## Lie Groups and Representation Theory

Seminar information archive ～04/18｜Next seminar｜Future seminars 04/19～

Date, time & place | Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
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**Seminar information archive**

### 2017/09/26

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Representations of Semisimple Lie Groups and Penrose Transform (Japanese)

**Hideko Sekiguchi**(The University of Tokyo)Representations of Semisimple Lie Groups and Penrose Transform (Japanese)

[ Abstract ]

The classical Penrose transform is generalized to an intertwining operator on Dolbeault cohomologies of complex homogeneous spaces $X$ of (real) semisimple Lie groups.

I plan to discuss a detailed analysis when $X$ is an indefinite Grassmann manifold.

To be more precise, we determine the image of the Penrose transform, from the Dolbeault cohomology group on the indefinite Grassmann manifold consisting of maximally positive $k$-planes in ${\mathbb{C}}^{p,q}$ ($1 \le k \le \min(p,q)$) to the space of holomorphic functions over the bounded symmetric domain.

Furthermore, we prove that there is a duality between Dolbeault cohomology groups in two indefinite Grassmann manifolds, namely, that of positive $k$-planes and that of negative $k$-planes.

The classical Penrose transform is generalized to an intertwining operator on Dolbeault cohomologies of complex homogeneous spaces $X$ of (real) semisimple Lie groups.

I plan to discuss a detailed analysis when $X$ is an indefinite Grassmann manifold.

To be more precise, we determine the image of the Penrose transform, from the Dolbeault cohomology group on the indefinite Grassmann manifold consisting of maximally positive $k$-planes in ${\mathbb{C}}^{p,q}$ ($1 \le k \le \min(p,q)$) to the space of holomorphic functions over the bounded symmetric domain.

Furthermore, we prove that there is a duality between Dolbeault cohomology groups in two indefinite Grassmann manifolds, namely, that of positive $k$-planes and that of negative $k$-planes.

### 2017/03/10

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Satake compactifications and metric Schottky problems (English)

**Lizhen Ji**(University of Michigan, USA)Satake compactifications and metric Schottky problems (English)

[ Abstract ]

The quotient of the Poincare upper half plane by the modular group SL(2, Z) is a basic locally symmetric space and also the moduli space of compact Riemann surfaces of genus 1, and it admits two important classes of generalization:

(1) Moduli spaces M_g of compact Riemann surfaces of genus g>1,

(2) Arithmetic locally symmetric spaces \Gamma \ G/K such as the Siegel modular variety A_g, which is also the moduli of principally polarized abelian varieties of dimension g.

There have been a lot of fruitful work to explore the similarity between these two classes of spaces, and there is also a direct interaction between them through the Jacobian (or period) map J: M_g --> A_g.

In this talk, I will discuss some results along these lines related to the Stake compactifications and the Schottky problems on understanding the image J(M_g) in A_g from the metric perspective.

The quotient of the Poincare upper half plane by the modular group SL(2, Z) is a basic locally symmetric space and also the moduli space of compact Riemann surfaces of genus 1, and it admits two important classes of generalization:

(1) Moduli spaces M_g of compact Riemann surfaces of genus g>1,

(2) Arithmetic locally symmetric spaces \Gamma \ G/K such as the Siegel modular variety A_g, which is also the moduli of principally polarized abelian varieties of dimension g.

There have been a lot of fruitful work to explore the similarity between these two classes of spaces, and there is also a direct interaction between them through the Jacobian (or period) map J: M_g --> A_g.

In this talk, I will discuss some results along these lines related to the Stake compactifications and the Schottky problems on understanding the image J(M_g) in A_g from the metric perspective.

### 2016/07/05

17:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Shimizu's lemma for general linear groups

**Toshiaki Hattori**(Tokyo Institute of Technology)Shimizu's lemma for general linear groups

[ Abstract ]

We find two versions of Shimizu's lemma for subgroups of GL(n, C). By using these results, we study the noncompact classical groups and the associated Riemannian symmetric spaces. This is a continuation of the talk at this seminar in July 2015.

We find two versions of Shimizu's lemma for subgroups of GL(n, C). By using these results, we study the noncompact classical groups and the associated Riemannian symmetric spaces. This is a continuation of the talk at this seminar in July 2015.

### 2016/04/12

17:00-18:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Universal Gysin formulas for flag bundles

**Piotr Pragacz**(Institute of Mathematics, Polish Academy of Sciences)Universal Gysin formulas for flag bundles

[ Abstract ]

We give generalizations of the formula for the push-forward of a power of the hyperplane class in a projective bundle to flag bundles of type A, B, C, D. The formulas (and also the proofs) involve only the Segre classes of the original vector bundles and characteristic classes of universal bundles. This is a joint work with Lionel Darondeau.

