Speaker : Prof. R.P. Langlands
Title : On the passage from two-dimensional models in statistical mechanics to representations of the Virasoro algebra: a mathematician’s experience and hopes.
Abstract: Not given

Speaker : Prof. David Vogan
Title: Cutting and pasting unitary representations
Abstract: Building on ideas going back to Fourier and many others, Gelfand showed in the 1940s how to use unitary representations - linear actions of a group on a nice topological vector space, preserving a positive Hermitian form - to treat a range of analytical problems. Understanding the unitary representations of a group can therefore be a powerful first step toward solving a variety of analytic and geometric problems.
In many cases, it turns out to be relatively easy to understand representations that preserve a (possibly indefinite) Hermitian form. What’s difficult is understanding when the form is definite. On a one-dimensional vector space, any Hermitian form is definite, because there is no room for a signature that has both a positive and a negative part. It follows that any one-dimensional Hermitian representation is unitary. I’ll explain a way to apply the same idea to infinite-dimensional representations, by understanding them as glued together from indivisible pieces. This understanding is still incomplete: even a proper definition of the ”indivisible pieces” is lacking in general. But enough is known to make calculations possible in examples, and so to prove that certain interesting representations are unitary.

Speaker: Prof. A. El Kacimi
Title: Towards a basic index theory.
Abstract: In this talk, we will give and explain the statement of the Hodge decomposition theorem for a transversely elliptic operator on a Riemannian foliation. Such an operator is Fredholm and therefore admits an index. This brings us to formulate the following problem: can this index be computed in terms of (topological) transverse invariants like in the Atiyah-Singer Index theorem? Some examples of transversely elliptic operators whose indices are topologically invariant will be given, in particular the basic Euler-Poincare number and the signature of a Riemannian foliation.

Speaker : Prof. Oda Takayuki
Title: Secondary spherical functions and the associated Eisenstein-Poincaré series
Abstract : Eisenstein-Poincaré series is a fundamental tool to construct automorphic forms. Normally for some spherical function φ on a certain semisimple Lie group G, which is left-invariant under some closed subgroup R of G, we consider an infinite sum

[E_φ(g) := ∑ _{γ∈R∩Γ\Γ}φ(γg) (g ∈ G).]

Here Γ is a discrete (arithmetic) subgroup of G, which is either cocompact, or more generally of finite covolume. When φ is of L¹ on G mod R, E _φ(g) converges absolutely.

But it is quite useful and important to consider non-L¹ φ’s. Then we have to regard E_φ as a distribution on G. However when φ is a usual spherical function on G, which, say, belongs to a principal series representation of G, the series E_φ does not converge even as a distribution. For this difficulty, a known remedy is to replace φ by “secondary spherical functions”, i.e., either ’resolvent kernels’, or Harish-Chandra’s hypergeometric series on G, which appears in the asymptotic expansions of various spherical functions.

Then φ = φ_s has a system of parameters to define the principal series representations, and the residue or the special value of the Eisenstein-Poincaré series E_φs(s) at a nice point s = ρ often has geometric meaning (e.g. Harder-Schwermer’s Eisenstein cohomology classes).

This kind of Eisenstein-Poincaré series played an important role in the work of Gross-Zagier on Arakelov intersection theory of modular curves. Also Miatello-Wallach investigated the case of R-rank 1 group G systematically.

For groups G of highher R-rank, there seems to be many problems to fix. The speaker is going to report on two projetcs in progress, related with the above problem:
(A): Explicit calculus on principal series (secondary) Whittaker functions on GL(3,C) (a joint work with Miki Hirano of Ehime University), and
(B): Green currents for modular cycles of higher codimension on certain modular varieties (a joint work with Masao Tsuzuki of Sophia University).

These are developments of recent former papers refered below:

References

Manabe, H., Ishii, T., Oda, T: Principal series Whittaker functions on SL(3,R), Japanese J. Math., 30 (2004), 183–226.

Oda, T., Tsuzuki, M.: Automorphic Green functions associated with the secondary spherical functions 39 (2003), 451–533.

Speaker : Prof. Joseph Wolf
Title: Cycle Spaces in Representation Theory
Abstract: Admissible representations of semisimple Lie groups can generally be realized on cohomologies of partially holomorphic vector bundles over real group orbits in complex flag manifolds. For tempered representa- tions this can be done in such a way that the cohomology carries a corresponding invariant hermitian inner product, so the representation is realized as a unitary representation. For non-tempered unitary representations the inner product is usually lost, though there are some special situations where one keeps track of it. But in general, one needs a realization for which growth information is available. One possibility is the transfer of representations from flag domains to cycle spaces by means of double fibration transforms. This talk will be a survey of current developments on the structure of cycle spaces and double fibration transforms.

