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Lie Groups and Representation Theory Seminar 2007

List of speakers:
Yiannis Sakellaridis, Ali Baklouti, Taro Yoshino (#1), Gen Mano (#1), Joseph Bernstein (#1), Joseph Bernstein (#2) (4 lectures), Tilmann Wurzbacher (6 lectures), Peter Trapa, Herve Sabourin, R. Stanton, Taro Yoshino (#2), Masatoshi Iida, Tomoyuki Arakawa, Gen Mano (#2), Chifune Kai, Nobutaka Boumuki, Soji Kaneyuki, Karl-Hermann Neeb, Yoshishige Haraoka, Salem Ben Saïd, Pablo Ramacher, Michaël Pevzner (#1), Michaël Pevzner (#2) (4 lectures), Hisayosi Matumoto, Masayasu Moriwaki, Kyo Nishiyama, Junko Inoue, Noriyuki Abe,

Seminar at RIMS (until March 2007)

Date: February 6 (Tue), 2007, 16:30-17:30
Room: RIMS, Kyoto University : Room 402
Speaker: Yiannis Sakellaridis (Tel Aviv)
Title: On the spectrum of spherical varieties over p-adic fields
Abstract:
[pdf]
Spherical varieties are a very important class of spaces which includes all symmetric varieties. Over a p-adic field, their representation theory seems to admit some description through a "Langlands dual" group. I will discuss results on the unramified component of their spectrum which point to this direction. If time permits, I will show how this theory can be used to obtain a completely general Casselman-Shalika formula for eigenfunctions of the Hecke algebra on an arbitrary spherical variety.
 
Date: February 8 (Thu), 2007, 10:00-11:00
Room: RIMS, Kyoto University : Room 402
Speaker: Ali Baklouti (Sfax)
Title: On Hardy's Theorem on nilpotent Lie groups
Abstract:
[pdf]
It is well known that Hardy's uncertainty principle for Rn was generalized to connected and simply connected nilpotent Lie groups. In this work, we extend it further to connected nilpotent Lie groups with non-compact centre. We show however that Hardy's theorem fails for a connected nilpotent Lie group which is not simply connected.
 
Date: February 8 (Thu), 2007, 11:15-12:15
Room: RIMS, Kyoto University : Room 402
Speaker: Taro Yoshino (吉野太郎) (RIMS)
Title: Deformation spaces of compact Clifford-Klein forms of homogeneous spaces of Heisenberg groups
Abstract:
[pdf]
T. Kobayashi introduced the deformation space of Clifford-Klein forms, which is a natural generalization of deformation spaces of geometric structures. Selberg-Weil's local rigidity theorem claims that the deformation space is discrete for Riemannian irreducible symmetric spaces M if the dimension d(M) ≥ 3. In contrast to this theorem, Kobayashi proved that local rigidity does not hold (even in higher dimensional case) in the non-Riemannian case. Then, this opens a new problem to find explicity such deformation spaces in high dimensions. However, such explicit forms have not been obtained except for a few cases now. In this talk, I will give an explicit description of the deformation spaces of compact Clifford-Klein forms of homogeneous spaces of Heisenberg groups.
 
Date: February 8 (Thu), 2007, 12:15-13:00
Room: RIMS, Kyoto University : Room 402
Speaker: Gen Mano (真野元) (RIMS)
Title: The unitary inversion operator for the minimal representation of the indefinite orthogonal group O(p,q)
Abstract:
[pdf]
The L2-model of the minimal representation of the indefinite orthogonal group O(p,q) (p+q even, greater than four) was established by Kobayashi-Ørsted (2003). In this talk, we present an explicit formula for the unitary inversion operator, which plays a key role for the analysis on this L2-model. Our proof uses the Radon transform of distributions supported on the light cone.
 
Date: February 14 (Wed), 2007, 10:00-11:30
Room: RIMS, Kyoto University : Room 005
Speaker: Joseph Bernstein (Tel Aviv)
Title: On the support of the Plancherel measure
Abstract:
[pdf]
In 1970-s Harish Chandra finished his work on harmonic analysis on real reductive groups G. In particular, he proved the Plancherel formula for G which describes the decomposition of the regular representation of G as an integral of irreducible unitary representations of the group G × G (Plancherel decomposition).

