## Seminar information archive

Seminar information archive ～02/06｜Today's seminar 02/07 | Future seminars 02/08～

### 2014/09/17

#### PDE Real Analysis Seminar

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

On the abstract evolution equations of hyperbolic type (JAPANESE)

**Kentarou Yoshii**(Faculty of Science Division I, Tokyo University of Science)On the abstract evolution equations of hyperbolic type (JAPANESE)

[ Abstract ]

This talk deals with the abstract Cauchy problem for linear evolution equations of hyperbolic type in a Hilbert space. We will discuss the existence and uniqueness of its classical solution and apply the results to linear Schrödinger equations with time dependent potentials.

This talk deals with the abstract Cauchy problem for linear evolution equations of hyperbolic type in a Hilbert space. We will discuss the existence and uniqueness of its classical solution and apply the results to linear Schrödinger equations with time dependent potentials.

### 2014/09/12

#### FMSP Lectures

10:30-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Stochastic homogenization for first order Hamilton-Jacobi equations(III) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

**Hung V. Tran**(The University of Chicago)Stochastic homogenization for first order Hamilton-Jacobi equations(III) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

### 2014/09/10

#### FMSP Lectures

10:30-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Stochastic homogenization for first order Hamilton-Jacobi equations(II) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

**Hung V. Tran**(The University of Chicago)Stochastic homogenization for first order Hamilton-Jacobi equations(II) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

### 2014/09/09

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

**Hatem Zaag**(CNRS / University of Paris Nord)Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

[ Abstract ]

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

#### FMSP Lectures

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

**Hatem Zaag**(CNRS/University of Paris Nord)Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

[ Abstract ]

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

### 2014/09/08

#### FMSP Lectures

10:30-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Stochastic homogenization for first order Hamilton-Jacobi equations(I) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

**Hung V. Tran**(The University of Chicago)Stochastic homogenization for first order Hamilton-Jacobi equations(I) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

### 2014/09/04

#### Lectures

12:10-13:00 Room #470 (Graduate School of Math. Sci. Bldg.)

"X-ray imaging of moving objects" (ENGLISH)

**Samuli Siltanen**(University of Helsinki, Finland)"X-ray imaging of moving objects" (ENGLISH)

### 2014/08/28

#### thesis presentations

10:00-11:15 Room #128 (Graduate School of Math. Sci. Bldg.)

On the C1 stabilization of homoclinic tangencies for diffeomorphisms in dimension three(3次元の微分同相写像に対するホモクリニック接触のC1安定化について) (JAPANESE)

**李 曉龍**(東京大学大学院数理科学研究科)On the C1 stabilization of homoclinic tangencies for diffeomorphisms in dimension three(3次元の微分同相写像に対するホモクリニック接触のC1安定化について) (JAPANESE)

### 2014/08/06

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

The stochastic SIS epidemic model in a periodic environment (ENGLISH)

**Nicolas Bacaer**(Insitut de Recherche pour le Developpement (IRD))The stochastic SIS epidemic model in a periodic environment (ENGLISH)

[ Abstract ]

In the stochastic SIS epidemic model with a contact rate a,

a recovery rate bT is such that (log T)/N converges to c=b/a-1-log(b/a) as N grows to

infinity. We consider the more realistic case where the contact rate

a(t) is a periodic function whose average is bigger than b. Then (log

T)/N converges to a new limit C, which is linked to a time-periodic

Hamilton-Jacobi equation. When a(t) is a cosine function with small

amplitude or high (resp. low) frequency, approximate formulas for C

can be obtained analytically following the method used in [Assaf et

al. (2008) Population extinction in a time-modulated environment. Phys

Rev E 78, 041123]. These results are illustrated by numerical

simulations.

In the stochastic SIS epidemic model with a contact rate a,

a recovery rate bT is such that (log T)/N converges to c=b/a-1-log(b/a) as N grows to

infinity. We consider the more realistic case where the contact rate

a(t) is a periodic function whose average is bigger than b. Then (log

T)/N converges to a new limit C, which is linked to a time-periodic

Hamilton-Jacobi equation. When a(t) is a cosine function with small

amplitude or high (resp. low) frequency, approximate formulas for C

can be obtained analytically following the method used in [Assaf et

al. (2008) Population extinction in a time-modulated environment. Phys

Rev E 78, 041123]. These results are illustrated by numerical

simulations.