Speaker : Prof. Gregg Zuckerman
Title : “Harmonic Algebra on Semisimple Lie Groups.”
Abstract:  "Harmonic algebra" is our term for a synthesis of harmonic  analysis  and homological algebra.  We are particularly motivated by the interaction of  the theory of admissible representations of semisimple Lie groups and the theory  of homogeneous vector bundles and differential operators.  For example, we  would like to better understand the Dirac equation on a pseudoRiemannian  homogeneous space.

 Frobenius Reciprocity and Shapiro's Lemma play an essential role in harmonic  algebra and lead to a new application of the so-called Zuckerman derived  functors.  Ultimately, we formulate harmonic algebra in the derived  category of  modules over the universal enveloping algebra.  For example, we formulate a  homotopy theory of complexes of homogeneous differential operators.

Speaker : Prof. Wilfried Schmid
Title: The Rankin-Selberg method for automorphic distributions
Abstract: Rankin and Selberg separately proved the holomorphic continuation and functional equation of what, in retrospect, can be called the Langlands L-function of the tensor product of two holomorphic discrete series representations of SL(2,R). The method has been adapted to the GL(n)XGL(m), exterior square, and symmetric square L-functions for GL(n). After recounting the existing results, I shall describe an adaptation of the Rankin-Selberg method to automorphic distributions, which has led to new results. This is joint work with Steve Miller.

Speaker : Prof. Adam Koranyi
Title : New Remarks on the Satake Compactifications.
Abstract: The first remark gives a simple direct construction of the Satake-Furstenberg compactifications of a non-compact globally symmetric Riemannian space. The second is about a natural construction of a family of imbeddings of such a space as a convex domain in a vector space. For each one of the Satake compactifications there is an imbedding with the property that the boundary components are exactly the flat faces of the domain. The Harish-Chandra realization of Hermitian symmetric spaces is a special case.

Speaker: Prof. Birgit Speh
Title: Pseudo dual pairs and restrictions of cohomologica lly unitary induced representations.
Abstract: Let G be a semisimple Lie group with Cartan involution Θ. For an involution τ₁ of G which commutes with τ₁ consider the related involution τ₂ = τ₁ ⋅ Θ with fix point sets H ₁ and H ₂ respectively. I will discuss some special cases of the restriction of cohomologically induced representations to H₁ and H₂. Some applications to automorphic forms will be wil l be indicated.

Speaker: Prof. D. Milicic
Title: Geometry and representations of real semisimple Lie groups
Abstract:  Harish-Chandra studied representations of real semisimple Lie groups by studying analytic objects attached to representations, like  their characters or matrix coefficients. Continuation of this approach lead to the classification of irreducible representations in the work
of Langlands, Knapp and Zuckerman.

Localization theory of Beilinson and Bernstein gives a simple framework for the classification of irreducible representations using D-module theory. Roughly speaking, derived categories of representations are equivalent to derived categories of D-modules (generalizing the
classical Borel-Weil-Bott theorem). In a particular chamber, this equivalence is induced by the equivalence of the cores of these derived categories corresponding to the natural t-structure.
This generalizes the Borel-Weil theorem. In the other chambers, the situation is more complicated since the natural t-structure on representation side corresponds to an "exotic" t-structure on the D-module side. The D-module categories in different chambers
are related by intertwining functors. We shall discuss how, by analyzing the action of intertwining functors, one can extract analytic information about representatiHarish-Chandra studied representations of real semisimple Lie groups by studying analytic objects attached to representations, like  their characters or matrix coefficients. Continuation of this approach
lead to the classification of irreducible representations in the work of Langlands, Knapp and Zuckerman.

Localization theory of Beilinson and Bernstein gives a simple framework for the classification of irreducible representations using D-module theory. Roughly speaking, derived categories of representations are equivalent to derived categories of D-modules (generalizing the
classical Borel-Weil-Bott theorem). In a particular chamber, this equivalence is induced by the equivalence of the cores of these derived categories corresponding to the natural t-structure.
This generalizes the Borel-Weil theorem. In the other chambers, the situation is more complicated since the natural t-structure on representation side corresponds to an "exotic" t-structure on the D-module side. The D-module categories in different chambers  are related by intertwining functors. We shall discuss how, by analyzing the action of intertwining functors, one can extract analytic information about representations. This allows to rederive, by  algebro-geometric methods, the results on classification of representations by parabolic induction (Harish-Chandra, Langlands, Knapp and Zuckerman) and by cohomological induction (Vogan and Zuckerman).

This is a continuation of a joint project with Hecht, Schmid and Wolf.

Speaker:  Prof. T. Kobayashi
Title: Restrictions of Unitary Representations of Real Reductive Groups
Abstract: The branching problem asks how a given irreducible representation of a group
decomposes when restricted to a subgroup.

