## GCOE lecture series

Seminar information archive ～02/01｜Next seminar｜Future seminars 02/02～

**Seminar information archive**

### 2009/02/13

15:00-16:00 Room #370 (Graduate School of Math. Sci. Bldg.)

ASYMPTOTIC EXPANSIONS FOR SOME HYPERBOLIC EQUATIONS 第3講

**Vladimir Romanov**(Sobolev Instutite of Mathematics)ASYMPTOTIC EXPANSIONS FOR SOME HYPERBOLIC EQUATIONS 第3講

[ Abstract ]

For a linear second-order hyperbolic equation with variable coefficients the fundamental solution for the Cauchy problem is considered. An asymptotic expansion of this solution in a neighborhood of the characteristic cone is introduced and explicit formulae for coefficients of this expansion are derived. Similar questions are discussed for the elasticity equations related to an inhomogeneous isotropic medium.

For a linear second-order hyperbolic equation with variable coefficients the fundamental solution for the Cauchy problem is considered. An asymptotic expansion of this solution in a neighborhood of the characteristic cone is introduced and explicit formulae for coefficients of this expansion are derived. Similar questions are discussed for the elasticity equations related to an inhomogeneous isotropic medium.

### 2009/02/06

15:00-16:00 Room #370 (Graduate School of Math. Sci. Bldg.)

ASYMPTOTIC EXPANSIONS FOR SOME HYPERBOLIC EQUATIONS 第2講

**Vladimir Romanov**(Sobolev Instutite of Mathematics)ASYMPTOTIC EXPANSIONS FOR SOME HYPERBOLIC EQUATIONS 第2講

[ Abstract ]

For a linear second-order hyperbolic equation with variable coefficients the fundamental solution for the Cauchy problem is considered. An asymptotic expansion of this solution in a neighborhood of the characteristic cone is introduced and explicit formulae for coefficients of this expansion are derived. Similar questions are discussed for the elasticity equations related to an inhomogeneous isotropic medium.

For a linear second-order hyperbolic equation with variable coefficients the fundamental solution for the Cauchy problem is considered. An asymptotic expansion of this solution in a neighborhood of the characteristic cone is introduced and explicit formulae for coefficients of this expansion are derived. Similar questions are discussed for the elasticity equations related to an inhomogeneous isotropic medium.

### 2009/01/26

17:15-18:15 Room #470 (Graduate School of Math. Sci. Bldg.)

ASYMPTOTIC EXPANSIONS FOR SOME HYPERBOLIC EQUATIONS 第1講

**Vladimir Romanov**(Sobolev Instutite of Mathematics)ASYMPTOTIC EXPANSIONS FOR SOME HYPERBOLIC EQUATIONS 第1講

[ Abstract ]

For a linear second-order hyperbolic equation with variable coefficients the fundamental solution for the Cauchy problem is considered. An asymptotic expansion of this solution in a neighborhood of the characteristic cone is introduced and explicit formulae for coefficients of this expansion are derived. Similar questions are discussed for the elasticity equations related to an inhomogeneous isotropic medium.

For a linear second-order hyperbolic equation with variable coefficients the fundamental solution for the Cauchy problem is considered. An asymptotic expansion of this solution in a neighborhood of the characteristic cone is introduced and explicit formulae for coefficients of this expansion are derived. Similar questions are discussed for the elasticity equations related to an inhomogeneous isotropic medium.

### 2009/01/23

17:00-18:00 Room #370 (Graduate School of Math. Sci. Bldg.)

The spectral category of Hecke algebras and applications 第4講 Example: Lusztig's unipotent representations for classical groups.

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2009.html#20090108opdam

**Eric Opdam**(University of Amsterdam )The spectral category of Hecke algebras and applications 第4講 Example: Lusztig's unipotent representations for classical groups.

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2009.html#20090108opdam

### 2009/01/22

17:00-18:00 Room #370 (Graduate School of Math. Sci. Bldg.)

The spectral category of Hecke algebras and applications 第3講 The spectral category and correspondences of tempered representations.

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2009.html#20090108opdam

**Eric Opdam**(University of Amsterdam)The spectral category of Hecke algebras and applications 第3講 The spectral category and correspondences of tempered representations.

