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Journées SL2R

5–6 décembre 2019, Université du Luxembourg

Programme

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Jeudi 5 décembre 2019 Salle MSA 3.350, Campus Belval

 13:00 Inscription 13:45–14:45 Ines Kath Spectra of compact quotients of the oscillator group 14:45–15:45 Salah Mehdi Representation theoretic spectrum of locally symmetric spaces 15:45–16:15 Pause café 16:15–17:15 Rupert Yu On invariant semisimple CR structures on compact Lie groups 17:15–18:15 Sourav Ghosh Moduli space of affine Anosov representations 19:30 Dîner de conférence au restaurant Postkutsch 8 rue Xavier Brasseur, L-4040 Esch-sur-Alzette

Vendredi 6 décembre 2019 Salle MSA 3.350, Campus Belval

 09:00–10:00 Bernhard Krötz Discrete series for real spherical spaces 10:00–10:30 Pause café 10:30–11:30 Wolfgang Bertram Associative geometries: the arithmetic case 11:30–12:30 Erik van den Ban Fourier inversion for Whittaker functions on a real reductive group 12:45 Déjeuner à la cantine de l'université 'Food House'

Résumés

• Erik van den Ban (Universiteit Utrecht):

Fourier inversion for Whittaker functions on a real reductive group

A Whittaker function on a real reductive group G is a smooth function on G that transforms from the right according to a (regular) unitary character of a maximal unipotent subgroup. I will describe two new results in the harmonic analysis of these functions which are closely related to the Whittaker-Plancherel theorem. The first is a Fourier inversion theorem. The second is a closely related Paley-Wiener theorem.

• Wolfgang Bertram (Université de Lorraine - Nancy):

Géométrie associative : le cas arithmétique

Associative geometries have been designed to be, for associative algebras, what Lie groups are for Lie algebras (cf. B.-Kinyon, arxiv). The construction applies to an even greater variety of cases, among them the arithmetic case, i.e., the lattice of integers with operations lcm (lowest common multiple) and gcd (greatest common divisor). To our surprise, this case gives rise to a purely lattice theoretic formula describing the whole construction. We will explain this formula and its proof, along with examples (producing large families of semigroups of integers), and discuss some further questions.

• Sourav Ghosh (Université du Luxembourg):

Moduli space of affine Anosov representations

In this talk I will define affine Anosov representations and explain their relation to proper affine actions of a word hyperbolic group. Moreover, I will explain certain recent results about the geometry of the moduli space of such objects.

• Ines Kath (Universität Greifswald):

Spectra of compact quotients of the oscillator group

We consider the oscillator group $\mathcal{O}$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of $\mathcal{O}$ up to inner automorphisms of $\mathcal{O}$. For every lattice $L$ in $\mathcal{O}$, we compute the decomposition of the right regular representation of $\mathcal{O}$ on $L^2(L\backslash\mathcal{O})$ into irreducible unitary representations. This is joint work with Mathias Fischer.

Discrete series for real spherical spaces

no one

• Salah Mehdi (Université de Lorraine - Metz):

Representation theoretic spectrum of locally symmetric spaces

We consider locally symmetric spaces $\Gamma\backslash G/H$ where $G/H$ is a semisimple symmetric space and $\Gamma$ is a discrete subgroup of $G$. It is a fact that the algebra $D(G/H)$ of $G$-invariant differential operators on $G/H$ is commutative. I will discuss some features of the joint spectrum of $D(G/H)$ acting, as unbounded operators, on the Hilbert space $L^2(\Gamma\backslash G/H)$ of square integrable complex functions on $\Gamma\backslash G/H$. Though the group $G$ does not act on $L^2(\Gamma\backslash G/H)$, I will explain how representations of $G$ enter into its spectral decomposition. As a byproduct, and if time allows, I will present a result on the admissibiliy of $G$-representations with respect to non-compact subgroups. These results are joint with Martin Olbrich.

• Rupert Yu (Université de Reims):

On invariant semisimple CR structures on compact Lie groups

Let $G_{0}$ be a compact Lie group and $\mathfrak{g}_{0}$ its Lie algebra. An invariant CR structure on $G_{0}$ is a subbundle $U$ of the complexified tangent bundle such that $[U,U]\subset U$, $U\cap \overline{U}= 0$ and $U$ is stable under translation by $G$. Such structures are determined by its fibre at the identity element of $G$ which is a complex Lie subalgebra $\mathfrak{h}$ of the complexification of $\mathfrak{g}_{0}$ verifying $\mathfrak{h}\cap \mathfrak{g}_{0}=\{ 0\}$. This allows us to study these structures in a purely Lie algebra setting. In this talk, we shall present some results on such structures of maximal rank, in particular, those whose fibre at the identity element is a semisimple Lie algebra.
This is a joint work in progress with Hella Ounaies-Khalgui.