Theory of Groups and Extension of Har- monic Analysis (edited by T. Kawazoe).
[37] 小林俊行.きっかけはいろんなこと.数学まなびはじめ 第2集, pp. 198–221.
2006.
[38] T. Kobayashi and S. Nasrin. Deformation of properly discontinuous ac- tions of Zk on Rk+1. RIMS Preprint-1537 (March 2006), to appear in Internat. J. Math.
[39] T. Kobayashi. Multiplicity-free theorems of the restrictions of uni- tary highest weight modules with respect to reductive symmetric pairs.
preprint. to appear in Progress in Mathematics, Birkh¨auser.
[40] T. Kobayashi. On discontinuous group actions on non-Riemannian homo- geneous spaces. RIMS Preprint-1536 (March 2006), to appear in Sugaku Expositions, Amer. Math. Soc.; a translation of the original article in Japanese.
[41] T. Kobayashi. Introduction to visible actions on complex manifolds and multiplicity-free representations. Surikaiseki Kokyuroku, RIMS. Develop- ments of Cartan Geometry and Related Mathematical Problems (edited by T. Morimoto), submitted.
[42] T. Kobayashi, W. Schmid, and J.-H. Yang, editors. Representation The- ory and Automorphic Forms. Progress in Mathematics. Birkh¨auser. in preparation.
[43] T. Kobayashi. Propagation of multiplicity-free property for holomorphic vector bundles. preprint.
[44] T. Kobayashi. Visible actions on symmetric spaces. preprint.
[45] T. Kobayashi. A generalized cartan decomposition for the double coset space (U(n1)×U(n2)×U(n3))\U(n)/(U(p)×U(q)). preprint.
[46] T. Kobayashi and G. Mano. The inversion formula and holomorphic ex- tension of the minimal representation of the conformal group. preprint.
講演（2000–2006）
[1] A Note on Caratheodory Metric, 1999年度表現論ワークショップ(世話人＝
野村隆昭・ 田川裕之), 紀州加太(和歌山), 6–8 January 2000.
[2] Geometric Approach to Discretely Decomposable Restrictions, Workshop on the Geometry of Nilpotent Orbits and Representation Theory, Kyoto University, Japan, 31 January–3 February 2000.
[3] A Generalization of the Kostant-Schmid Formula,リー群論・表現論セミ ナー,東京大学, 18 April 2000.
[4] Branching Laws of Unitary Highest Weight Modules with Respect to Semisimple Symmetric Pairs, (opening lecture), Representation Theory and Complex Analysis, Oberwolfach, Germany, 23–29 April 2000.
[5] Discrete Decomopsable Restrictions and Geometry of Locally Symmetric Spaces, Workshop in Representation Theory and Automorphic Forms (or- ganized by J.-S. Li), Hangzhou, China, June 2000.
[6] Branching Problems and Unitary Representations, (8 lectures), European School on Group Theory, SDU-Odense University, Denmark, 14–26 Au- gust 2000.
[7] Discontinuous Subgroups for Non-Riemannian Homogeneous Spaces, MIT Lie Groups Seminar (organized by David Vogan), USA, 25 October 2000.
[8] Discrete Decomposable Restrictions of Unitary Representations, collo- quium, Oklahoma State University, USA, 1 December 2000.
[9] Discrete Groups for Pseudo-Riemannian Homogeneous Spaces, Lie Group Seminar (organized by Korany, cosponsored by Baruch), City University of New York, USA, 12 December 2000.
[10] Discontinuous Groups for Non-Riemannian Homogeneous Manifolds and Deformations of Clifford-Klein Forms, (2 lectures), The 2000 Twente Con- ference on Lie Groups, University of Twente, Enschede, the Netherlands, 18–20 December 2000.
[11] Conformal Geometry and Analysis on the Minimal Representation of O(p, q), MIT Lie Groups Seminar (organized by David Vogan), USA, 4 April 2001.
[12] Restriction of Representations to Reductive Subgroups, (24 lectures), Har- vard University, USA, April–May 2001.
[13] Discretely Decomposable Restriction of Unitary Representations and a Vanishing Theorem for Modular Symbols, 2001 Midwest Workshop in Lie Theory, Representation Theory and Automorphic Forms (organized by Moy), University of Michigan in Ann Arbor, USA, 11–13 May 2001.
[14] Conformal Geometry and Analysis on Minimal Representations, Work- shop on Integral Geometry in Representation Theory, MSRI, Berkeley, USA, 8–12 October 2001.