We give generalizations of the formula for the push-forward of a power of the hyperplane class in a projective bundle to flag bundles of type A, B, C, D. The formulas (and also the proofs) involve only the Segre classes of the original vector bundles and characteristic classes of universal bundles. This is a joint work with Lionel Darondeau.

### 2015/11/26

17:00-18:45 Room # (Graduate School of Math. Sci. Bldg.)

Introduction to the cohomology of discrete groups and modular symbols 2 (English)

**Birgit Speh**(Cornell University)Introduction to the cohomology of discrete groups and modular symbols 2 (English)

[ Abstract ]

The course is an introduction to the cohomology of torsion free discrete subgroups $\Gamma \subset G $ of a semi simple group $G$. The discrete group $\Gamma$ acts freely on the symmetric space $X= G/K$ and we will always assume that $\Gamma \backslash G/K$ is compact or has finite volume. An example is a torsion free subgroup $\Gamma_n $ of finite index n in Sl(2,Z) acting on $Sl(2.R)/SO(2) \simeq {\mathcal H}=\{z=x+iy \in C| y >0 \}$ by fractional linear transformations. $\Gamma_n \backslash {\mathcal H}$ can be determined explicitly and it can be visualized as an area in the upper half plane glued at the boundary. It is easy to see that it has some nice compactifications.

The cohomology $H^*(\Gamma, C)$ of the group $\Gamma$ is equal to the deRham cohomology $H^*_{deRham}(\Gamma \backslash X, C)$ of the manifold $\Gamma\backslash X$. This cohomology is studied by proving that it is isomorphic to the $H^*(g,K,{\mathcal A}(\Gamma \backslash G))$. Here ${\mathcal A}(\Gamma \backslash G)$ of automorphic functions on $\Gamma \backslash G$. In the case $\Gamma_n \subset Sl(2,Z)$ the space ${\mathcal A}(\Gamma \backslash G)$ is the space of classical automorphic functions on the upper half plane containing holomorphic cusp form, Eisenstein series, Maass forms and it is often introduced in an introductory course in analytic number theory.

On the geometric side we will construct some of the cycles (modular symbols) in the homology $H_*(\Gamma\backslash X)$ which are dual to the cohomology classes we constructed. In our example $\Gamma_n\backslash Sl(2,R)/SO(2)$ these cycles correspond to geodesics and can easily be visualized.

In this course I will explain these results and show how to use them to prove vanishing and non vanishing theorem for $H^*_{deRham}(\Gamma \backslash X)$. I will state the results in full generality, but I will prove them only in the classical case: G=SL$(2,R)$ and the subgroup $\Gamma= \Gamma_n$ a congruence subgroup. Some familiarity with Lie groups and Lie algebras is only prerequisite for the course.

The course is an introduction to the cohomology of torsion free discrete subgroups $\Gamma \subset G $ of a semi simple group $G$. The discrete group $\Gamma$ acts freely on the symmetric space $X= G/K$ and we will always assume that $\Gamma \backslash G/K$ is compact or has finite volume. An example is a torsion free subgroup $\Gamma_n $ of finite index n in Sl(2,Z) acting on $Sl(2.R)/SO(2) \simeq {\mathcal H}=\{z=x+iy \in C| y >0 \}$ by fractional linear transformations. $\Gamma_n \backslash {\mathcal H}$ can be determined explicitly and it can be visualized as an area in the upper half plane glued at the boundary. It is easy to see that it has some nice compactifications.

The cohomology $H^*(\Gamma, C)$ of the group $\Gamma$ is equal to the deRham cohomology $H^*_{deRham}(\Gamma \backslash X, C)$ of the manifold $\Gamma\backslash X$. This cohomology is studied by proving that it is isomorphic to the $H^*(g,K,{\mathcal A}(\Gamma \backslash G))$. Here ${\mathcal A}(\Gamma \backslash G)$ of automorphic functions on $\Gamma \backslash G$. In the case $\Gamma_n \subset Sl(2,Z)$ the space ${\mathcal A}(\Gamma \backslash G)$ is the space of classical automorphic functions on the upper half plane containing holomorphic cusp form, Eisenstein series, Maass forms and it is often introduced in an introductory course in analytic number theory.