### 2014/07/29

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Asymptotic structure of free Araki-Woods factors (ENGLISH)

**Cyril Houdayer**(ENS Lyon)Asymptotic structure of free Araki-Woods factors (ENGLISH)

### 2014/07/28

#### Numerical Analysis Seminar

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Computer assisted analysis of Craik’s and Pehlivan’s 3D dynamical systems (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Tomoyuki Miyaji**(RIMS, Kyoto University)Computer assisted analysis of Craik’s and Pehlivan’s 3D dynamical systems (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

### 2014/07/25

#### Colloquium

16:30-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Mathematical modelling of Tumor Immune System Interaction (JAPANESE)

**Yasuhiro Takeuchi**(Aoyama Gakuin University)Mathematical modelling of Tumor Immune System Interaction (JAPANESE)

[ Abstract ]

We study the dynamical behavior of a tumor-immune system (T-IS) interaction model with two discrete delays,

namely the immune activation delay for effector cells (ECs) and activation delay for Helper T cells (HTCs).

By analyzing the characteristic equations, we establish the stability of two equilibria (tumor-free equilibrium and immune-control equilibrium) and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter.

Our results exhibit that both delays do not affect the stability of tumor-free equilibrium.

However, they are able to destabilize the immune-control equilibrium and cause periodic solutions.

We numerically illustrate how these two delays can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumors.

The numerical simulation results show that the immune activation delay for HTCs can induce heteroclinic cycles to connect the tumor-free equilibrium and immune-control equilibrium.

Furthermore, we observe that the immune activation delay for HTCs can even stabilize the unstable immune-control equilibrium.

We study the dynamical behavior of a tumor-immune system (T-IS) interaction model with two discrete delays,

namely the immune activation delay for effector cells (ECs) and activation delay for Helper T cells (HTCs).

By analyzing the characteristic equations, we establish the stability of two equilibria (tumor-free equilibrium and immune-control equilibrium) and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter.

Our results exhibit that both delays do not affect the stability of tumor-free equilibrium.

However, they are able to destabilize the immune-control equilibrium and cause periodic solutions.

We numerically illustrate how these two delays can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumors.

The numerical simulation results show that the immune activation delay for HTCs can induce heteroclinic cycles to connect the tumor-free equilibrium and immune-control equilibrium.

Furthermore, we observe that the immune activation delay for HTCs can even stabilize the unstable immune-control equilibrium.

#### thesis presentations

10:30-11:45 Room #128 (Graduate School of Math. Sci. Bldg.)

On the study of front propagation in nonlinear free boundary problems(非線形自由境界問題における波面の伝播の研究) (JAPANESE)

**周 茂林**(東京大学大学院数理科学研究科)On the study of front propagation in nonlinear free boundary problems(非線形自由境界問題における波面の伝播の研究) (JAPANESE)

### 2014/07/24

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Decomposition of the Mobius energy (JAPANESE)

**Nagasawa Takeyuki**(Saitama University)Decomposition of the Mobius energy (JAPANESE)

### 2014/07/23

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

The Cuntz semigroup---a critical component for classification? (ENGLISH)

**George Elliott**(Univ. Toronto)The Cuntz semigroup---a critical component for classification? (ENGLISH)

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Resting-state brain networks, their energy landscapes, and sleep (JAPANESE)

**Naoki Masuda**(University of Bristol, Department of Engineering Mathematics)Resting-state brain networks, their energy landscapes, and sleep (JAPANESE)

### 2014/07/22

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The Index Map and Reciprocity Laws for Contou-Carrere Symbols (ENGLISH)

**Jesse Wolfson**(Northwestern University)The Index Map and Reciprocity Laws for Contou-Carrere Symbols (ENGLISH)

[ Abstract ]

In the 1960s, Atiyah and Janich constructed the families index as a natural map from the space of Fredholm operators to the classifying space of topological K-theory, and showed it to be an equivalence. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. Building on recent work of Sho Saito, we show this provides an analogue of Atiyah and Janich's equivalence. More significantly, the index map allows us to relate the Contou-Carrere symbol, a local analytic invariant of schemes, to algebraic K-theory. Using this, we provide new proofs of reciprocity laws for Contou-Carrere symbols in dimension 1 (first established by Anderson--Pablos Romo) and 2 (established recently by Osipov--Zhu). We extend these reciprocity laws to all dimensions.

In the 1960s, Atiyah and Janich constructed the families index as a natural map from the space of Fredholm operators to the classifying space of topological K-theory, and showed it to be an equivalence. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. Building on recent work of Sho Saito, we show this provides an analogue of Atiyah and Janich's equivalence. More significantly, the index map allows us to relate the Contou-Carrere symbol, a local analytic invariant of schemes, to algebraic K-theory. Using this, we provide new proofs of reciprocity laws for Contou-Carrere symbols in dimension 1 (first established by Anderson--Pablos Romo) and 2 (established recently by Osipov--Zhu). We extend these reciprocity laws to all dimensions.

