Let \(\mathbb{B}\) be a Lie group admitting a left-invariant negatively curved Kählerian structure. Consider a strongly continuous action \(\alpha\) of \(\mathbb{B}\) on a Fréchet algebra \(\mathcal{A}\). Denote by \(\mathcal{A}^\infty\) the associated Fréchet 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 Fréchet algebra structures \(\{\star_{\theta}^\alpha\}_{\theta\in\mathbb{R}}\) on \(\mathcal{A}^\infty\). When \(\mathcal{A}\) is a \(C^*\)-algebra, every deformed Fréchet 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 Kählerian 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 Calderón-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.