On the geometric side we will construct some of the cycles (modular symbols) in the homology $H_*(\Gamma\backslash X)$ which are dual to the cohomology classes we constructed. In our example $\Gamma_n\backslash Sl(2,R)/SO(2)$ these cycles correspond to geodesics and can easily be visualized.

In this course I will explain these results and show how to use them to prove vanishing and non vanishing theorem for $H^*_{deRham}(\Gamma \backslash X)$. I will state the results in full generality, but I will prove them only in the classical case: G=SL$(2,R)$ and the subgroup $\Gamma= \Gamma_n$ a congruence subgroup. Some familiarity with Lie groups and Lie algebras is only prerequisite for the course.

### 2015/11/24

17:00-18:45 Room # (Graduate School of Math. Sci. Bldg.)

Introduction to the cohomology of discrete groups and modular symbols 1 (English)

**Birgit Speh**(Cornell University)Introduction to the cohomology of discrete groups and modular symbols 1 (English)

[ Abstract ]

The course is an introduction to the cohomology of torsion free discrete subgroups $\Gamma \subset G $ of a semi simple group $G$. The discrete group $\Gamma$ acts freely on the symmetric space $X= G/K$ and we will always assume that $\Gamma \backslash G/K$ is compact or has finite volume. An example is a torsion free subgroup $\Gamma_n $ of finite index n in Sl(2,Z) acting on $Sl(2.R)/SO(2) \simeq {\mathcal H}=\{z=x+iy \in C| y >0 \}$ by fractional linear transformations. $\Gamma_n \backslash {\mathcal H}$ can be determined explicitly and it can be visualized as an area in the upper half plane glued at the boundary. It is easy to see that it has some nice compactifications.

The cohomology $H^*(\Gamma, C)$ of the group $\Gamma$ is equal to the deRham cohomology $H^*_{deRham}(\Gamma \backslash X, C)$ of the manifold $\Gamma\backslash X$. This cohomology is studied by proving that it is isomorphic to the $H^*(g,K,{\mathcal A}(\Gamma \backslash G))$. Here ${\mathcal A}(\Gamma \backslash G)$ of automorphic functions on $\Gamma \backslash G$. In the case $\Gamma_n \subset Sl(2,Z)$ the space ${\mathcal A}(\Gamma \backslash G)$ is the space of classical automorphic functions on the upper half plane containing holomorphic cusp form, Eisenstein series, Maass forms and it is often introduced in an introductory course in analytic number theory.

On the geometric side we will construct some of the cycles (modular symbols) in the homology $H_*(\Gamma\backslash X)$ which are dual to the cohomology classes we constructed. In our example $\Gamma_n\backslash Sl(2,R)/SO(2)$ these cycles correspond to geodesics and can easily be visualized.

In this course I will explain these results and show how to use them to prove vanishing and non vanishing theorem for $H^*_{deRham}(\Gamma \backslash X)$. I will state the results in full generality, but I will prove them only in the classical case: G=SL$(2,R)$ and the subgroup $\Gamma= \Gamma_n$ a congruence subgroup. Some familiarity with Lie groups and Lie algebras is only prerequisite for the course.

The course is an introduction to the cohomology of torsion free discrete subgroups $\Gamma \subset G $ of a semi simple group $G$. The discrete group $\Gamma$ acts freely on the symmetric space $X= G/K$ and we will always assume that $\Gamma \backslash G/K$ is compact or has finite volume. An example is a torsion free subgroup $\Gamma_n $ of finite index n in Sl(2,Z) acting on $Sl(2.R)/SO(2) \simeq {\mathcal H}=\{z=x+iy \in C| y >0 \}$ by fractional linear transformations. $\Gamma_n \backslash {\mathcal H}$ can be determined explicitly and it can be visualized as an area in the upper half plane glued at the boundary. It is easy to see that it has some nice compactifications.

The cohomology $H^*(\Gamma, C)$ of the group $\Gamma$ is equal to the deRham cohomology $H^*_{deRham}(\Gamma \backslash X, C)$ of the manifold $\Gamma\backslash X$. This cohomology is studied by proving that it is isomorphic to the $H^*(g,K,{\mathcal A}(\Gamma \backslash G))$. Here ${\mathcal A}(\Gamma \backslash G)$ of automorphic functions on $\Gamma \backslash G$. In the case $\Gamma_n \subset Sl(2,Z)$ the space ${\mathcal A}(\Gamma \backslash G)$ is the space of classical automorphic functions on the upper half plane containing holomorphic cusp form, Eisenstein series, Maass forms and it is often introduced in an introductory course in analytic number theory.

