Let $mathbb{B}$ be a Lie group admitting a left-invariant negatively curved Kahlerian structure. Consider a strongly continuous action $alpha$ of $mathbb{B}$ on a Frechet algebra $mathcal{A}$. Denote by $mathcal{A}^infty$ the associated Frechet algebra of smooth vectors for this action. In the Abelian case $mathbb{B}=mathbb{R}^{2n}$ and $alpha$ isometric, Marc Rieffel proved that Weyl's operator symbol composition formula (the so called Moyal product) yields a deformation through Frechet algebra structures ${star_{theta}^alpha}_{thetainmathbb{R}}$ on $mathcal{A}^infty$. When $mathcal{A}$ is a $C^*$-algebra, every deformed Frechet algebra $(mathcal{A}^infty,star^alpha_theta)$ admits a compatible pre-$C^*$-structure, hence yielding a deformation theory at the level of $C^*$-algebras too.

In this memoir, the authors prove both analogous statements for general negatively curved Kahlerian groups. The construction relies on the one hand on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geom,etrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a non-Abelian version of the Calderon-Vaillancourt Theorem. In particular, the authors give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.