C*-algebras determine spaces of irreducible Hilbert space representations in the same way that the algebras of regular functions on affine varieties determine those varieties. I shall attempt to illustrate this by explaining the very close connection between the reduced C*-algebra of a reductive group and the work of Harish-Chandra and others in tempered representation theory. Taking this as a starting point, I shall then introduce some new C*-algebras that are associated to the harmonic analysis of reductive symmetric spaces.

In the first lecture we highlight “multiplicities” of irreducible representations occurring in the space of functions. We discuss what kind of geometry guarantees a “good control” of the transformation group on the function space, and the answer brings us naturally to the notion of spherical varieties/real spherical manifolds.

The main approach of Lecture 1 is PDEs.

We are interested here in certain cones which describe how an orbit of a Lie group decomposes, after projection, into a union of orbits for a subgroup.

This topic has been widely studied in the context of adjoint orbits. Let us cite the pioneering works of Schur, Horn, Kostant and Heckman, and the extension of these results to the symplectic framework by Atiyah, Guillemin-Sternberg, Kirwan, Hilgert-Neeb-Planck, et al.

Here we will discuss the "symplectic with involution" framework which was first considered by Duistermaat (abelian case) and O'Shea-Sjamaar (non-abelian case). This will allow us to study the orbits of isotropic representations of symmetric spaces.

We will explain how techniques from the Geometric Invariant Theory (due to Klyachko, Belkale-Kumar, Berenstein-Sjamaar and Ressayre) allow to describe the inequalities that define these cones.

The local Gross-Prasad conjecture was introduced by Gross and Prasad in the 1990s. The conjecture for tempered parameters over non-archimedean local fields was proved by Waldspurger using the trace formula and twisted formula, and the conjecture for generic parameters over non-archimedean local fields was later proved by Mœglin and Waldspurger. I will present my proof for the conjecture for generic parameters over archimedean local fields, and part of the work (tempered cases) is joint with Z. Luo. In tempered cases, we followed Waldspurger’s approach and simplified some steps of the proof (in particular we do not need the twisted trace formula). This simplification only works over archimedean local fields. In generic cases, I use representation theory results parallel to Moeglin and Waldspurger's proof and some harmonic analyses slightly different from Mœglin and Waldspurger's approach due to the absence of Casselman's canonical pairing.

We discuss asymptotic behaviors of matrix coefficients on $L^2(X)$ beyond (real) spherical spaces, and explain a criterion to detect for which homogeneous space $X$ the regular representation $L^2(X)$ is tempered. If time permits, we discuss the notion of tempered subgroups à la Margulis and its connection with discontinuous groups.

The main approach of Lecture 2 is dynamical system and unitary representation theory.

Noncommutative geometry attempts to develop geometric perspectives on C*-algebras, as if those C*-algebras were the algebras of continuous functions on geometric spaces. I shall describe two basic constructions of this type within the realm of real reductive groups - the Dirac operator and the dual Dirac operator. They were invented by Gennadi Kasparov, and they lead directly towards the Connes-Kasparov isomorphism.

The theory of crystal bases, due to Kashiwara and Lusztig, is a means of simplifying the representation theory of semisimple Lie algebras by passing through quantum groups. Varying the parameter q of the quantized enveloping algebras, we pass from the classical theory at q=1 through the Drinfeld-Jimbo algebras at 0 < q < 1 to the crystal limit at q=0, where the main features of the representation theory (matrix coefficients, Clebsch-Gordan coefficients, branching rules) crystallize into purely combinatorial data described by crystal graphs. In this talk, we will describe what happens to the C*-algebra of functions on a compact semisimple Lie group under the crystallization process, leading to a of higher-rank graph algebras. This is joint work with Marco Matassa.

In this talk I shall explain how the Schrödinger equation of the hydrogen atom gives rise to an algebraic family of Harish-Chandra pairs that is related to SO(4), SO(3,1) and the corresponding Cartan motion group. I will show that solutions of the Schrödinger equation form an algebraic family of Harish-Chandra modules. If time permits, I will explain how the infinitesimally unitary Jantzen quotients of the family of solutions are related to harmonic analysis of a certain homogeneous space.

We discuss how Delorme’s proof of Arthur’s Paley-Wiener theorem for real reductive groups leads to a new proof of the Connes-Kasparov isomorphism.

Pseudodifferential operator theory may seem to be an unlikely topic for this meeting. But pseudodifferential operators have been central to index theory, and index theory is in turn central to the Connes-Kasparov isomorphism. Additionally, pseudodifferential operators played an important role in the revival of Mackey’s ideas about a correspondence between the irreducible representations of a real reductive group and those of its Cartan motion group. Finally, new perspectives in pseudodifferential operator theory, which I shall explain, make it natural to look for new ways to use pseudodifferential operators in representation theory and noncommutative geometry.

Let G be a real reductive algebraic group. Every Harish-Chandra module V for G admits a globalization V subset Vmin, which is the union over R>0 of its subspaces C^infty_R(G)*V. It was shown by Kashiwara and Schmid that Vmin is isomorphic to the topological vector space of analytic vectors for V. We explore the relation between the size of R and the size of a neighborhood of G in its complexification, to which the analytic vectors extend holomorphically.

