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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.