### 2014/07/19

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Hurwitz integrality of the power series expansion of the sigma function at the origin (JAPANESE)

種数 3 の trigonal curve から来る Kummer 多様体の定義方程式と Coble の超平面 (JAPANESE)

**Yoshihiro Onishi**(Miejyo University) 13:30-14:30Hurwitz integrality of the power series expansion of the sigma function at the origin (JAPANESE)

**Yoshihiro Onishi**(Meijyo University) 15:00-16:00種数 3 の trigonal curve から来る Kummer 多様体の定義方程式と Coble の超平面 (JAPANESE)

### 2014/07/17

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

The space of compact shrinking solutions to Lagrangian mean curvature flow in $C^2$ (ENGLISH)

**Jingyi Chen**(University of British Columbia)The space of compact shrinking solutions to Lagrangian mean curvature flow in $C^2$ (ENGLISH)

[ Abstract ]

We will discuss compactness and rigidity of compact surfaces which are shrinking solutions to Lagrangian mean curvature flow. This is recent joint work with John Ma.

We will discuss compactness and rigidity of compact surfaces which are shrinking solutions to Lagrangian mean curvature flow. This is recent joint work with John Ma.

### 2014/07/16

#### Infinite Analysis Seminar Tokyo

10:30-12:00 Room #002 (Graduate School of Math. Sci. Bldg.)

From the Hilbert scheme to m/n Pieri rules (ENGLISH)

**Andrei Negut**(Columbia University, Department of Mathematics)From the Hilbert scheme to m/n Pieri rules (ENGLISH)

[ Abstract ]

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

### 2014/07/15

#### Infinite Analysis Seminar Tokyo

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

From the shuffle algebra to the Hilbert scheme (ENGLISH)

**Andrei Negut**(Columbia University, Department of Mathematics)From the shuffle algebra to the Hilbert scheme (ENGLISH)

[ Abstract ]

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

### 2014/07/14

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Higher dimensional analogues of fake projective planes (ENGLISH)

**Gopal Prasad**(University of Michigan)Higher dimensional analogues of fake projective planes (ENGLISH)

[ Abstract ]

A fake projective plane is a smooth projective complex algebraic surface which is not isomorphic to the complex projective plane but whose Betti numbers are that of the complex projective plane. The fake projective planes are algebraic surfaces of general type and have smallest possible Euler-Poincare characteristic among them. The first fake projective plane was constructed by D. Mumford using p-adic uniformization, and it was known that there can only be finitely many of them. A complete classification of the fake projective planes was obtained by Sai-Kee Yeung and myself. We showed that there are 28 classes of them, and constructed at least one explicit example in each class. Later, using long computer assisted computations, D. Cartwright and Tim Steger found that the 28 families altogether contain precisely 100 fake projective planes. Using our work, they also found a very interesting smooth projective complex algebraic surface whose Euler-Poincare characteristic is 3 but whose first Betti

number is 2. We have a natural notion of higher dimensional analogues of fake projective planes and to a large extent determined them. My talk will be devoted to an exposition of this work.

A fake projective plane is a smooth projective complex algebraic surface which is not isomorphic to the complex projective plane but whose Betti numbers are that of the complex projective plane. The fake projective planes are algebraic surfaces of general type and have smallest possible Euler-Poincare characteristic among them. The first fake projective plane was constructed by D. Mumford using p-adic uniformization, and it was known that there can only be finitely many of them. A complete classification of the fake projective planes was obtained by Sai-Kee Yeung and myself. We showed that there are 28 classes of them, and constructed at least one explicit example in each class. Later, using long computer assisted computations, D. Cartwright and Tim Steger found that the 28 families altogether contain precisely 100 fake projective planes. Using our work, they also found a very interesting smooth projective complex algebraic surface whose Euler-Poincare characteristic is 3 but whose first Betti

number is 2. We have a natural notion of higher dimensional analogues of fake projective planes and to a large extent determined them. My talk will be devoted to an exposition of this work.

### 2014/07/13

#### Infinite Analysis Seminar Tokyo

14:00-15:00 Room #002 (Graduate School of Math. Sci. Bldg.)

From vertex operators to the shuffle algebra (ENGLISH)

**Andrei Negut**(Columbia University, Department of Mathematics)From vertex operators to the shuffle algebra (ENGLISH)

[ Abstract ]

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

### 2014/07/12

#### Lie Groups and Representation Theory

13:20-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Perverse sheaves on hyperplane arrangements (ENGLISH)

Upper global nasis, cluster algebra and simplicity of tensor products of simple modules (ENGLISH)

Branching Problems of Representations of Real Reductive Groups (ENGLISH)

**Mikhail Kapranov**(Kavli IPMU) 13:20-14:20Perverse sheaves on hyperplane arrangements (ENGLISH)

[ Abstract ]

Given an arrangement of hyperplanes in $R^n$, one has the complexified arrangement in $C^n$ and the corresponding category of perverse sheaves (smooth along the strata of the natural stratification).

