「部分多様体の幾何学における

有限次元および無限次元リー理論的方法」

Ernst Heintze 教授 集中講義

「Kac-Moody対称空間の理論」

組織者：

大仁田　義裕（大阪市立大学）

酒井　高司（大阪市立大学 数学研究所）

集中講義　：

平成１９年１０月２日（火）-５日（金)，３回

Ernst Heintze 教授 (Univ. of Augsburg, Germany)

"Theory of Kac-Moody Symmetric Spaces"

Lecture 1: Finite dimensional symmetric spaces and their isotropy representations.

We review basic properties of finite dimensional Riemannian symmetric spaces and consider their isotropy representations. We show that this is polar and that principal orbits of polar representations are isoparametric. By results of Dadok, Palais, Terng and Thorbergsson "symmpetric spaces", "polar representations" and "isoparametric submanifolds" of R^n are essentially in bijection to each other.

Lecture 2: Affine Kac-Moody groups and corresponding infinite dimensional symmetric spaces

A compact Lie group with biinvariant metric is one of the simplest examples of a finite dimensional symmetric space. Probably the most direct generalization to infinite dimensions is an affine Kac-Moody group. We describe affine Kac-Moody algebras and their corresponding groups and show that they can be viewed as a symmetric space. More generally we consider symmetric spaces G/K where G is an affine Kac-Moody group. We show that their isotropy representations are polar and are closely related to Terng's P(G,H)-actions on Hilbert space by gauge transformations.

Lecture 3: Classification and geometry of Kac-Moody symmetric spaces

We show that these spaces are classified by pairs of automorphisms of order less or equal two of finite dimensional simple Lie algebras (up to a certain equivalence relation) and indicate that several basic geometric properties of finite dimensional symmetric spaces carry over to this new situation. We conjecture that these spaces, polar actions on Hilbert spaces and isoparametric submanifolds of Hilbert spaces are essentially also in bijection to each other. Finally we address some of the unsolved problems in this theory.

微分幾何学特別セミナー：

Ernst Heintze (Univ. of Augsburg, Germany)

"Polar actions on Hilbert spaces"

Jorge Lauret (Univ. Nat. Cordoba, Argentina) PDF ファイル

"A canonical Riemannian metric for geometric structures on nilmanifolds"

田丸　博士（広島大理) PDF ファイル

"Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds"

小池　直之（理科大理) PDF ファイル

"Isoparametric submanifolds in a symmetric space of non-compact type and infinite dimensional anti-Kählerian geometry"

酒井　高司（阪市大数学研)

"Weakly reflective orbits and tangentially degenerate orbits of s-representations"

坊向　伸隆（阪市大数学研) PDF ファイル

"The classification of simple irreducible pseudo-Hermitian symmetric spaces : from a viewpoint of elliptic orbits"

北川　友美子（阪市大数学研) PDF ファイル

"On infinitesimal automorphisms of sub-Riemannian manifolds"

David Brander（神戸大理 学振特別研究員)

"Special submanifolds from Grassmann geometries in loop group homogeneous spaces"

笹木　集夢（早稲田大理工 D4) PDF ファイル

"Visible actions on multiplicity-free spaces"

馬場　蔵人（理科大理 D2) PDF ファイル

"Local orbit types of s-representations of semi-simple pseudo-Riemannian symmetric spaces"

昆　万佑子（北大理 D3) PDF ファイル

"Compact minimal CR submanifolds of a complex projective space with positive Ricci curvature"

大仁田　義裕（阪市大理)

"Differential geometry of Lagrangian submanifolds and related variational problem"

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