The remarkable feature of this formula was the fact that only some of the unitary representations of the group Gcontributed to this formula (so called tempered representations). In fact this phenomenon was already known in PDE. Namely in this case it was known that one can describe the spectral decomposition of an elliptic self-adjoint differential operator D in terms of eigenfunctions which have moderate growth (i.e. they almost lie in L2). The general result of this sort was proven by Gelfand and Kostyuchenko in 1955.

In my paper "On the support of Plancherel measure" (1988) I have applied the ideas of Gelfand and Kostyuchenko and gave an a priori proof of the fact that only tempered representations contribute to the Plancherel decomposition.

Moreover, I have shown that a similar statement holds for decompositions of L2(X) for a large class of homogeneous G-spaces X.

Examples are:

(i) X = G/K, where K is the maximal compact subgroup

(ii) more generally, X = G/H, where H is a symmetric subgroup (subgroup of fixed points of some involution of G);

(iii) X = G/Γ, where Γ is a discrete lattice in G.

(iv) G a reductive p-adic group, X = G/H, where H is either an open compact subgroup or a symmetric subgroup.

I have discovered that the corresponding statement depends on some geometric structure on the space X (I called it "the structure of large scale space") and that this structure has the same properties in all the cases listed above.

In my lecture I will discuss all these questions.

 
Date: February 9 (Fri), 2007, 10:00-11:30
February 13 (Tue), 2007, 13:00-14:30
February 16 (Fri), 2007, 10:00-11:30
February 20 (Tue), 2007, 13:00-14:30
Room: RIMS, Kyoto University : Room 402
Speaker: Joseph Bernstein (Tel Aviv)
Title: Applications of representation theory to problems in analytic number theory (mini course)
Abstract:
[pdf]
In this minicourse I will describe a general approach which allows to use methods of analytic representation theory in order to prove some highly non-trivial estimates in analytic number theory.

This minicourse is based on my works with Andre Reznikov.

I will study representations of the group G = SL(2,R) (and closely related groups) in the space of functions on the automorphic space X = Γ\G.

My aim is to describe relations of this problem to analytic number theory and to show how using methods of representation theory one can get very powerful estimates of different quantities important in number theory.

The plan of the minicourse.

Lecture 1. Automorphic forms on the upper half-pane.

Abstract. In this lecture I will discuss automorphic forms and Maass forms on upper half-plane.

I will show that many problems about such forms are better expressed in the language of automorphic representations.

I will illustrate this on the model example which gives bounds for Sobolev norms of the automorphic functional.

Lecture 2. Triple product problem. Convexity bound.

Abstract. I will discuss the problem of estimates for triple product of automorphic functions and its connection to estimates of automorphic L-functions.

Using the language of automorphic representations described in first lecture I will show how to explain the exponentially decaying factor in the triple product and then I will describe how to prove the convexity bound for these products.

Lecture 3. Subconvexity bound for triple products.

I will continue the investigation of triple products and show how one can prove a non-trivial subconvexity bound for triple products using a combination of geometric and spectral estimates.

 
Date: March 7 (Wed), 2007, 10:00-11:30 (Room 005)
March 8 (Tur), 2007, 10:00-11:30 (Room 005)
March 9 (Fri), 2007, 10:00-11:30 (Room 402)
March 13 (Tue), 2007, 10:00-11:30 (Room 402)
March 14 (Wed), 2007, 10:00-11:30 (Room 005)
March 16 (Fri), 2007, 10:00-11:30 (Room 402)
Room: RIMS, Kyoto University : Room 005 and 402
Speaker: Tilmann Wurzbacher (Metz)
Title: Co-isotropic actions (mini course)
Abstract:
[pdf]
The first aim of this course is to explain the context and the basic properties of co-isotropic actions of Lie groups on symplectic manifolds (i.e. actions having generically co-isotropic orbits), as well as of spherical varietes (i.e. complex-algebraic varieties with an action of a complex reductive Lie group such that all Borel subgroups thereof have an open orbit). After interludes on geometric quantization resp. on lagrangian actions, we prove the equivalence of the two above conditions in the complex-algebraic set-up. Finally, we give applications of this theorem to, e.g., geometric quantization of Kähler manifolds and remark on connections to related subjects.

