共形幾何と分岐則

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(14) . 共形幾何と表現論小林俊行（東京大学大学院数理科学研究科）共形幾何学において自然に現れる表現論と部分群への分岐則に関して、最近活発に進展している話題から２つのテーマＡ，Ｂを取り上げて概説する。. A. 任意の擬リーマン多様体に対して、その共形変換群の表現を自然な形で構成し、その部分群である等長変換群への分岐則を手法として、大域解の空間を理解する。例えば、 • ユニタリ化 vs 微分方程式の保存量の存在 • 共形同相だ等長ではない幾何モデルの活用 • フーリエ変換の一般化と変形などがこの話題に関連して自然な形で登場する。. B. 擬リーマン多様体 X とその部分多様体 Y が与えられたとき、２つの共形変換群が定義される。この組に関して自然に生じる分岐則の問題を紹介する。 1.

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(467) References • References to Topic A—General scheme / Open problem. [1] T. Kobayashi, Special functions in minimal representations, In: Perspectives in Representation Theory in honor of Igor Frenkel on his 60th birthday, Comtemp. Math., 610, pp.253–266. Amer. Math. Soc., 2014. [2] T. Kobayashi. Geometric analysis on minimal representations. In J. Matsuzawa and S. Tsunoda, editors, Ninth Oka Symposium Lecture Notes, pages 27-61. Department of Mathematics, Faculty of Science, Nara Women’s University, 2011. [3] T. Kobayashi. Algebraic analysis of minimal representations. Publ. RIMS (Publications of the Research Institute for Mathematical Sciences), 47(2), pp.585-611, 2011. Special issue in commemoration of the golden jubilee of algebraic analysis. [4] T. Kobayashi, Conformal geometry and global solutions to the Yamabe equations on classical pseudo-Riemannian manifolds, Proceedings of the 22nd Winter School “Geometry and Physics” (Srn´ı, 2002), Rend. Circ. Mat. Palermo (2) Suppl. 71, 2003, pp.15–40. • References to Topic A—further papers on geometric analysis of “small representations” with emphasis on branching problems [5] T. Kobayashi, Y. Oshima, Classification of symmetric pairs with discretely decomposable restrictions of (g, K)-modules, Journal f¨ ur die reine und angewandte Mathematik, 2015, (2015), no.703, pp.201– 223. [6] J. Hilgert, T. Kobayashi, J. Möllers, and B. Ørsted. Fock model and Segal-Bargmann transform for minimal representations of Hermitian Lie groups. Journal of Functional Analysis, 263(11) pp.3492– 3563, 2012. [7] J. Hilgert, T. Kobayashi, and J. Möllers, Minimal representations via Bessel operators, Journal of Mathematical Society of Japan, 66, (2014), 349–414.. 20.

(468) [8] S. Ben Sa¨ıd, T. Kobayashi, and B. Ørsted. Laguerre semigroup and Dunkl operators. Compositio Mathematica, 148, (2012), 1265– 1336. [9] T. Kobayashi, G. Mano, The Schrödinger model for the minimal representation of the indefinite orthogonal group O(p, q), Mem. Amer. Math. Soc. (2011), 212, no. 1000, vi+132 pages. [10] T. Kobayashi, B. Ørsted, M. Pevzner, Geometric analysis on small unitary representations of GL(n, R), J. Funct. Anal., 260, (2011) 1682–1720. [11] J. Hilgert, T. Kobayashi, G. Mano, and J. Möllers. Orthogonal polynomials associated to a certain fourth order differential equation. Ramanujan Journal, 26, 295–310, 2011; Special functions associated to a certain fourth order differential equation. Ramanujan Journal, 26, 1–34, 2011. [12] T. Kobayashi and G. Mano. The inversion formula and holomorphic extension of the minimal representation of the conformal group. In Jian-Shu Li, Eng-Chye Tan, Nolan Wallach, and Chen-Bo Zhu, editors, Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory: In Honour of Roger E. Howe, pages 159–223. Singapore University Press and World Scientific Publishing, 2007. [13] T. Kobayashi, B. Ørsted, Analysis on the minimal representation of O(p, q). Part I, Adv. Math., 180, (2003), pp.486–512; Part II, ibid, pp.513–550; Part III, ibid, pp.551–595. • Books for Topic B [15] A. Juhl, Families of Conformally Covariant Differential Operators, Q-Curvature and Holography, Prog. Math., 275, 2009. [16] T. Kobayashi and B. Speh. Symmetry Breaking for Representations of Rank One Orthogonal Groups, Memoirs of American Mathematical Society 238, 2015. 118 pp. ISBN: 978-1-4704-1922-6. [17] T. Kobayashi, T. Kubo, and M. Pevzner, Conformal symmetry breaking operators for differential forms on spheres, viii+192 pages. Lecture Notes in Mathematics, vol. 2170, 2016. viii+192pages ISBN: 978-981-10-2656-0.. 21.

(469) • Some papers related to Topic B Reference–General Scheme/Classification Theory/Open Problems [18] T. Kobayashi. A program for branching problems in the representation theory of real reductive groups. Progr. Math. 312, pp.277–322, 2015. (Special volume for Vogan’s 60th Birthday). [19] T. Kobayashi. Shintani functions, real spherical manifolds, and symmetry breaking operators. Developments in Mathematics, 37, pp.127–159, 2014. [20] T. Kobayashi and T. Matsuki. Classification of finite-multiplicity symmetric pairs, Transformation Groups, 19, (2014), pp.457–493. Special Issue in honour of Professor Dynkin for his 90th birthday [21] T. Kobayashi and T. Oshima. Finite multiplicity theorems for induction and restriction. Advances in Mathematics, 248, (2013), pp.921–944. • F-method to find differential symmetry breaking operators [22] T. Kobayashi. F-method for constructing equivariant differential operators. Contemporary Mathematics, 598, pp.141–148. Amer. Math. Soc., 2013. (Special volume for Helgason’s 85th birthday). [23] T. Kobayashi. F-method for symmetry breaking operators. Differential Geometry and its Applications, 33, (2014), pp.272–289, (Special issue for Michael Eastwood’s 60th birthday). [24] T. Kobayashi, B. Ørsted, P. Somberg, and V. Souˇcek. Branching laws for Verma modules and applications in parabolic geometry. I. Advances in Mathematics, 285, (2015), pp.1796–1852. [25] T. Kobayashi and M. Pevzner. Differential symmetry breaking operators. I. General theory and F-method. Selecta Mathematica, 22, (2016), pp.801–845, [26] T. Kobayashi and M. Pevzner. Differential symmetry breaking operators. II. Rankin–Cohen operators for symmetric pairs, Selecta Mathematica 22, (2016), pp.847–911.. 22.

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