On the geometric side we will construct some of the cycles (modular symbols) in the homology $H_*(\Gamma\backslash X)$ which are dual to the cohomology classes we constructed. In our example $\Gamma_n\backslash Sl(2,R)/SO(2)$ these cycles correspond to geodesics and can easily be visualized.

In this course I will explain these results and show how to use them to prove vanishing and non vanishing theorem for $H^*_{deRham}(\Gamma \backslash X)$. I will state the results in full generality, but I will prove them only in the classical case: G=SL$(2,R)$ and the subgroup $\Gamma= \Gamma_n$ a congruence subgroup. Some familiarity with Lie groups and Lie algebras is only prerequisite for the course.

### 2015/07/28

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

On g,K-modules over arbitrary fields and applications to special values of L-functions

**Fabian Januszewski**(Karlsruhe Institute of Technology)On g,K-modules over arbitrary fields and applications to special values of L-functions

[ Abstract ]

I will introduce g,K-modules over arbitrary fields of characteristic 0 and discuss their fundamental properties and constructions, including Zuckerman functors. This may be applied to produce models of certain standard modules over number fields, which has applications to special values of automorphic L-functions, and also furnishes the space of regular algebraic cusp forms of GL(n) with a natural global Q-structure.

I will introduce g,K-modules over arbitrary fields of characteristic 0 and discuss their fundamental properties and constructions, including Zuckerman functors. This may be applied to produce models of certain standard modules over number fields, which has applications to special values of automorphic L-functions, and also furnishes the space of regular algebraic cusp forms of GL(n) with a natural global Q-structure.

### 2015/07/21

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

GEOMETRIC STRUCTURE IN SMOOTH DUAL

**Paul Baum**(Penn State University)GEOMETRIC STRUCTURE IN SMOOTH DUAL

[ Abstract ]

Let G be a connected split reductive p-adic group. Examples are GL(n, F) , SL(n, F) , SO(n, F) , Sp(2n, F) , PGL(n, F) where n can be any positive integer and F can be any finite extension of the field Q_p of p-adic numbers. The smooth (or admissible) dual of G is the set of equivalence classes of smooth irreducible representations of G. This talk will first review the theory of the Bernstein center. According to this theory, the smooth dual of G is the disjoint union of subsets known as the Bernstein components. The talk will then explain the ABPS (Aubert-Baum-Plymen-Solleveld) conjecture which states that each Bernstein component is a complex affine variety. Each of these complex affine varieties is explicitly identified as the extended quotient associated to the given Bernstein component.

The ABPS conjecture has been proved for GL(n, F), SO(n, F), and Sp(2n, F).

Let G be a connected split reductive p-adic group. Examples are GL(n, F) , SL(n, F) , SO(n, F) , Sp(2n, F) , PGL(n, F) where n can be any positive integer and F can be any finite extension of the field Q_p of p-adic numbers. The smooth (or admissible) dual of G is the set of equivalence classes of smooth irreducible representations of G. This talk will first review the theory of the Bernstein center. According to this theory, the smooth dual of G is the disjoint union of subsets known as the Bernstein components. The talk will then explain the ABPS (Aubert-Baum-Plymen-Solleveld) conjecture which states that each Bernstein component is a complex affine variety. Each of these complex affine varieties is explicitly identified as the extended quotient associated to the given Bernstein component.

The ABPS conjecture has been proved for GL(n, F), SO(n, F), and Sp(2n, F).

### 2015/07/21

15:30-16:30 Room #122 (Graduate School of Math. Sci. Bldg.)

(Japanese)

**Toshiaki Hattori**(Tokyo Institute of Technology)(Japanese)

### 2015/07/14

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

MORITA EQUIVALENCE REVISITED

**Paul Baum**(Penn State University)MORITA EQUIVALENCE REVISITED

[ Abstract ]

Let X be a complex affine variety and k its coordinate algebra. A k- algebra is an algebra A over the complex numbers which is a k-module (with an evident compatibility between the algebra structure of A and the k-module structure of A). A is not required to have a unit. A k-algebra A is of finite type if as a k-module A is finitely generated. This talk will review Morita equivalence for k-algebras and will then introduce --- for finite type k-algebras ---a weakening of Morita equivalence called geometric equivalence. The new equivalence relation preserves the primitive ideal space (i.e. the set of isomorphism classes of irreducible A-modules) and the periodic cyclic homology of A. However, the new equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence.