I. Symplectic reminders

II. Geometric quantization in 90 minutes

III. Lagrangian actions

IV. Co-isotropic actions

V. Applications, remarks and outlook

For details, see the pdf file.

 
Date: March 13 (Tue), 2007, 16:30-17:30
Room: RIMS, Kyoto University : Room 402
Speaker: Peter Trapa (Utah, USA)
Title: On the Matsuki correspondence for sheaves
Abstract:
[pdf]
Suppose G is a real reductive group with maximal compact subgroup K. Let X denote the flag variety for the complexified Lie algebra of G, and let KC denote the complexification of K. Nearly thirty years ago, Matsuki established an order-reversing bijection between the sets of KC and G orbits on X. Later Mirkovic-Uzawa-Vilonen extended this to an equivalence of KC-equivariant and G-equivariant sheaves on X (a result originally conjectured by Kashiwara). Meanwhile, to each such kind of sheaf, Kashiwara showed how to attach a Lagrangian cycle in the cotangent bundle of X. Composing this characteristic cycle construction with the Mirkovic-Uzawa-Vilonen equivalence, one obtains an isomorphism between the top-dimensional homology of the conormal variety for KC orbits on X and the top-dimensional homology of the conormal variety for G orbits on X. Schmid and Vilonen proved that this isomorphism is compatible with the Kostant-Sekiguchi correspondence of nilpotent orbits. The purpose of this talk is to give a finer explicit computation of a suitable "leading part" of the isomorphism in homology.
 
Date: March 20 (Tue), 2007, 16:30-17:30
Room: RIMS, Kyoto University : Room 402
Speaker: Herve Sabourin (Universite de Poitiers)
Title: Unipotent representations of a real simple Lie group attached to small nilpotent orbits
Abstract:
[pdf]
It is a classical idea of Kirillov and Kostant that irreducible representations of a real simply connected Lie group G are related to the orbits of G in the dual g* of its Lie algebra. When G is nilpotent, we know that there is a bijection between the set of G-coadjoint orbits and the unitary dual \widehat{G} of G. When G is solvable, a similar correspondence is due to Auslander and Kostant. For other groups, there are complications even with regard to what is true. Let us suppose now that G is simple and let O be a coadjoint orbit. If O is semi-simple, there is a natural way to associate to O an unitary representation Π(O), but the problem is much more difficult if O is nilpotent. Nevertheless, when O is a minimal nilpotent orbit, one can define a notion of representation "associated" to O and develop a strategy to construct explicitly Π(O). Our goal is to show how this strategy can be extended to the non minimal case and what kind of new results it yields.
 
Date: March 23 (Fri), 2007, 10:30-11:30
Room: RIMS, Kyoto University : Room 402
Speaker: R. Stanton (Ohio)
Title: Symplectic constructions for extraspecial parabolics
Abstract:
[pdf]
The minimal nilpotent orbit in a simple, say, complex Lie algebra has interaction with several topics. In work joint with M. Slupinski, we are investigating the Heisenberg grading associated to any element of the orbit. Röhrle ['93] referred to the corresponding Jacobson-Morozov parabolic as an extraspecial parabolic, and parametrized the orbits of the Levi subgroup acting on the nilradical modulo the center. Using exclusively methods from symplectic geometry, we shall re-examine this representation of the Levi subgroup. We shall classify orbits using the moment map; examine the symplectic nature of each of the orbits; give symplectic constructions of distinguished subgroups that occur in Rubenthaler's list of reductive dual pairs. In particular, we give a symplectic construction of the exceptional simple group G2.