[15] Multiplicity One Theorem of Branching Laws, MSRI seminar (organized by Gindikin), Berkeley, USA, 20 November 2001.
[16] 分岐則の重複度1定理, (3回講演), 東京大学数理科学研究科, 5, 12, 19 December 2001.
[17] Multiplicity One Theorem of Branching Laws,リー群論・表現論セミナー, 東京大学, 18 December 2001.
[18] Canonical Hilbert Space of Ultrahyperbolic Solutions, 幕張ワークショッ
プ(世話役＝野村隆昭，熊原啓作), 放送大学セミナーハウス(千葉), 6–8
January 2002.
[19] Conformal Geometry and Analysis on Minimal Representations, (3 ple- nary lectures), The 22th Winter School on Geometry and Physics, Srni, Czech Republic, 12–19 January 2002.
[20] 山辺作用素の解空間における不変な内積について,シンポジウム「種々の 幾何構造の発展」(研究代表者＝佐々木武),日本工業大学, 6–8 March 2002.
[21] 無限次元表現の分岐則, 集中講義(5回), 東京大学数理科学研究科, 17–21 June 2002.
[22] Canonical Hilbert Spaces of Ultrahyperbolic Solutions,談話会,東京大学数 理科学研究科, 21 June 2002.
[23] 山辺作用素の解空間における不変な内積について,研究集会「非可換代数 系の表現と調和解析」(研究代表者＝太田琢也),京都大学数理解析研究所, 23–26 July 2002.
[24] Conformal Geometry and Analysis on Minimal Representations ofO(p, q), Workshop on Representation Theory of Lie Groups, Special Year Program on Representation Theory of Lie Groups (organized by Jeffrey Adams, Jian-Shu Li, Kyo Nishiyama, Dipendra Prasad, Gordan Savin, Eng-Chye Tan and Chen-Bo Zhu), Institute for Mathematical Sciences (IMS), Sin- gapore, 13 August 2002.
[25] Branching Problems of Unitary Representations, (invited lecture), Inter- national Congress of Mathematicians (ICM 2002), Beijing, China, 20–28 August 2002.
[26] Conformal Geometry and Global Solutions to the Yamabe Equations on Classical Pseudo-Riemannian Manifolds, Special Year Program on Repre- sentation Theory of Lie Groups (organized by Eng-Chye Tan and Chen-Bo Zhu), Institute for Mathematical Sciences (IMS), Singapore, 24 Septem- ber 2002.
[27] Restriction of Unitary Representations, Special Year Program on Repre- sentation Theory of Lie Groups (organized by Eng-Chye Tan and Chen-Bo Zhu), Institute for Mathematical Sciences (IMS), Singapore, 1 October 2002.
[28] Complex Geometry and Multiplicity One Theorem in Branching Laws, Seminar on Lie Groups and Representation Theory, National University of Singapore, Singapore, 22 October 2002.
[29] Branching Problems of Unitary Representations, (series lectures), 表現論 シンポジウム(世話人＝佐野茂，飯田正敏，本田龍央),富士ハイツ(静岡), 12–15 November 2002.
[30] Multiplicity One Theorem in the Branching Laws, Workshop on Represen- tations of Lie Groups, Harmonic Analysis on Homogeneous Spaces and Quantization (organized by G. van Dijk and V. F. Molchanov), Lorentz Center, Leiden University, the Netherlands, 9–13 December 2002.
[31] Conformal Geometry and Analysis on Minimal Representations ofO(p, q), The 2002 Twente Conference on Lie Groups, Twente University, the Netherlands, December 2002.
[32] O(p, q)の極小ユニタリ表現のSchr¨odingerモデル,短期共同研究集会「IV 型対称領域上の保型形式の研究」(研究代表者＝織田孝幸), 京都大学数理 解析研究所, 24–26 December 2002.
[33] Lorentz群の極小表現のSchr¨odingerモデル, 幕張ワークショップ, 放送大 学セミナーハウス(千葉), 6–8 January 2003.
[34] Analysis on Minimal Representations and Conformal Geometry, S´eminaire d’Analyse Harmonique, Institut de Math´ematiques ´Elie Car- tan, Universit´e Henri Poincar´e, France, 15 May 2003.
[35] Branching Problems of Representations of Lie Groups, colloquium, Uni- versit´e de Reims, France, 12 June 2003.
