"Differential Equations and Symmetric Spaces"

January 13-16, 2009

Date: | January 13 (Tue) - January 16 (Fri) |

Banquet on January 15 (Thu) 18:00- | |

Venue: | Graduate School of Mathematical Sciences, The University of Tokyo, Japan [ Access ] |

Invited speakers

Program

Abstracts

- Erik van den Ban (Utrecht University)
- Patrick Delorme (University of Aix-Marseille II)
- Mogens Flensted-Jensen (University of Copenhagen)
- Masaki Kashiwara (”Œ´³Ž÷) (RIMS)
- Toshiyuki Kobayashi (¬—Ñrs) (The University of Tokyo)
- Toshihiko Matsuki (¼–Ø•q•F) (Kyoto University)
- Hisayosi Matumoto (¼–{‹v‹`) (The University of Tokyo)
- Hiroyuki Ochiai (—Ž‡Œ[”V) (Nagoya University)
- Yasunori Okada (‰ª“c–õ‘¥) (Chiba University)
- Eric Opdam (University of Amsterdam)
- Toshio Oshima (‘å“‡—˜—Y) (The University of Tokyo)
- Yoshiki Otobe (‰³•”ŒµŒÈ) (Shinshu University)
- Hideko Sekiguchi (ŠÖŒû‰pŽq) (The University of Tokyo)
- Jiro Sekiguchi (ŠÖŒûŽŸ˜Y) (Tokyo University of Agriculture and Technology)
- Nobukazu Shimeno (Ž¦–ìMˆê) (Okayama University of Science)

Jan 13 (Tue) | |

13:30-13:40 | Opening |

13:40-14:40 | Hisayosi Matumoto (The University of Tokyo) / ¼–{‹v‹` (“Œ‹ž‘åŠw) |

On homomorphisms between scalar generalized Verma modules | |

15:00-16:00 | Hideko Sekiguchi (The University of Tokyo) / ŠÖŒû‰pŽq (“Œ‹ž‘åŠw) |

Penrose transform between symmetric spaces | |

16:20-17:20 | Nobukazu Shimeno (Okayama University of Science) / Ž¦–ìMˆê (‰ªŽR—‰È‘åŠw) |

Matrix-valued commuting differential operators associated with symmetric spaces | |

Jan 14 (Wed) | |

09:30-10:30 | Jiro Sekiguchi (Tokyo University of Agriculture and Technology) / ŠÖŒûŽŸ˜Y (“Œ‹ž”_H‘åŠw) |

Unitary reflection groups and uniformization differential equations | |

11:00-12:00 | Eric Opdam (University of Amsterdam) |

[ abstract ] | Spectral transfer category of affine Hecke algebras |

13:30-14:30 | Toshihiko Matsuki (Kyoto University) / ¼–Ø•q•F (‹ž“s‘åŠw) |

Analysis and geometry on homogeneous spaces | |

15:00-16:00 | Erik van den Ban (Utrecht University) |

Relations in the Paley—Wiener space of a semisimple Lie group | |

16:30-17:30 | Patrick Delorme (University of Aix-Marseille II) |

[ abstract ] | A Paley-Wiener theorem for Whittaker functions on a reductive p-adic group |