Seminar at the University of Tokyo (since April 2007)

 
Date: April 24 (Tue), 2007, 16:30-18:00
Place:Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Taro Yoshino (吉野太郎) (University of Tokyo)
Title: Existence problem of compact Clifford-Klein forms of the infinitesimal homogeneous space of indefinite Stiefle manifolds
Abstract:
[ pdf ]

The existence problem of compact Clifford-Klein forms is important in the study of discrete groups. There are several open problems on it, even in the reductive cases, which is most studied. For a homogeneous space of reductive type, one can define its 'infinitesimal' homogeneous space. This homogeneous space is easier to consider the existence problem of compact Clifford-Klein forms.

In this talk, we especially consider the infinitesimal homogeneous spaces of indefinite Stiefel manifolds. And, we reduce the existence problem of compact Clifford-Klein forms to certain algebraic problem, which was already studied from other motivation.

Date: May 1 (Tue), 2007, 16:30-18:00
Place:Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Masatoshi Iida (飯田正敏) (Josai University)
Title: Harish-Chandra expansion of the matrix coefficients of PJ Principal series Representation of Sp(2,R)
Abstract:
[ pdf ]

Asymptotic expansion of the matrix coefficents of class-1 principal series representation was considered by Harish-Chandra. The coefficient of the leading exponent of the expansion is called the c-function which plays an important role in the harmonic analysis on the Lie group.

In this talk, we consider the Harish-Chandra expansion of the matrix coefficients of the standard representation which is the parabolic induction with respect to a non-minimal parabolic subgroup of Sp(2,R).

This is the joint study with Professor T. Oda.

Date: May 8 (Tue), 2007, 17:00-18:00
Place:Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Tomoyuki Arakawa (荒川知幸) (Nara Women's University)
Title: Affine W-algebras and their representations
Abstract:
[ pdf ]

The W-algebras are an interesting class of vertex algebras, which can be understood as a generalization of Virasoro algebra. It was originally introduced by Zamolodchikov in his study of conformal field theory. Later Feigin-Frenkel discovered that the W-algebras can be defined via the method of quantum BRST reduction. A few years ago this method was generalized by Kac-Roan-Wakimoto in full generality, producing many interesting vertex algebras. Almost at the same time Premet re-discovered the finite-dimensional version of W-algebras (finite W-algebras), in connection with the modular representation theory.

In the talk we quickly recall the Feigin-Frenkel theory which connects the Whittaker models of the center of U(g) and affine (principal) W-algebras, and discuss their representation theory. Next we recall the construction of Kac-Roan-Wakimoto and discuss the representation theory of affine W-algebras associated with general nilpotent orbits. In particular, I explain how the representation theory of finite W-algebras (=the endmorphism ring of the generalized Gelfand-Graev representation) applies to the representation of affine W-algebras.

Remark: この週は同氏による集中講義 14:40-16:40 があります。 セミナーの時刻はいつもと違いますのでご注意ください。
Date: May 17 (Thu), 2007, 15:00-16:30
Place: Room 002, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Gen Mano (真野 元) (University of Tokyo)
Title: The unitary inversion operator for the minimal representation of the indefinite orthogonal group O(p,q)
Abstract:
[ pdf ]
The indefinite orthogonal group O(p,q) (p+q even, greater than four) has a distinguished infinite dimensional irreducible unitary representation called the 'minimal representation'. Among various models, the L2-model of the minimal representation of O(p,q) was established by Kobayashi-Ørsted (2003). In this talk, we focus on and present an explicit formula for the unitary inversion operator, which plays a key role for the analysis on this L2-model as well as understanding the G-action on L2(C). Our proof uses the Radon transform of distributions supported on the light cone. This is a joint work with T. Kobayashi.
Remark: いつもと時刻・部屋が違います。ご注意ください。
Date: May 22 (Tue), 2007, 16:30-18:00
Place:Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Chifune Kai (甲斐千舟) (Kyushu University)
Title: A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps
Abstract:
[ pdf ]

When a regular open convex cone is given, a natural partial order is introduced into the ambient vector space. If we consider the cone of positive numbers, this partial order is the usual one, and is reversed by taking inverse numbers in the cone. In general, for every symmetric cone, the inverse map of the associated Jordan algebra reverses the order.