[36] A Multiplicity Theorem in the Branching Laws of Representations, Th´eorie des Repr´esentations, Th´eorie des Groupes, Universit´e de Paris VII, France, 16 June 2003.
[37] Multiplicity One Theorem on Branching Laws and Geometry of Complex Manifolds, 研究集会「Lie Theory のひろがりと新たな進展」(研究代表 者＝有木 進), 京都大学数理解析研究所, 22–25 July 2003.
[38] 非リーマン等質空間の不連続群について,日本数学会秋季総合分科会(2003 年9月)幾何学分科会特別講演, 千葉大学, September 2003.
[39] Restriction of Unitary Representations, 学術振興会日欧科学協力事業・日 独セミナー, Lie群：解析と幾何,日独シンポジウム,京都大学, 6–11 October 2003.
[40] Survey Lecture on Representation Theory of Lie Groups; Restriction of Unitary Representations, (2 lectures), Seoul National University, Republic of Korea, November 2003.
[41] 重複度1の表現について,大談話会, 京都大学, 26 November 2003.
[42] 非リーマン等質空間の不連続群について, 大岡山談話会, 東京工業大学, 8 December 2003.
[43] 不定値計量をもつ等質空間への群作用,集中講義,東京工業大学, 8–12 De- cember 2003.
[44] 両側剰余空間U(n1)×U(n2)×U(n3)\U(n)/U(p)×U(q) — 作用の可視性，
Cartan 分解の一般化, 2003年度幕張ワークショップ(世話役：野村隆昭，
熊原啓作),放送大学セミナーハウス(千葉), 6–8 January 2004.
[45] 複素多様体における可視的な作用と表現論への応用について,集中講義(5 回), 東京大学, 19–23 January 2004.
[46] Visible Actions on Complex Manifolds, and their Applications to Rep- resentation Theory, Mathematisches Kolloquium, Technische Universit¨at Clausthal, Germany, 30 January 2004.
[47] Visible Actions on Complex Manifolds and Multiplicity One Theorems, (opening lecture), Finite and Infinite Dimensional Complex Geometry and Representation Theory, Oberwolfach, Germany, 1–7 February 2004.
[48] On Complex Geometry and Multiplicity-free Representations, Fourth Workshop on Nilpotent Orbits and Representation Theory, Nagoya Uni- versity, Japan, 21–24 February 2004.
[49] 対称性と幾何 — 連続群と不連続群, 現代の数学と数理解析 — 基礎概念 とその諸科学への広がり, 京都大学数理解析研究所, 28 May 2004.
[50] Restriction of Unitary Representations of Reductive Lie Groups, Inter- national Symposium on Representation Theory and Harmonic Analysis, Urumqi, Xinjiang, China, 2–8 August 2004.
[51] Visible Actions on Complex Manifolds and Multiplicity-free Representa- tions, Conference on Lie Groups and Representation Theory, Odense, Denmark, 10–13 August 2004.
[52] Schr¨odinger Model of the Minimal Representation of the Lorentz Group O(p, q), Workshop on Harmonic Analysis and Homogeneous Spaces (in honor of Professor G. van Dijk), Lorentz Center, Leiden University, the Netherlands, 23–26 August 2004.
[53] Restriction of Unitary Representations — Discrete and continuous spec- trum, (plenary lecture), Sixth Pan-African Congress of Mathematicians
(PACOM2004), Institut National des Sciences Appliqu´ees et de la Tech- nologie (INSAT), Tunis, Tunisia, September 2004.
[54] Conformal Geometry and Analysis on Minimal Representation of O(p, q), PACOM2004, Parallel Session: Lie Groups and Representation Theory, Tunis, Tunisia, September 2004.
[55] Restriction of Unitary Representations of Real Reductive Groups, 第6回 冪零軌道と表現論研究集会(プログラム責任者＝関口次郎),河口湖富士桜 荘, 6–10 September 2004.
[56] 極小表現の極小K-typeのFourier変換の公式について, RIMS研究集会
「Sp(2, R)とSU(2,2)上の保型形式, III」(研究代表者＝織田孝幸),京都大 学数理解析研究所, September 2004.
[57] 重複度1の表現について,談話会, 大阪市立大学, 2 February 2005.
[58] Multiplicity-free Representations and Visible Actions on Complex Mani- folds, 集中講義(4回),大阪市立大学, 31 January–4 February 2005.
[59] Visible Actions on Complex Manifolds and Multiplicity-free Theorems, (2 lectures), International Symposium on Representation Theory and Au- tomorphic Forms, Seoul National University, Seoul, Republic of Korea, 14–17 February 2005.