Let G be a connected split reductive p-adic group, The ABPS (Aubert- Baum-Plymen-Solleveld) conjecture states that the finite type algebra which Bernstein assigns to any given Bernstein component in the smooth dual of G, is geometrically equivalent to the coordinate algebra of the associated extended quotient. The second talk will give an exposition of the ABPS conjecture.

Let X be a complex affine variety and k its coordinate algebra. A k- algebra is an algebra A over the complex numbers which is a k-module (with an evident compatibility between the algebra structure of A and the k-module structure of A). A is not required to have a unit. A k-algebra A is of finite type if as a k-module A is finitely generated. This talk will review Morita equivalence for k-algebras and will then introduce --- for finite type k-algebras ---a weakening of Morita equivalence called geometric equivalence. The new equivalence relation preserves the primitive ideal space (i.e. the set of isomorphism classes of irreducible A-modules) and the periodic cyclic homology of A. However, the new equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence.

Let G be a connected split reductive p-adic group, The ABPS (Aubert- Baum-Plymen-Solleveld) conjecture states that the finite type algebra which Bernstein assigns to any given Bernstein component in the smooth dual of G, is geometrically equivalent to the coordinate algebra of the associated extended quotient. The second talk will give an exposition of the ABPS conjecture.

### 2015/06/30

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Random subgroups and representation theory

**Anatoly Vershik**(St. Petersburg Department of Steklov Institute of Mathematics)Random subgroups and representation theory

[ Abstract ]

The following problem had been appeared independently in different teams and various reason:

to describe the Borel measures on the lattice of all subgroups of given group, which are invariant with respect to the action of the group by conjugacy. The main interest of course represents nonatomic measures which exist not for any group.

I will explain how these measures connected with characters and representations of the group, and describe the complete list of such measures for infinite symmetric group.

The following problem had been appeared independently in different teams and various reason:

to describe the Borel measures on the lattice of all subgroups of given group, which are invariant with respect to the action of the group by conjugacy. The main interest of course represents nonatomic measures which exist not for any group.

I will explain how these measures connected with characters and representations of the group, and describe the complete list of such measures for infinite symmetric group.

### 2015/05/26

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Local functional equations of Clifford quartic forms and homaloidal EKP-polynomials

**Takeyoshi Kogiso**(Josai University)Local functional equations of Clifford quartic forms and homaloidal EKP-polynomials

[ Abstract ]

It is known that one can associate local functional equation to the irreducible relative invariant of an irreducible regular prehomogeneous vector spaces. We construct Clifford quartic forms that cannot obtained from prehomogeneous vector spaces, but, for which one can associate local functional equations. The characterization of polynomials which satisfy local functional equations is an interesting problem. In relation to this characterization problem (in a more general form), Etingof, Kazhdan and Polishchuk raised a conjecture. We make a counter example of this conjecture from Clifford quartic forms. (This is based on the joint work with F.Sato)

It is known that one can associate local functional equation to the irreducible relative invariant of an irreducible regular prehomogeneous vector spaces. We construct Clifford quartic forms that cannot obtained from prehomogeneous vector spaces, but, for which one can associate local functional equations. The characterization of polynomials which satisfy local functional equations is an interesting problem. In relation to this characterization problem (in a more general form), Etingof, Kazhdan and Polishchuk raised a conjecture. We make a counter example of this conjecture from Clifford quartic forms. (This is based on the joint work with F.Sato)

### 2015/05/19

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

RoCK blocks, wreath products and KLR algebras (English)

**Anton Evseev**(University of Birmingham)RoCK blocks, wreath products and KLR algebras (English)

[ Abstract ]

The so-called RoCK (or Rouquier) blocks play an important role in representation theory of symmetric groups over a finite field of characteristic $p$, as well as of Hecke algebras at roots of unity. Turner has conjectured that a certain idempotent truncation of a RoCK block is Morita equivalent to the principal block $B_0$ of the wreath product $S_p\wr S_d$ of symmetric groups, where $d$ is the "weight" of the block. The talk will outline a proof of this conjecture, which generalizes a result of Chuang-Kessar proved for $d < p$. The proof uses an isomorphism between a Hecke algebra at a root of unity and a cyclotomic Khovanov-Lauda-Rouquier algebra, the resulting grading on the Hecke algebra and the ideas behind a construction of R-matrices for modules over KLR algebras due to Kang-Kashiwara-Kim.