In this talk, we investigate this order-reversing property in the class of homogeneous convex cones which is much wider than that of symmetric cones. We show that a homogeneous convex cone is a symmetric cone if and only if the order is reversed by the Vinberg's *-map, which generalizes analytically the inverse maps of Jordan algebras associated with symmetric cones. Actually, our main theorem is formulated in terms of the family of pseudoinverse maps including the Vinberg's *-map as a special one. While our result is a characterization of symmetric cones, also we would like to mention O. Güler's result that for every homogeneous convex cone, some analogous pseudoinverse maps always reverse the order.

Date: May 25 (Fri), 2007, 14:30-16:00
Place:Room 122, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Nobutaka Boumuki (坊向伸隆) (Osaka City University, Advanced Mathematical Institute)
Title: The classification of simple irreducible pseudo-Hermitian symmetric spaces: from a view of elliptic orbits
Abstract:
[ pdf ]
In this talk, we call a special elliptic element an Spr-element, we create an equivalence relation on the set of Spr-elements of a real form of a complex simple Lie algebra, and we classify Spr-elements of each real form of all complex simple Lie algebras under our equivalence relation. Besides, we demonstrate that the classification of Spr-elements under our equivalence relation corresponds to that of simple irreducible pseudo-Hermitian symmetric Lie algebras under Berger's equivalence relation. In terms of the correspondence, we achieve the classification of simple irreducible pseudo-Hermitian symmetric Lie algebras without Berger's classification.
Date: May 25 (Fri), 2007, 16:00-17:30
Place:Room 122, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Soji Kaneyuki (金行壯二)
Title: Causalities, G-structures and symmetric spaces
Abstract:
[ pdf ]

Let M be an n-dimensional smooth manifold, F(M) the frame bundle of M, and let G be a Lie subgroup of GL(n,R). We say that M has a G-structure, if there exists a principal subbundle Q of F(M) with structure group G. Let C be a causal cone in Rn, and let AutC denote the automorphism group of C.

Starting from a causal structure \mathcal{C} on M with model cone C, we construct an AutC-structure Q(\mathcal{C}). Several concepts on causal structures can be interpreted as those on AutC-structures. As an example, the causal automorphism group Aut(M,\mathcal{C}) of M coincides with the automorphism group Aut(M,Q(\mathcal{C})) of the AutC-structure.

As an application, we will consider the unique extension of a local causal transformation on a Cayley type symmetric space M to the global causal automorphism of the compactification of M.

