The classical Rankin-Cohen brackets are bi-differential operators from $C^\infty(\mathbb R)\times C^\infty(\mathbb R)$ into $ C^\infty(\mathbb R)$. They are covariant for the diagonal action of ${\rm SL}(2,\mathbb R)$ through principal series representations. We construct generalizations of these operators, replacing $\mathbb R$ by $\mathbb R^n,$ the group ${\rm SL}(2,\mathbb R)$ by the...

Let $G$ be a semisimple Lie group with finite centre, $K$ a maximal compact subgroup and $P$ a parabolic subgroup of $G$. We present a new construction of Poisson transforms between vector bundle valued differential forms on the homogeneous parabolic geometry $G/P$ and its corresponding Riemannian symmetric space $G/K$ which is tailored to the exterior calculus and can be fully described by...

The Baum-Connes conjecture on the K-theory of group C*-algebras is a difficult open problem since the beginning of the 1980’s. In the last 30 years a programme has been developed to prove the Baum-Connes conjecture with coefficients for semi-simple Lie groups. The tools involved are: the flag manifolds, the BGG complex, and L2 cohomology of symmetric spaces.

Locally compact quantum group (LCQG) in the setting of von Neumann algebras (aka Kustermans-Vaes quantum groups), is believed to give the correct notion of symmetries of quantum spaces (in the setting of operator algebras). While this theory is fast growing, there are very few examples of (non-compact) LCQG.

In this talk, I will explain how the good old Kohn-Nirenberg quantization allows to...

Let $G$ be a Lie group, $H$ a closed subgroup of $G$ and $\Gamma$ a discontinuous subgroup for the homogeneous space $\mathscr{X}=G/H$, which means that $\Gamma$ is a discrete subgroup of $G$ acting properly discontinuously and fixed point freely on $\mathscr{X}$. For any deformation of $\Gamma$, the deformed discrete subgroup may fail to act discontinuously on $\mathscr{X}$, except for the...

We introduce a notion of symplectic reduction for symplectic symmetric spaces as a means to the study of their structure theory. We show that any such space can be written as a direct product of a semisimple and a completely symplectically reducible one. Underlying symplectic reduction is a notion of so-called pre-Lie triple system. We will explain how these are related to étale affine...

Abstract: We describe a translation principle for the Dirac index of virtual $({\mathfrak g},K)$-modules. To each coherent family of such modules we attach a polynomial, on the dual of the compact Cartan subalgebra, which expresses the dependence of the leading term in the Taylor expansion of the character of the modules. Finally we will explain how this polynomial is related to the...