The so-called RoCK (or Rouquier) blocks play an important role in representation theory of symmetric groups over a finite field of characteristic $p$, as well as of Hecke algebras at roots of unity. Turner has conjectured that a certain idempotent truncation of a RoCK block is Morita equivalent to the principal block $B_0$ of the wreath product $S_p\wr S_d$ of symmetric groups, where $d$ is the "weight" of the block. The talk will outline a proof of this conjecture, which generalizes a result of Chuang-Kessar proved for $d < p$. The proof uses an isomorphism between a Hecke algebra at a root of unity and a cyclotomic Khovanov-Lauda-Rouquier algebra, the resulting grading on the Hecke algebra and the ideas behind a construction of R-matrices for modules over KLR algebras due to Kang-Kashiwara-Kim.

### 2015/04/28

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Restricting automorphic forms to geodesic cycles (English)

**Bent Orsted**(Aarhus University and the University of Tokyo)Restricting automorphic forms to geodesic cycles (English)

[ Abstract ]

We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles in terms of their expansion into eigenfunctions of the Laplacian. Our method resembles earlier work on products of automorphic forms by Bernstein and Reznikov, and it uses Kobayashi's new symmetry-breaking kernels. This is joint work with Jan M\"o{}llers.

We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles in terms of their expansion into eigenfunctions of the Laplacian. Our method resembles earlier work on products of automorphic forms by Bernstein and Reznikov, and it uses Kobayashi's new symmetry-breaking kernels. This is joint work with Jan M\"o{}llers.

### 2015/04/21

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Norm computation and analytic continuation of vector valued holomorphic discrete series representations

(English)

**Ryosuke Nakahama**(the University of Tokyo, Department of Mathematical Sciences)Norm computation and analytic continuation of vector valued holomorphic discrete series representations

(English)

[ Abstract ]

The holomorphic discrete series representations is realized on the space of vector-valued holomorphic functions on the complex bounded symmetric domains. When the parameter is sufficiently large, then its norm is given by the converging integral, but when the parameter becomes small, then the integral does not converge. However, if once we compute the norm explicitly, then we can consider its analytic continuation, and can discuss its properties, such as unitarizability. In this talk we treat the results on explicit norm computation.

The holomorphic discrete series representations is realized on the space of vector-valued holomorphic functions on the complex bounded symmetric domains. When the parameter is sufficiently large, then its norm is given by the converging integral, but when the parameter becomes small, then the integral does not converge. However, if once we compute the norm explicitly, then we can consider its analytic continuation, and can discuss its properties, such as unitarizability. In this talk we treat the results on explicit norm computation.

### 2015/04/14

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

Visible actions of compact Lie groups on complex spherical varieties (English)

**Yuichiro Tanaka**(Institute of Mathematics for Industry, Kyushu University)Visible actions of compact Lie groups on complex spherical varieties (English)

[ Abstract ]

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the theory of visible actions on complex manifolds.

In this talk we consider visible actions of a compact real form U of a connected complex reductive algebraic group G on G-spherical varieties. Here a complex G-variety X is said to be spherical if a Borel subgroup of G has an open orbit on X. The sphericity implies the multiplicity-freeness property of the space of polynomials on X. Our main result gives an abstract proof for the visibility of U-actions. As a corollary, we obtain an alternative proof for the visibility of U-actions on linear multiplicity-free spaces, which was earlier proved by A. Sasaki (2009, 2011), and the visibility of U-actions on generalized flag varieties, earlier proved by Kobayashi (2007) and T- (2013, 2014).

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the theory of visible actions on complex manifolds.

In this talk we consider visible actions of a compact real form U of a connected complex reductive algebraic group G on G-spherical varieties. Here a complex G-variety X is said to be spherical if a Borel subgroup of G has an open orbit on X. The sphericity implies the multiplicity-freeness property of the space of polynomials on X. Our main result gives an abstract proof for the visibility of U-actions. As a corollary, we obtain an alternative proof for the visibility of U-actions on linear multiplicity-free spaces, which was earlier proved by A. Sasaki (2009, 2011), and the visibility of U-actions on generalized flag varieties, earlier proved by Kobayashi (2007) and T- (2013, 2014).

### 2015/04/07

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

Branching laws and elliptic boundary value problems

(English)

**Bent Orsted**(Aarhus University)Branching laws and elliptic boundary value problems

(English)

[ Abstract ]

Classically the Poisson transform relates harmonic functions in the complex upper half plane to their boundary values on the real axis. In

some recent work by Caffarelli et al. some new generalizations of this appears in connection with the fractional Laplacian. In this lecture we

shall explain how the symmetry-breaking operators introduced by T. Kobayashi for studying branching laws may shed new light on the situation for elliptic boundary value problems. This is based on joint work with J. M\"o{}llers and G. Zhang.

