• Alexandre Afgoustidis (Metz & Nancy)

Title : Local Langlands and lowest K-types.

Abstract : For the irreducible representations of a real reductive groups, various parametrizations are known. Depending on the chosen parametrization, useful invariants of the representations can be easy to read on the corresponding parameters, or more difficult to extract. My talk will adress a special case of this question: does the local Langlands correspondence encode lowest K-types in an easily accessible manner? I will describe a simple connection between the lowest K-types of an irreducible admissible representation and its Langlands parameters (in the ‘refined’ version which includes the internal parametrization of L-packets in terms of component groups attached to L-homomorphisms). This is joint work with Jeffrey Adams.

• Anne-Marie Aubert (Paris)

Titre : Blocs stables de paramètres de Langlands enrichis.

Résumé : La décomposition en blocs de Bernstein de la catégorie des représentations lisses d'un groupe p-adique G admet un analogue galoisien en termes de paramètres de Langlands enrichis. Le rôle des représentations supercuspidales y est joué par les paramètres de Langlands enrichis « cuspidaux », une notion qui trouve son origine dans la construction par Lusztig de la correspondance de Springer généralisée. Nous formulerons des conditions sur la correspondance de Langlands pour les représentations supercuspidales des sous-groupes de Levi de G qui permettent de construire une bijection canonique entre les blocs associés de part et d'autre. Nous expliquerons ensuite comment regrouper les blocs de Bernstein, de sorte que ces nouveaux blocs soient des unions de L-paquets.

• Marc van Leeuwen (Poitiers)

Title (Atlas) : More about the Atlas software.

Abstract : The Atlas software library includes a wide range of capabilities that can be used in representation theory calculations or elsewhere. Examples include linear algebra (solving systems of equations with integer or rational coordinates), combinatorics (factorials, binomial coefficients, partition functions, lexicographic sorting of permutations), enumeration of special Lie algebra elements, Weyl group representations, Kazhdan-Lusztig polynomials, and more. This lecture will talk about how to find such tools, and offer more details about the {\tt axis} scripting language used in the software.

• Martin Olbrich (Belval)

Title : Restriction problems related to harmonic analysis on non-Riemannian locally symmetric spaces.

Abstract : Most known compact locally symmetric spaces Gamma\G/H modeled on a semsisimple symmetric space G/H are standard: there exists a reductively embedded connected subgroup L of G acting properly and transitively on G/H, and Gamma is a lattice in L. In this talk we discuss consequences of results on the restrictions to L of H-spherical G-representations for the spectral theory of certain standand non-Riemannian locally symmeytric spaces. This is joint work with Salah Mehdi.

• Pavle Pandzic (Zagreb)

Title : SL(2) theory and quantum computing.

Abstract : We consider the action of G=GL_2(C) x GL_2(C) x GL_2(C) on the space P(V) of polynomials on V= C^2\otimes C^2\otimes C^2, where C^2 is the standard representation of GL_2(C). We describe explicit highest weight vectors, as well as the multiplicities, of all irreducible submodules of P(V). We apply this to obtain a description of the orbit decomposition of V under the action of G, as well as under the action of the subgroup K=U(2)xU(2)xU(2) of G. The latter gives a classification of the three-cubit states in quantum computing. This is joint work with Jing-Song Huang and Soo Teck Lee.

• Paul-Emile Paradan (Montpellier)

Title : Convexity in Lie Theory: Some New Results.

Abstract : The following example will serve as a guide during the presentation. What is the relationship between the spectra s(X) and s(Y) of an n-square Hermitian matrix X and that of its real part Y = Re(X)? We will see that the set formed by the pairs (s(X),s(Y)) is a polyhedral cone that can be described recursively relative to « n ». Our previous question is part of the general problem of describing the projections of adjoint orbits. This topic has been widely studied since 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. We shall see that to answer our initial question, we have to work in the « symplectic with involution » framework first considered by Duistermaat (abelian case) and O'Shea-Sjamaar (non-abelian case).

• David Renard (Palaiseau)

Title : Arthur packets and discrete series of real classical symmetric spaces.

Abstract : I will explain some aspects of the Sakellaridis-Venkatesh conjectures on the spectrum of a spherical variety G/H, more precisely the description of the spectrum of G/H in the Arthur-Langlands formalism. When G is a classical real group and G/H is a symmetric space, we establish the validity of the conjectures for the discrete part of the spectrum (joint work with C. Moeglin).

• David Vogan (Cambridge, MA)

Title (colloquium) : What's special about special?

Abstract : Both conjugacy classes of nilpotent matrices (of size n) and irreducible representations of the symmetric group Sn are indexed by partitions of n. For any complex reductive group, there is a (finite) collection of conjugacy classes of nilpotent Lie algebra elements, and a (finite) set of irreducible Weyl group representations, both enumerated by the 1950s. One might therefore hope for a relationship between these finite sets. I'll first explain Springer's (somewhat complicated) description of such a relationship, and then Lusztig's identification of a bijection between what he called special Weyl group representations and special nilpotent orbits. I'll explain how these ideas arise in the representation theory of real reductive groups, and what light that might shed on Lusztig's definition of special.


Title (Atlas): Introduction to the Atlas software.

Abstract : The Atlas software makes a wide variety of computations related to the (infinite-dimensional) representation theory of a real reductive Lie group G. The software was begun in 2004--06 by Fokko du Cloux, and since 2006 has been developed by Marc van Leeuwen. The morning session will offer a little information about how to download and start the software; then a demonstration of how to use it. The session is meant to be very interactive, so the choice of examples can be based on audience interest. The session will conclude with perhaps half an hour outlining some of the mathematics on which the software is based.


Titre (Atlas) : Introduction au logiciel Atlas.

Résumé : Le logiciel Atlas effectue une grande variété de calculs liés aux représentations (de dimension infinie) d'un groupe de Lie réel réductive G. Atlas a été initié dans les années 2004-06 par Fokko du Cloux, et est développé depuis 2006 par Marc van Leeuwen. L'atelier du matin sera consacré à l'installation du logiciel et à son démarrage, suivis d'une démonstration de son utilisation. La session sera interactive et les exemples qui seront traités dépendront des centres d'intérêts des participants. L'atelier se concluera par une decription des mathématiques mises en oeuvre dans le logiciel.