Making an observation on wild features of branching problems for non-compact subgroups in the general setting, we address the question of finding a criterion for discreteness of spectrum and finiteness of multiplicities in the irreducible decomposition of the restriction.

We also will indicate some applications of our theorems to topological results on modular varieties in locally symmetric spaces, and construction of new discrete series representations for
pseudo-Riemannian homogeneous spaces.

Speaker:  Prof. D.A. Vogan
Title:  Dirac operators and unitary representations: some of  the work of Professor Parthasarathy
Abstract: The representation theory of a compact Lie group K is well understood by classical work of Cartan and Weyl.  If T is a maximal torus in K,   then a representation of K is may be regarded as a sum of characters of T.  The homogeneous space K/T is a projective algebraic variety, and (thanks to Borel, Weil, and Bott) representations of K can be realized in sheaf cohomology of K/T with coefficients in algebraic line bundles.

Suppose G is a semisimple Lie group.  Inside G there is a maximal compact subgroup K, and the homogeneous space G/K is a Riemannian symmetric space.  Harish-Chandra showed in the 1950s and 1960s that  infinite-dimensional representations of G can be thought of as sums of
finite-dimensional representations of K, and used this point of view to establish many fundamental properties of harmonic analysis on G.    But his work did not show exactly which sums of K representations could be extended to G representations, and it provided few direct
clues about how to get detailed information about G representations from our knowledge of K.

The work of Parthasarathy changed that. He provided many of the tools that allow one to relate representations of G and of K almost as closely as one relates representations of K and of T in the classical theory. One of the first and most spectacular examples was his  introduction of a Dirac operator on G/K.  This operator plays the role of the del-bar operator (more precisely, of del-bar plus its adjoint) on K/T.  First of all this allowed him to realize discrete series  representations of G as solutions of the Dirac operator on G/K, in  analogy with the Borel-Weil-Bott realization of finite-dimensional representations. Just as the Bott-Borel-Weil theorem is the beginning of a still-growing body of understanding of compact groups, so Parthasarathy's Dirac operator has led over the past thirty-five years to some of the
deepest results on unitary representations of G.  Typical examples are the work of Parthasarathy's student Kumaresan describing unitary representations that can contribute to the cohomology of locally   symmetric spaces, and Parthasarathy's characterization of unitary 
holomorphic representations.

Speaker: Prof. Arvind Nair
Title: On the Hodge theory of Shimura varieties
Abstract: Shimura varieties or locally symmetric varieties which are not compact have canonical projective compactifications (due to Baily and Borel).  These are singular varieties and Mumford et. al.  constructed resolutions of them using the theory of torus embeddings.
I shall discuss a homological property of the fibres of these resolution morphisms which has applications to understanding the Hodge theory of such Shimura varieties.

Speaker: Prof. W. Casselman
Title: The Paley-Wiener Theorem for $SL(3,Z)\SL(3,R)$
Abstract: The Paley-Wiener theorem for arithmetic quotients for higher rank is decidedly m
ore complicated than one for rank one groups.  This talk will explain it for the simplest case, and sketch a proof.  I hope also to discuss some consequences.

Speaker:  Prof. S. Salamanca Riba
Title:  On the unitary Dual at Regular Infinitesimal Character
Abstract: In  this talk I will discuss a way to describe a certain class of  unitary representations ofa real reductive Lie group as induced from representations of smaller subgroups. In particular we will focus on the role  of Parthasarathy Dirac Operator Inequality in this process.

Speaker: Prof. L. Clozel
Title: Geometry and analysis on hyperbolic varities
Abstract: This is  joint work with Bergeron. For compact quotients of  hyperbolic spaces, we formulate generalizations of Selber's $\frac{\lambda}{4}$ conjecture. We show that they imply deep cohomological properties similar to the Kudla-Millson theory. In some small degrees these can be proven.

Names : BERGERON, SELBERG,  KUDLA-MILLSON

Speaker: Prof. J.S. Huang
Title: Dirac cohomology and representations of semisimple Lie groups
Abstract: The Dirac cohomology defined using the Dirac operators due to Parthasarathy, Vogan and Kostant is a new tool to study representations of semisimple Lie groups.  It relates the index theory in differential geometry to representation theory. Using the Dirac operators as a unifying scheme we show that some of the most important results in representation theory
fit together when viewed from this perspective.

Speaker: Prof. J. Sengupta
Title:  Beurling's theorem for Riemannian symmetric spaces
Abstract: The uncertainty principle in Harmonic analysis says that a function and  its Fourier transform cannot simultaneously decrease very rapidly at infinity. There are several quantitative versions of this principle like Hardy's Theorem, Cowling-Price Theorem etc.  In this we will give the formulation and a sketch of proof of Beurling's Theorem for Riemannian symmetric spaces which is the  ``master theorem " in this topic.