Classically the Poisson transform relates harmonic functions in the complex upper half plane to their boundary values on the real axis. In

some recent work by Caffarelli et al. some new generalizations of this appears in connection with the fractional Laplacian. In this lecture we

shall explain how the symmetry-breaking operators introduced by T. Kobayashi for studying branching laws may shed new light on the situation for elliptic boundary value problems. This is based on joint work with J. M\"o{}llers and G. Zhang.

### 2015/03/24

18:00-19:30 Room #126 (Graduate School of Math. Sci. Bldg.)

A Gysin formula for Hall-Littlewood polynomials

**Piotr Pragacz**(Institute of Mathematics, Polish Academy of Sciences)A Gysin formula for Hall-Littlewood polynomials

[ Abstract ]

Schubert calculus on Grassmannians is governed by Schur S-functions, the one on Lagrangian Grassmannians by Schur Q-functions. There were several attempts to give a unifying approach to both situations.

We propose to use Hall-Littlewood symmetric polynomials. They appeared implicitly in Hall's study of the combinatorial lattice structure of finite abelian p-groups and in Green's calculations of the characters of GL(n) over finite fields; they appeared explicitly in the work of Littlewood on some problems in representation theory.

With the projection in a Grassmann bundle, there is associated its Gysin map, induced by pushing forward cycles (topologists call it "integration along fibers").

We state and prove a Gysin formula for HL-polynomials in these bundles. We discuss its two specializations, giving better insights to previously known formulas for Schur S- and P-functions.

Schubert calculus on Grassmannians is governed by Schur S-functions, the one on Lagrangian Grassmannians by Schur Q-functions. There were several attempts to give a unifying approach to both situations.

We propose to use Hall-Littlewood symmetric polynomials. They appeared implicitly in Hall's study of the combinatorial lattice structure of finite abelian p-groups and in Green's calculations of the characters of GL(n) over finite fields; they appeared explicitly in the work of Littlewood on some problems in representation theory.

With the projection in a Grassmann bundle, there is associated its Gysin map, induced by pushing forward cycles (topologists call it "integration along fibers").

We state and prove a Gysin formula for HL-polynomials in these bundles. We discuss its two specializations, giving better insights to previously known formulas for Schur S- and P-functions.

### 2015/01/27

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Representations of quantized function algebras and the transition matrices from Canonical bases to PBW bases (JAPANESE)

**Hironori Oya**(The University of Tokyo)Representations of quantized function algebras and the transition matrices from Canonical bases to PBW bases (JAPANESE)

[ Abstract ]

Let $G$ be a connected simply connected simple complex algebraic group of type $ADE$ and $\mathfrak{g}$ the corresponding simple Lie algebra. In this talk, I will explain our new algebraic proof of the positivity of the transition matrices from the canonical basis to the PBW bases of $U_q(\mathfrak{n}^+)$. Here, $U_q(\mathfrak{n}^+)$ denotes the positive part of the quantized enveloping algebra $U_q(\ mathfrak{g})$. (This positivity, which is a generalization of Lusztig's result, was originally proved by Kato (Duke Math. J. 163 (2014)).) We use the relation between $U_q(\mathfrak{n}^+)$ and the specific irreducible representations of the quantized function algebra $\mathbb{Q} _q[G]$. This relation has recently been pointed out by Kuniba, Okado and Yamada (SIGMA. 9 (2013)). Firstly, we study it taking into account the right $U_q(\mathfrak{g})$-algebra structure of $\mathbb{Q}_q[G]$. Next, we calculate the transition matrices from the canonical basis to the PBW bases using the result obtained in the first step.

Let $G$ be a connected simply connected simple complex algebraic group of type $ADE$ and $\mathfrak{g}$ the corresponding simple Lie algebra. In this talk, I will explain our new algebraic proof of the positivity of the transition matrices from the canonical basis to the PBW bases of $U_q(\mathfrak{n}^+)$. Here, $U_q(\mathfrak{n}^+)$ denotes the positive part of the quantized enveloping algebra $U_q(\ mathfrak{g})$. (This positivity, which is a generalization of Lusztig's result, was originally proved by Kato (Duke Math. J. 163 (2014)).) We use the relation between $U_q(\mathfrak{n}^+)$ and the specific irreducible representations of the quantized function algebra $\mathbb{Q} _q[G]$. This relation has recently been pointed out by Kuniba, Okado and Yamada (SIGMA. 9 (2013)). Firstly, we study it taking into account the right $U_q(\mathfrak{g})$-algebra structure of $\mathbb{Q}_q[G]$. Next, we calculate the transition matrices from the canonical basis to the PBW bases using the result obtained in the first step.