[60] Analysis on Minimal Representation of Indefinite Orthogonal Groups, S´eminaire Th´eorie de Lie et Applications, Groupe de travail, Universit´e de Poitiers, France, 26 May 2005.
[61] Visible Actions on Complex Manifolds and Multiplicity-free Representa- tions, S´eminaire Th´eorie de Lie et applications, Universit´e de Poitiers, France, 2 June 2005.
[62] Analysis on Homogeneous Spaces Revisited — From Viewpoint of Branch- ing Laws of Unitary Representations, Harmonic Analysis on Lie Groups and Symmetric Spaces in honor of Jacques Faraut, Joint meeting of Sem- inar Sophus Lie, Nancy, France, 10–11 June 2005.
[63] Integral Formulas for the Minimal Representation of the Conformal Group, Seminar organized by Professor Faraut, Analyse Alg´ebrique, Uni- versit´e Pierre et Marie Curie, France, 22 June 2005.
[64] リー群と表現論入門, 現代の数学と数理解析 —基礎概念とその諸科学へ の広がり, 京都大学数理解析研究所, 8 July 2005.
[65] Multiplicity-free Theorem and Visible Actions on Complex Manifolds, The Asian Mathematical Conference (AMC2005), Singapore, 20–23 July 2005.
[66] Multiplicity-free Representations and Visible Actions on Complex Mani- folds: 1. Multiplicity-free theorem; 2. Visible actions on symmetric spaces;
3. Visible actions on non-symmetric spaces, (3 lectures), Workshop on Integral Transformations on Homogeneous Spaces (organized by Toshio Oshima), Tambara Institute of Mathematical Sciences, the University of Tokyo, Japan, 21–25 August 2005.
[67] Visible Actions on Complex Manifolds and Multiplicity-free Representa-
tions, RIMSシンポジウム「カルタン幾何の進化発展と関連する数学の諸
問題」(研究代表者＝森本 徹), 京都大学数理解析研究所, 24–27 October 2005.
[68] 重複度1の表現と複素多様体上の可視的な作用(概説講演),表現論シンポ ジウム, 掛川, 15–18 November 2005.
[69] Restrictions of Unitary Representations of Real Reductive Groups, Inter- national Conference on Representations of Real Reductive Groups, in honor of R. Parthasarathy, Mumbai, India, 2–5 January 2006.
[70] Multiplicity-free Representations and Visible Actions on Complex Mani- folds, (opening lecture), International Conference on Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory (on the Occasion of Professor Roger Howe’s 60th Birthday), Singapore, 9–11 January 2006.
[71] 複素多様体における可視的な作用について, 等質空間の幾何学的諸相(金 行壮二先生退職記念研究集会), 名城大学, 2–4 March 2006.
[72] 局所的に同じ対称性をもつ空間がとりうる大域的な形,談話会, 京都大学, 12 April 2006.
[73] 群作用とリー群の表現論, 集中講義(5回),広島大学, 24–28 April 2006.
[74] 可視的な作用と重複のない表現,談話会, 広島大学, 25 April 2006.
[75] 複素解析的手法によるユニタリ表現の話題から, 集中講義(5回), 東京大 学, 8–12 May 2006.
[76] 対称性の数学, 現代の数学と数理解析 —基礎概念とその諸科学への広が り, 数理解析研究所, 26 May 2006.
[77] Is the Universe Closed? — Existence Problem of Compact Clifford-Klein Forms of Symmetric Spaces, colloquium, Paderborn, Germany, 4 July 2006. promised.
[78] TBA, Strategic Workshop and Round Table Discussion on Perspectives in Representation Theory (organized by Steffen Koenig, Henning Krause, Peter Littelmann, and Gunter Malle), Physikzentrum Bad Honnef, Ger- many, 7–9 July 2006. promised.
[79] Is the Universe Closed? — Existence Problem of Compact Locally Sym- metric Spaces, colloquium, Darmstadt, Germany, 12 July 2006. promised.
[80] Unitary Representations, Restrictions, and their Applications, (course lec- tures), Summer School 2006 on Microlocal and Geometric Methods in Representation Theory, G¨unzburg, Germany, 17–28 July 2006. promised.
[81] TBA, (course lectures), Summer School on Geometry Topology, Analysis of Locally Symmetric Spaces and Discrete Groups, Beijing, Republic of China, 17 July–4 August 2006. promised.