Jan 15 (Thu) | |

09:30-10:30 | Yasunori Okada (Chiba University) / ‰ª“c–õ‘¥ (ç—t‘åŠw) |

A Massera type theorem in hyperfunctions | |

11:00-12:00 | Mogens Flensted-Jensen (University of Copenhagen) |

Harmonic analysis on semisimple symmetric spaces | |

—some highlights from the last 40 years | |

13:30-14:30 | Masaki Kashiwara (RIMS) / ”Œ´³Ž÷ (‹ž‘å”—Œ¤) |

Quantization of complex manifolds | |

15:00-16:00 | Toshiyuki Kobayashi (The University of Tokyo) / ¬—Ñrs (“Œ‹ž‘åŠw) |

[ abstract ] | Global geometry and analysis on locally symmetric spaces |

—beyond the Riemannian case | |

16:20-17:20 | Toshio Oshima (The University of Tokyo) / ‘å“‡—˜—Y (“Œ‹ž‘åŠw) |

[ abstract ] | Classification of Fuchsian systems and their connection problem |

18:00- | Banquet (Lever son Verre Komaba / ƒ‹ƒ”ƒF@ƒ\ƒ“@ƒ”ƒF[ƒ‹@‹îê) |

Jan 16 (Fri) | |

09:30-10:30 | Hiroyuki Ochiai (Nagoya University) / —Ž‡Œ[”V (–¼ŒÃ‰®‘åŠw) |

An algebraic transformation of Gauss hypergeometric function | |

11:00-12:00 | Yoshiki Otobe (Shinshu University) / ‰³•”ŒµŒÈ (MB‘åŠw) |

TeX in Japan and Professor Oshima / ‘å“‡æ¶‚Æ“ú–{‚ÌTeX |

**Erik van den Ban**

Relations in the Paley-Wiener space of a semisimple Lie group**Patrick Delorme**

A Paley-Wiener theorem for Whittaker functions on a reductive*p*-adic groupAbstract: We define a Fourier transform for functions on a reductive

*p*-adic group, which transforms by a nondegenerate character of a maximal unipotent subgroup, and with compact support modulo this unipotent subgroup. This transformation, as well as wave packets are studied using a theory of the constant term. Then, a result of V. Heiermann is used to characterize the image of this Fourier transform.**Mogens Flensted-Jensen**

Harmonic analysis on semisimple symmetric spaces - some highlights from the last 40 years**Masaki Kashiwara**(”Œ´³Ž÷)

Quantization of complex manifolds**Toshiyuki Kobayashi**(¬—Ñrs)

Global geometry and analysis on locally symmetric Spaces—beyond the Riemannian caseAbstract: The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry.

In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

In this talk, I plan to give an exposition on the recent developments on the question about the global natures of locally non-Riemannian homogeneous spaces, with emphasis on the existence problem of compact forms, rigidity and deformation.

**Toshihiko Matsuki**(¼–Ø•q•F)

Analysis and geometry on homogeneous spaces**Hisayoshi Matumoto**(¼–{‹v‹`)

On homomorphisms between scalar generalized Verma modules**Hiroyuki Ochiai**(—Ž‡Œ[”V)

An algebraic transformation of Gauss hypergeometric function**Yasunori Okada**(‰ª“c–õ‘¥)

A Massera type theorem in hyperfunctions**Eric Opdam**

Spectral transfer category of affine Hecke algebras

Abstract: Given affine Hecke algebras

*H*_{1},*H*_{2}we introduce the notion of a "spectral transfer map" from*H*_{1}to*H*_{2}. Such a map is not given by an algebra homomorphism from*H*_{1}to*H*_{2}but rather by a homomorphism from the center*Z*_{2}of*H*_{2}to the center*Z*_{1}of*H*_{1}which is required to be "compatible" in a certain way with the Harish-Chandra μ-functions on the centers*Z*_{1}and*Z*_{2}.The main property of such a transfer map is that it induces a correspondence between the tempered spectra of

*H*_{1}and*H*_{2}which respects the canonical spectral measures ("Plancherel measures"), up to a locally constant factor with values in the nonzero rational numbers.The category of smooth, unramified representations of a connected split simple p-adic group of adjoint type

*G*(*F*) is Morita equivalent, via Bernstein's functor, to a direct sum*R*of affine Hecke algebras. It is a remarkable fact that*R*admits an essentially unique spectral transfer map to the Iwahori-Matsumoto Hecke algebra of*G*which is equivariant with respect the natural action of the center*Z*(\hat{*G*}) of the Langlands dual group \hat{*G*}.Using these facts and results of joint work with Solleveld we discuss the combinatorial structure of unramified

*L*-packets of classical split groups.**Toshio Oshima**(‘å“‡—˜—Y)

Classification of Fuchsian systems and their connection problem

Abstract: We explain a classification of Fuchsian systems on the Riemann sphere together with Katz's middle convolution, Yokoyama's extension and their relation to a Kac-Moody root system discovered by Crawley-Boevey.

Then we present a beautifully unified connection formula for the solution of the Fuchsian ordinary differential equation without moduli and apply the formula to the harmonic analysis on a symmetric space.

**Yoshiki Otobe**(‰³•”ŒµŒÈ)

TeX in Japan and Professor Oshima**Hideko Sekiguchi**(ŠÖŒû‰pŽq)

Penrose transform between symmetric spaces**Jiro Sekiguchi**(ŠÖŒûŽŸ˜Y)

Unitary reflection groups and uniformization differential equations**Nobukazu Shimeno**(Ž¦–ìMˆê)

Matrix-valued commuting differential operators associated with symmetric spaces

Organizers: T. Kobayashi, H. Matumoto, H. Ochiai, H. Sekiguchi

© Toshiyuki Kobayashi