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2009.html#20090108opdam

### 2009/01/09

17:00-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

The spectral category of Hecke algebras and applications 第2講 Affine Hecke algebras and harmonic analysis.

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2009.html#20090108opdam

**Eric Opdam**(University of Amsterdam)The spectral category of Hecke algebras and applications 第2講 Affine Hecke algebras and harmonic analysis.

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2009.html#20090108opdam

### 2009/01/08

17:00-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

The spectral category of Hecke algebras and applications

第1講: Reductive p-adic groups and Hecke algebras

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2009.html#20090108opdam

**Eric Opdam**(University of Amsterdam)The spectral category of Hecke algebras and applications

第1講: Reductive p-adic groups and Hecke algebras

[ Abstract ]

Hecke algebras play an important role in the harmonic analysis of a p-adic reductive group. On the other hand, their representation theory and harmonic analysis can be described almost completely explicitly. This makes affine Hecke algebras an ideal tool to study the harmonic analysis of p-adic groups. We will illustrate this in this series of lectures by explaining how various components of the Bernstein center contribute to the level-0 L-packets of tempered representations, purely from the point of view of harmonic analysis.

We define a "spectral category" of (affine) Hecke algebras. The morphisms in this category are not algebra morphisms but are affine morphisms between the associated tori of unramified characters, which are compatible with respect to the so-called Harish-Chandra μ-functions. We show that such a morphism generates a Plancherel measure preserving correspondence between the tempered spectra of the two Hecke algebras involved. We will discuss typical examples of spectral morphisms.

We apply the spectral correspondences of affine Hecke algebras to level-0 representations of a quasi-split simple p-adic group. We will concentrate on the example of the special orthogonal groups $SO_{2n+1}(K)$. We show that all affine Hecke algebras which arise in this context admit a *unique* spectral morphism to the Iwahori-Matsumoto Hecke algebra, a remarkable phenomenon that is crucial for this method. We will recover in this way Lusztig's classification of "unipotent" representations.

[ Reference URL ]Hecke algebras play an important role in the harmonic analysis of a p-adic reductive group. On the other hand, their representation theory and harmonic analysis can be described almost completely explicitly. This makes affine Hecke algebras an ideal tool to study the harmonic analysis of p-adic groups. We will illustrate this in this series of lectures by explaining how various components of the Bernstein center contribute to the level-0 L-packets of tempered representations, purely from the point of view of harmonic analysis.

We define a "spectral category" of (affine) Hecke algebras. The morphisms in this category are not algebra morphisms but are affine morphisms between the associated tori of unramified characters, which are compatible with respect to the so-called Harish-Chandra μ-functions. We show that such a morphism generates a Plancherel measure preserving correspondence between the tempered spectra of the two Hecke algebras involved. We will discuss typical examples of spectral morphisms.

We apply the spectral correspondences of affine Hecke algebras to level-0 representations of a quasi-split simple p-adic group. We will concentrate on the example of the special orthogonal groups $SO_{2n+1}(K)$. We show that all affine Hecke algebras which arise in this context admit a *unique* spectral morphism to the Iwahori-Matsumoto Hecke algebra, a remarkable phenomenon that is crucial for this method. We will recover in this way Lusztig's classification of "unipotent" representations.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2009.html#20090108opdam

### 2008/11/28

14:40-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Circle-valued Morse theory

, Lecture 2

**Andrei Pajitnov**(Univ. de Nantes)Circle-valued Morse theory

, Lecture 2

### 2008/11/26

14:40-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Circle-valued Morse theory, Lecture 1

**Andrei Pajitnov**(Univ. de Nantes)Circle-valued Morse theory, Lecture 1

[ Abstract ]

Morse theory of circle-valued functions, initiated by S. P. Novikov in 1980-1982 is now a rapidly developing domain with applications and connections to many other fields of geometry and topology such as dynamical systems, Lagrangian intersections,

knots and links in three-dimensional sphere.

We will start with the basics of the theory, discuss the construction of the Novikov complex, relations with the dynamical zeta functions, and the knot theory. We will conclude with a list of the open problems of the theory.

Morse theory of circle-valued functions, initiated by S. P. Novikov in 1980-1982 is now a rapidly developing domain with applications and connections to many other fields of geometry and topology such as dynamical systems, Lagrangian intersections,

knots and links in three-dimensional sphere.