By this we obtain a new proof of the theorem of Kashiwara and Schmid, and thereby also a new proof of Helgason's conjecture for the Poisson transform of G/K (joint work with H. Gimperlein, B. Krötz, and J. Kuit).

KMS states and the corresponding vectors (obtained from the GNS construction) are well-known in the context of operator algebras. However, their role in the representation theory of semisimple Lie groups is much less clear. They exist in two flavors: First, they arise from Gibbs states associated to unitary lowest weight representations for certain elliptic one-parameter groups, but second, and more universally, they arise for certain hyperbolic one-parameter groups as distributional generators of nets of standard subspaces on causal homogeneous spaces. The second type conjecturally occurs in all unitary representations of the corresponding groups. In the talk we explain how standard subspaces of complex Hilbert spaces are related to KMS conditions (a closed real subspace V is standard if V ∩ iV = 0 and V + iV is dense) and how they can be used to construct nets of local algebras as they occur in Algebraic Quantum Field Theory.

We address a question which reductive homogeneous space is tempered. It turns out that this question is subtle even for semisimple symmetric spaces, however, quite surprisingly, one can give a complete description of (reductive) non-tempered homogeneous spaces. Lecture 3 focuses on some basic ideas in the classification theory of non-tempered homogeneous spaces.

The main approach of Lecture 3 is combinatorics for convex polyhedra.

Continuing on what Nigel Higson told us in his lectures, I will describe C*-algebras that can be associated to reductive symmetric pairs. I will focus on describing these C*-algebras and their K-theory groups for some of the well-studied rank-one examples (and rank-two examples depending on time). I will describe what are non-trivial questions, what we can expect and what are new phenomena relative to the classical group case (i.e. the reduced group C*-algebras).

\emph{Jun Yang (Harvard)}\quad-\quad{\small Jan. 19 at 2:00pm.}

We start with a formula proved by Atiyah-Schmid which connects formal dimensions of discrete series representations and von Neumann dimensions. We generalize this to the Plancherel measure of the unitary representations of an adelic group and the von Neumann algebra of a semi-simple group over a number field. This generalization is motivated by the real Lie groups with holomorphic discrete series and their lattices. Besides, we observe that cusp forms always intertwine with actions of the lattice on any two discrete series. We further present the cusp forms generate the entire commutant of the lattice.

The wavefront set of a distribution on a manifold describes the set of points having no neighbourhood where the distribution is smooth and the direction in which the singularity occurs. Let $\Psi_m$ be the positive definite kernels that arises from reflection positivity on sphere. The kernel $\Psi_m$ can be extended analytically to a complex domain called crown domain. This is a key tool in obtaining irreducible unitary spherical representations of the orthochronous Lorentz group $O_{1,n}(\mathbb{R})^{\uparrow}$. One of the boundary components of the crown is a one sheeted hyperboloid called de-Sitter space. We will look at the wavefront sets of distributions that are boundary values of sesquiholomorphic kernels $\Psi_m$ on de-Sitter space. We use the wavefront set to show that the limit distribution does not vanish on any non-empty open set.

Invented by Jean-Michel Bismut, the hypoelliptic Laplacian is a new type of index theory. It provides a remarkable trace formula, and further reveals completely new insights in representation theory. However, the main difficulty of this subject lies in analysis. Especially, the hypoellipticity of the operator makes the analysis challenging. In this talk, using representation of nilpotent Lie group, I will show the hypoelliptic Laplacian is actually elliptic, in the sense that it possess nice elliptic estimates. This will significantly reduce the complexity of analysis.

This is part of a joint program with N. Higson, E. MacDonald, F. Sukochev, and D. Zanin.

For a connected real reductive Lie group G and a maximal compact subgroup K, we introduce a certain C*-category assembled from classical order 0, G-equivariant pseudodifferential operators on G/K. Using categorical analogues of classical results for C*-algebras, we compute the K-theory of this C*-category explicitly in some examples, and show that in these cases we obtain a natural isomorphism between the K-theory of the C*-category and the representation ring of K.

In this talk, we will discuss one application of Bismut’s orbital integral formula to representation theory. In particular, I will talk about the larger time behavior of the orbital integral associated to the heat operator of the Casimir on symmetric spaces.

We discuss a bird's eye view on tempered homogeneous spaces in the variety of all Lie subalgebras in a fixed Lie algebra. We also discuss tempered homogeneous spaces from the viewpoint of “orbit philosophy”. These viewpoints enrich the concept of “tempered homogeneous spaces” not only from analysis but also from topology and geometry. If time permits, we explain some open problems.

The main disciplines of Lecture 4 are topology (collapsing Lie algebras) and geometric quantization.

Let G be a real or a p-adic connected reductive group. We will recall the description of the connected components of the tempered dual of G in terms of certain subalgebras of its reduced C*-algebra.

Each connected component comes with a torus equipped with a finite group action. We will see that, under a certain geometric assumption on the structure of stabilizers for that action (that is always satisfied for real groups), the component has a simple geometric structure which encodes the reducibility of the associate parabolically induced representations.

We will provide a characterization of the connected components for which the geometric assumption is satisfied, in the case when G is a p-adic symplectic group. It is joint work with Alexandre Afgoustidis.