Date: May 29 (Tue), 2007, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Karl-Hermann Neeb (Technische Universität Darmstadt)
Title: A host algebra for the regular representations of the canonical commutation relations
Abstract:
[ pdf ]
We report on joint work with H. Grundling (Sydney). The concept of a host algebra generalises that of a group C *-algebra to groups which are not locally compact in the sense that its non-degenerate representations are in one-to-one correspondence with representations of the group under consideration. A full host algebra covering all continuous unitary representations exist for an abelian topological group if and only if it (essentially) has a locally compact completion. Therefore one has to content oneselves with certain classes of representations covered by a host algebra. We show that there exists a host algebra for the set of continuous representations of the countably dimensional Heisenberg group corresponding to a non-zero central character.
Remark: Neeb 教授は5月26日(土)・27日(日)に 東京大学で行われる第2回高木レクチャーで招待講演をされます。 こちらもどうぞご参加ください。
Date: June 19 (Tue), 2007, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Yoshishige Haraoka (原岡喜重) (Kumamoto University)
Title: Rigid local systems, integral representations of their sections and connection coefficients
Abstract:
[ pdf ]
A local system on CP1-{finite points} is called physically rigid if it is uniquely determined up to isomorphisms by the local monodromies. We explain two algorithms to construct every physically rigid local systems. By applying the algorithms we obtain integral representations of solutions of the corresponding Fuchsian differential equation. Moreover we can express connection coefficients of the equation in terms of the integrals. These results will be applied to several differential equations arising from the representation theory.
Date: June 29 (Fri), 2007, 15:30-16:30, 16:45-17:45
Place: Room 122, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Salem Ben Saïd (Université Henri Poincaré - Nancy 1)
Title: On the theory of Bessel functions associated with root systems
Abstract:
[ pdf ]
The theory of spherical functions on Riemannian symmetric spaces G/K and on non-compactly causal symmetric spaces G/H has a long and rich history. In this talk we will show how one can use a limit transition approach to obtain generalized Bessel functions on flat symmetric spaces via the spherical functions. A similar approach can be applied to the theory of Heckman-Opdam hypergeometric functions to investigate generalized Bessel functions related to arbitrary root system. We conclude the talk by giving a conjecture about the nature and order of the singularities of the Bessel functions related to non-compactly causal symmetric spaces.
Date: October 2 (Tue), 2007, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Pablo Ramacher (Göttingen University)
Title: Invariant integral operators on affine G-varieties and their kernels
Abstract:
[ pdf ]
We consider certain invariant integral operators on a smooth affine variety M carrying the action of a reductive algebraic group G, and assume that G acts on M with an open orbit. Then M is isomorphic to a homogeneous vector bundle, and can locally be described via the theory of prehomogenous vector spaces. We then study the Schwartz kernels of the considered operators, and give a description of their singularities using the calculus of b-pseudodifferential operators developed by Melrose. In particular, the restrictions of the kernels to the diagonal can be described in terms of local zeta functions.
Date: October 9 (Tue), 2007, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Michaël Pevzner (Université de Reims and University of Tokyo)
Title: Rankin-Cohen brackets and covariant quantization
Abstract:
[ pdf ]
The particular geometric structure of causal symmetric spaces permits the definition of a covariant quantization of these homogeneous manifolds. Composition formulae (#-products) of quantizad operators give rise to a new interpretation of Rankin-Cohen brackets and allow to connect them with the branching laws of tensor products of holomorphic discrete series representations.
Date: October 25 (Thu), 2007, 16:30-18:00
October 30 (Tue), 2007, 15:00-16:30
November 1 (Thu), 2007, 16:30-18:00
November 6 (Tue), 2007, 15:00-16:30
Place: Room 126 (on Tuesdays) or 052 (on Thursdays), Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Michaël Pevzner (Université de Reims and University of Tokyo)
Title: Quantization of symmetric spaces and representation
Abstract:
[ pdf ]

The goal of this series of lectures will be to describe and compare two intimately related but nevertheless fundamentally different methods of quantization of symmetric spaces : on the one hand deformation quantization and symbolic calculus on the other hand. We shall also discuss interesting connections with the representation theory of semi-simple Lie groups.

Undergraduate students are welcome.

  • General setting and historic background : mathematical formulation of the Quantization procedure : Weyl symbolic calculus and Moyal star product.
  • Kontsevich' deformation quantization of a smooth Poisson mani-fold.
  • Quantization of a linear Poisson structure and Duflo isomorphism.
  • Non flat case, covariant symbolic calculus on co-adjoint orbits of conformal Lie algebras.
  • Quantization of para-Hermitian symmetric spaces : spectral approach.
  • Particular case of causal symmetric spaces of Cayley type and Rankin-Cohen brackets.
Lecture 1:

The first and introductory lecture of a series of four will be devoted to the discussion of fundamental principles of the Quantum mechanics and their mathematical formulation. This part is not essential for the rest of the course but it might give a global vision of the subject to be considered.

We shall introduce the Weyl symbolic calculus, that relates classical and quantum observables, and will explain its relationship with the so-called deformation quantization of symplectic manifolds.

Afterwards, we will pay attention to a more algebraic question of formal deformation of an arbitrary smooth Poisson manifold and will define the Kontsevich star-product.

Lecture 2:

Back to Mathematics. Two methods of quantization.

We'll start with a discussion on

  • Weyl symbolic calculus on a symplectic vector space and its asymptotic behavior.
In the second part, as a consequence of previous considerations, we'll define the notion of deformation quantization.

Lecture 3:

Kontsevich's formality theorem and applications in Representation theory.