### 2014/10/29

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Harmonic analysis on reductive p-adic symmetric spaces. (ENGLISH)

**Patrick Delorme**(UER Scientifique de Luminy Universite d'Aix-Marseille II )Harmonic analysis on reductive p-adic symmetric spaces. (ENGLISH)

[ Abstract ]

In this lecture we will review the Plancherel formula that

we got by looking to neighborhoods at infinity of the

symmetric spaces and then using the method of Sakellaridis-Venkatesh

for spherical varieties for a split group. For us the group

is not necessarily split. We will try to show what questions

are raised by this work for real spherical varieties.

We will present in the last part a joint work with Pascale

Harinck and Yiannis Sakellaridis in which we prove Paley-Wiener

theorems for symmetric spaces.

In this lecture we will review the Plancherel formula that

we got by looking to neighborhoods at infinity of the

symmetric spaces and then using the method of Sakellaridis-Venkatesh

for spherical varieties for a split group. For us the group

is not necessarily split. We will try to show what questions

are raised by this work for real spherical varieties.

We will present in the last part a joint work with Pascale

Harinck and Yiannis Sakellaridis in which we prove Paley-Wiener

theorems for symmetric spaces.

### 2014/07/12

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.

### 2014/07/12

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.

### 2014/07/01

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

WONDERFUL VARIETIES. REGULARIZED TRACES AND CHARACTERS (ENGLISH)

**Pablo Ramacher**

(Marburg University)WONDERFUL VARIETIES. REGULARIZED TRACES AND CHARACTERS (ENGLISH)

[ Abstract ]

Let G be a connected reductive complex algebraic group with split real form $G^\\sigma$.

In this talk, we introduce a distribution character for the regular representation of $G^\\sigma$ on the real locus of a strict wonderful G-variety X, showing that on a certain open subset of $G^\\sigma$ of transversal elements it is locally integrable, and given by a sum over fixed points.

Let G be a connected reductive complex algebraic group with split real form $G^\\sigma$.

In this talk, we introduce a distribution character for the regular representation of $G^\\sigma$ on the real locus of a strict wonderful G-variety X, showing that on a certain open subset of $G^\\sigma$ of transversal elements it is locally integrable, and given by a sum over fixed points.

### 2014/06/17

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

SINGULAR EQUIVARIANT ASYMPTOTICS AND THE MOMENTUM MAP. RESIDUE FORMULAE IN EQUIVARIANT COHOMOLOGY (ENGLISH)

**Pablo Ramacher**(Marburg University)SINGULAR EQUIVARIANT ASYMPTOTICS AND THE MOMENTUM MAP. RESIDUE FORMULAE IN EQUIVARIANT COHOMOLOGY (ENGLISH)

[ Abstract ]

Let M be a smooth manifold and G a compact connected Lie group acting on M by isometries. In this talk, we study the equivariant cohomology of the cotangent bundle of M, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case of compact symplectic manifolds with a Hamiltonian G-action, similar residue formulae were derived by Jeffrey, Kirwan et al..

Let M be a smooth manifold and G a compact connected Lie group acting on M by isometries. In this talk, we study the equivariant cohomology of the cotangent bundle of M, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case of compact symplectic manifolds with a Hamiltonian G-action, similar residue formulae were derived by Jeffrey, Kirwan et al..

### 2014/05/27

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On the structure of Schubert modules and filtration by Schubert modules

(JAPANESE)

**Masaki Watanabe**(the University of Tokyo, Graduate School of Mathematical Sciences)On the structure of Schubert modules and filtration by Schubert modules

(JAPANESE)

[ Abstract ]

One of the methods for studying Schubert polynomials is using

Schubert modules introduced by Kraskiewicz and Pragacz.

In this seminar I will talk about a new result on the structure of

Schubert modules, and give a criterion for a module to have a filtration by Schubert modules.

I will also talk about a problem concerning Schubert polynomials

which motivated this research.

One of the methods for studying Schubert polynomials is using

Schubert modules introduced by Kraskiewicz and Pragacz.

In this seminar I will talk about a new result on the structure of

Schubert modules, and give a criterion for a module to have a filtration by Schubert modules.

I will also talk about a problem concerning Schubert polynomials

which motivated this research.