We will start with the basics of the theory, discuss the construction of the Novikov complex, relations with the dynamical zeta functions, and the knot theory. We will conclude with a list of the open problems of the theory.

### 2008/10/27

16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Holomorphic extensions of highest weight representations to Olshanskii semigroups

**Joachim Hilgert**(Paderborn University)Holomorphic extensions of highest weight representations to Olshanskii semigroups

[ Abstract ]

In this lecture I will present a proof of Olshanskii's Theorem, which says that

for a simple group of Hermitean type unitarizable highest weight

representations can be holomorphically extended to contractive representations

of a complex semigroup containing the group in its boundary.

In this lecture I will present a proof of Olshanskii's Theorem, which says that

for a simple group of Hermitean type unitarizable highest weight

representations can be holomorphically extended to contractive representations

of a complex semigroup containing the group in its boundary.

### 2008/10/17

15:00-16:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Holomorphic extensions of unitary representations その4 Applications and open problems

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

**Joachim Hilgert**(Paderborn University)Holomorphic extensions of unitary representations その4 Applications and open problems

[ Abstract ]

In this lecture we present further applications of the given extension results and describe some open problems. In particular, we will mention estimates for automorphic forms (Krötz-Stanton), random matrices (Huckleberry-Püttmann-Zirnbauer), unitarizability of highest weight representation with non-scalar lowest K-type, and infinite dimensional groups.

[ Reference URL ]In this lecture we present further applications of the given extension results and describe some open problems. In particular, we will mention estimates for automorphic forms (Krötz-Stanton), random matrices (Huckleberry-Püttmann-Zirnbauer), unitarizability of highest weight representation with non-scalar lowest K-type, and infinite dimensional groups.

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

### 2008/10/16

15:00-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Holomorphic extensions of unitary representations その3 Highest weight representations

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

**Joachim Hilgert**(Paderborn University)Holomorphic extensions of unitary representations その3 Highest weight representations

[ Abstract ]

In this lecture we explain the extension results in a little more detail and explain how they lead to geometric realizations of singular highest weight representations on nilpotent coadjoint orbits.

[ Reference URL ]In this lecture we explain the extension results in a little more detail and explain how they lead to geometric realizations of singular highest weight representations on nilpotent coadjoint orbits.

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

### 2008/10/15

15:00-16:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Holomorphic extensions of unitary representations その2 Geometric background

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2008.html#20081014hilgert

**Joachim Hilgert**(Paderborn University)Holomorphic extensions of unitary representations その2 Geometric background

[ Abstract ]

In this lecture we will explain the complex geometry needed to understand the phenomena described in the first lecture. The key words here are Olshanski semigroups, invariant cones in Lie algebras, Akhiezer-Gindikin domain, and coadjoint orbits of convex type.

[ Reference URL ]In this lecture we will explain the complex geometry needed to understand the phenomena described in the first lecture. The key words here are Olshanski semigroups, invariant cones in Lie algebras, Akhiezer-Gindikin domain, and coadjoint orbits of convex type.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2008.html#20081014hilgert

### 2008/10/14

15:00-16:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Holomorphic extensions of unitary representations" その1 "Overview and Examples"

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2008.html#20081014hilgert

**Joachim Hilgert**(Paderborn University)Holomorphic extensions of unitary representations" その1 "Overview and Examples"

[ Abstract ]

In this lecture we present the Gelfand-Gindikin program of decomposing $L^2$-spaces into families of irreducible representations using complex geometry. We then briefly outline results due to Olshanski, Hilgert-Olafsson-Orsted, Hilgert-Neeb-Orsted, Krotz-Stanton and others in this direction. In particular, we will explain holomorphic extensions of holomorphic discrete series representations and their relation to Hardy and weighted Bergman spaces.

[ Reference URL ]In this lecture we present the Gelfand-Gindikin program of decomposing $L^2$-spaces into families of irreducible representations using complex geometry. We then briefly outline results due to Olshanski, Hilgert-Olafsson-Orsted, Hilgert-Neeb-Orsted, Krotz-Stanton and others in this direction. In particular, we will explain holomorphic extensions of holomorphic discrete series representations and their relation to Hardy and weighted Bergman spaces.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2008.html#20081014hilgert