Φ provides a convolution homomorphism between the Euclidean convolution structure on g and the group convolution on G, that is

In a recent paper Andler, Sahi and Torrosian have extended the Duflo isomorphism to arbitrary Lie groups. Their results give a version of equation which holds for germs of hyperfunctions with support at the identity.

Our result can be viewed as a statement that, for compact Lie groups, the results of hold for invariant distributions of compact support, and hold globally in the sense that the restriction that the supports are compact is needed only in order to ensure that the convolutions exist. This observation allows one to develop calculational tools for invariant harmonic analysis based on convolutions of orbits and distributions in the Euclidean space g.

We have extended these ideas to semi-direct product groups G = V \rtimes K, where V is a vector space and K a compact group. There are several significant differences between this case and the compact case — firstly, there is no identification between the adjoint and coadjoint pictures as the Killing form is indefinite, and secondly, perhaps more significantly, the fact that the orbits are no longer compact means that there are few Ad-invariant distributions of compact support — so the convolutions in formula (2) need careful interpretation.

I will present a simple proof of this result using K-projectives and some basic properties of triangulated categories. This requires the group K to be reductive, but this assumption is easily eliminated. No boundedness assumptions are necessary in my approach.

The heat equation is

The Heat equation has a natural generalization to all Riemannian manifolds. The solution is again given by the Heat transform

An exception is the class of symmetric spaces on noncompact type. In this talk, we start by a short discussion of the Heat transform on Rⁿ to motivate the main part of the talk and introduce the concepts and ideas that are needed for the Riemannian symmetric spaces of the form G/K where G is a connected noncompact semisimple Lie group and K a maximal compact subgroup.

We introduce the natural G-invariant complexification of G/K, called the crown, and describe the image of the Segal-Bargmann transform as a Hilbert space of holomorphic functions on the crown. If time allows, then we will give a different realization of the image space of L²(G/K)^K. That results has a natural formulation for the Heckmann-Opdam setting for positive multiplicity functions.

The main tools here are the spherical Fourier transform and the Abel transform.

We shall also discuss some geometric problems about the Radon transform on symmetric spaces, including refinements of its support properties.

For each highest weight λ which appears in the irreducible decomposition of \mathcal{P}(V) there exist, up to a scalar, a unique K-invariant polynomial p_λ(z,\overline{z}) and a unique K-invariant differential operator p_λ}(z,∂). In this talk, we describe all K-invariant polynomials {p_λ(z,\overline{z})} and K-invariant differential operators {p_λ(z,∂)} for a rank 3 multiplicity-free action (K,V) which is not derived from a Hermitian symmetric space. Moreover, we give two 'symmetric' slices for visibility of the action (K,V). We show that the action of the stabilizer of one indicates the symmetry of the K-invariant polynomials, and that of the other indicates the symmetry of the eigenvalues of the K-invariant differential operators.

• Hopf's conjecture on the sign of Euler characteristic of compact negatively curved manifolds (specificly, the Kahler case).
• Cannon's conjecture on hyperbolic groups whose ideal boundary is a 2-sphere.
• Optimal sectional curvature pinching for rank one symmetric spaces.

Let X=G/K be a connected Riemannian homogenous space of a real Lie group G. Then (G,K) is called a Gelfand pair and the homogeneous space X is said to be commutative if the algebra D(X)^G of G-invariant differential operators on X is commutative; or, equivalently, if the representation of G on L²(X) is multiplicity free. In this lectures we will consider other characterisations of Gelfand pairs. For example, X is commutative if and only if the action of G on the cotangent bundle T*X is coisotropic with respect to the standard G-invariant symplectic structure.

In 1956, Gelfand and Selberg independently introduced two sufficient conditions for commutativity. These conditions turned out to be equivalent and can be formulated as follows:

(GS) there is a group antiautomorphism σ of G such that each double coset of K is σ-stable.

If condition (GS) is satisfied, then X is said to be weakly symmmetric. This condition is not necessary for commutativity. In 1998, Lauret constructed the first example of a commutative but not weakly symmetric homogeneous space.

Suppose that G is reductive. Then, by a result of Akhiezer and Vinberg, (G,K) is a Gelfand pair if and only if X is weakly symmetric. Moreover, (G,K) is a Gelfand pair if and only if the complexification X(C) of X is a spherical G(C)-variety. Spherical variety is a well-studied object of algebraic geometry. If X(C) is homogeneous and affine, then the complete classification was obtained by Krämer, Brion, and Mikityuk. Classification of Gelfand pairs with reductive G easily follows from their results.

In general, if (G,K) is commutative, then, up to a local isomorphism, G has a factorisation G = N \rtimes L, where L is reductive, N is commutative or two-step nilpotent, and K ⊂ L. We will present an effective commutativity criterion in terms of representations of L and K on n = Lie N; and discuss main ideas of the classification of Gelfand pairs.

The notion of a Gelfand pair can be generalised in different ways. If K is not compact, then one can give various reasonable definitions that are not equivalent. In this way, we will obtain different objects, which belong to several areas of mathematics, like spherical varieties in case G is reductive (invariant theory, algebraic geometry); generalised Gelfand pairs (harmonic analysis); coisotropic actions (symplectic geometry, integrable Hamiltonian systems).

Then, the categorical quotient by O(U) × O(U) of the resolution turns out to be a conormal bundle of a certain closed GL(V)-orbit in the Lagrangean Grassmannian. The moment map image of the conormal bundle is the closure of a spherical nilpotent orbit, which is the theta lift of the trivial nilpotent orbit for a certain indefinite orthogonal group in the stable range.

We will explain the construction in detail, and relationships with representation theory and prehomogeneous spaces.

Beyond the formal definition and basic properties of such "pseudo-differential analyses" we shall focus on some fruitful applications of these techniques to the harmonic analysis on PHSS and more broadly to the representation theory of their transformation groups.

Main topics that will be discussed are:

• Composition formul&ae;, and branching laws for tensor products of highest weight modules - an alternative definition of Rankin-Cohen brackets.
• H*-algebra structure of L² spaces on PHSS.
• Square root method and the Plancherel formula for the rank-one PHSS.
• Star-products versus sharp-products.

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.