Symmetric spaces, non-formal star-products and Drinfel’d twists

Abstract

These notes present old and new results of mine and collaborators in the field of non-formal deformation quantization of symplectic symmetric spaces. I first review an explicit construction of oscillatory integral formulae for symmetry-invariant non-formal star-products on symplectic symmetric spaces. In a second step, I present a method for explicitly constructing non-formal invariant star-products on a large class of homogeneous symplectic spaces. In a third step, I review a joint work with V. Gayral where we explicitly define non-formal universal deformation formulae (Drinfel’d twists) for actions of non-Abelian Lie groups. At last, I present a class of symmetry-invariant non-formal star-product function algebras on the hyperbolic plane. These algebras are represented by sub-$\ast$-algebras of compact operators acting in the projective holomorphic discrete series of $\mathrm{SL}(2,R)$.