We shall first give

  • an explicit construction of an associative star-product on an arbitrary smooth finite-dimensional Poisson manifold.
As application, we'll consider in details the crucial example of the dual of a finite-dimensional Lie algebra and will sketch a generalization of the Duflo isomorphism describing the set of infinitesimal characters of irreducible unitary representations of the corresponding Lie group.

Lecture 4:

The last lecture will be devoted to following subjects:

  • Application of Kontsevich's star-product to the case of the dual of a Lie algebra.
  • Duflo Isomorphism through deformation quantization.
  • Covariant symbolic calculus on symmetric spaces.
We shall finish by discussing merits and demerits of these two approaches to the quantization problem of homogeneous spaces.
Date: October 30 (Tue), 2007, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Hisayosi Matumoto (松本久義) (University of Tokyo)
Title: On Weyl groups for parabolic subalgebras
Abstract:
[ pdf ]
Let ${\mathfrak g}$ be a complex semisimple Lie algebra. We call a parabolic subalgebra ${\mathfrak p}$ of ${\mathfrak g}$ normal, if any parabolic subalgebra which has a common Levi part with ${\mathfrak p}$ is conjugate to ${\mathfrak p}$ under an inner automorphism of ${\mathfrak g}$. For a normal parabolic subalgebra, we have a good notion of the restricted root system or the little Weyl group. We have a comparison result on the Bruhat order on the Weyl group for ${\mathfrak g}$ and the little Weyl group. We also apply this result to the existence problem of the homomorphisms between scalar generalized Verma modules.
Date: November 6 (Tue), 2007, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Masayasu Moriwaki (森脇政泰) (Hiroshima University)
Title: Multiplicity-free decompositions of the minimal representation of the indefinite orthogonal group
Abstract:
[ pdf ]
A unitary representation of a reductive Lie group can decompose when restricted to a subgroup which is a symmetric pair with finite or infinite multiplicity. On the other hand, T. Kobayashi proved that an irreducible unitary highest weight representation of scalar type decomposes with multiplicity-free when restricted to any subgroup which is a semisimple symmetric pair, and R. Howe proved that the Weil representation decomposes with multiplicity-free when restricted to any subgroup which is a dual pair.

In this talk, with respect to the minimal representation of the indefinite orthogonal group which was constructed by Kazhdan, Kostant, Binegar-Zierau and Kobayashi-Ørsted, we will show that the multiplicity-free theorem holds when restricted to more subgroups than symmetric subgroups.

Date: November 20 (Tue), 2007, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Kyo Nishiyama (西山 享) (Kyoto University)
Title: Asymptotic cone for semisimple elements and the associated variety of degenerate principal series
Abstract:
[ pdf ]
Let a be a hyperbolic element in a semisimple Lie algebra over the real number field. Let K be the complexification of a maximal compact subgroup of the corresponding real adjoint group. We study the asymptotic cone of the semisimple orbit through a under the adjoint action by K. The resulting asymptotic cone is the associated variety of a degenerate principal series representation induced from the parabolic associated to a.
Date: December 11 (Tue), 2007, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Junko Inoue (井上順子) (Tottori University)
Title: Characterization of some smooth vectors for irreducible representations of exponential solvable Lie groups
Abstract:
[ pdf ]
This is a joint work with Jean Ludwig (University of Metz). Let G be an exponential solvable Lie group, and π be an irreducible unitary representation of G. By induction from a character on a connected subgroup H, π is realized on a Hilbert space of L2-functions on a homogeneous space G/H. We investigate a subspace SE  of C-vectors of π consisting of functions with a certain property of rapidly decreasing at infinity. We give a description of SE  as the space of C-vectors of an extension of π to an exponential solvable group containing G.
Date: December 18 (Tue), 2007, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Noriyuki Abe (阿部紀行) (University of Tokyo)
Title: On the existence of homomorphisms between principal series of complex semisimple Lie groups
Abstract:
[ pdf ]
The principal series representations of a semisimple Lie group play an important role in the representation theory. We study the principal series representation of a complex semisimple Lie group and determine when there exists a nonzero homomorphism between the representations.
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