Deformation Quantization for Actions of K\"ahlerian Lie Groups

Abstract

Let $\mathbb B$ be a Lie group admitting a left-invariant negatively curved K\"ahlerian structure. Consider a strongly continuous action $\alpha$ of $\mathbb B$ on a Fr\'echet algebra $\mathcal A$. Denote by $\mathcal A^\infty$ the associated Fr\'echet algebra of smooth vectors for the action $\alpha$. In the Abelian case $\mathbb B=\mathbb R^{2n}$ and $\alpha$ isometric, Marc Rieffel proved that Weyl's operator symbol composition formula yields a deformation through Fr\'echet algebra structures ${\star_{\theta}^\alpha}_{\theta\in\mathbb R}$ on $\mathcal A^\infty$. When $\mathcal A$ is a $C^\star$-algebra, every deformed algebra $(\mathcal A^\infty,\star^\alpha_\theta)$ admits a compatible pre-$C^\star$-structure. In this paper, we prove both analogous statements in the general negatively curved K\"ahlerian group and (non-isometric) "tempered" action case. The construction relies on the one hand on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geometrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a non-Abelian version of the Calder\`on-Vaillancourt Theorem. In particular, we give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.