The talk, based in a joint work with V. Schechtman, will present an explicit description of this category in terms of data associated to the face complex of the real arrangement. Such a description suggests a possibility of categorifying the concept of a oerverse sheaf in this and possibly in more general cases.

Given an arrangement of hyperplanes in $R^n$, one has the complexified arrangement in $C^n$ and the corresponding category of perverse sheaves (smooth along the strata of the natural stratification).

The talk, based in a joint work with V. Schechtman, will present an explicit description of this category in terms of data associated to the face complex of the real arrangement. Such a description suggests a possibility of categorifying the concept of a oerverse sheaf in this and possibly in more general cases.

**Masaki Kashiwara**(RIMS) 14:40-15:40Upper global nasis, cluster algebra and simplicity of tensor products of simple modules (ENGLISH)

[ Abstract ]

One of the motivation of cluster algebras introduced by

Fomin and Zelevinsky is

multiplicative properties of upper global basis.

In this talk, I explain their relations, related conjectures by Besrnard Leclerc and the recent progress by the speaker with Seok-Jin Kang, Myungho Kima and Sejin Oh.

One of the motivation of cluster algebras introduced by

Fomin and Zelevinsky is

multiplicative properties of upper global basis.

In this talk, I explain their relations, related conjectures by Besrnard Leclerc and the recent progress by the speaker with Seok-Jin Kang, Myungho Kima and Sejin Oh.

**Toshiyuki Kobayashi**(the University of Tokyo) 16:00-17:00Branching Problems of Representations of Real Reductive Groups (ENGLISH)

[ Abstract ]

Branching problems ask how irreducible representations π of groups G "decompose" when restricted to subgroups G'.

For real reductive groups, branching problems include various important special cases, however, it is notorious that "infinite multiplicities" and "continuous spectra" may well happen in general even if (G,G') are natural pairs such as symmetric pairs.

By using analysis on (real) spherical varieties, we give a necessary and sufficient condition on the pair of reductive groups for the multiplicities to be always finite (and also to be of uniformly bounded). Further, we discuss "discretely decomposable restrictions" which allows us to apply algebraic tools in branching problems. Some classification results will be also presented.

If time permits, I will discuss some applications of branching laws of Zuckerman's derived functor modules to analysis on locally symmetric spaces with indefinite metric.

Branching problems ask how irreducible representations π of groups G "decompose" when restricted to subgroups G'.

For real reductive groups, branching problems include various important special cases, however, it is notorious that "infinite multiplicities" and "continuous spectra" may well happen in general even if (G,G') are natural pairs such as symmetric pairs.

By using analysis on (real) spherical varieties, we give a necessary and sufficient condition on the pair of reductive groups for the multiplicities to be always finite (and also to be of uniformly bounded). Further, we discuss "discretely decomposable restrictions" which allows us to apply algebraic tools in branching problems. Some classification results will be also presented.

If time permits, I will discuss some applications of branching laws of Zuckerman's derived functor modules to analysis on locally symmetric spaces with indefinite metric.

#### Lie Groups and Representation Theory

09:30-11:45 Room #126 (Graduate School of Math. Sci. Bldg.)

Hypergeometric systems and Kac-Moody root systems (ENGLISH)

Representations of covering groups with multiplicity free K-types (ENGLISH)

**Toshio Oshima**(Josai University) 09:30-10:30Hypergeometric systems and Kac-Moody root systems (ENGLISH)

**Gordan Savin**(the University of Utah) 10:45-11:45Representations of covering groups with multiplicity free K-types (ENGLISH)

[ Abstract ]

Let g be a simple Lie algebra over complex numbers. McGovern has

described an ideal J in the enveloping algebra U such that U/J, considered as a g-module under the adjoint action, is a sum of all self-dual representations of g with multiplicity one. In a joint work with Loke, we prove that all (g,K)-modules annihilated by J have multiplicity free K-types, where K is defined by the Chevalley involution.

Let g be a simple Lie algebra over complex numbers. McGovern has

described an ideal J in the enveloping algebra U such that U/J, considered as a g-module under the adjoint action, is a sum of all self-dual representations of g with multiplicity one. In a joint work with Loke, we prove that all (g,K)-modules annihilated by J have multiplicity free K-types, where K is defined by the Chevalley involution.

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