Non-formal deformation quantizations of solvable Ricci-type symplectic symmetric spaces

Abstract

Ricci-type symplectic manifolds have been introduced and extensively studied by M. Cahen et al. [10, 11]. In this note, we describe their deformation quantizations in the split solvable symmetric case. In particular, we introduce the notion of non-formal tempered deformation quantization on such a space. We show that the set of tempered deformation quantizations is in one-to-one correspondence with the space of Schwartz operator multipliers on the real line. Moreover we prove that every invariant formal star product on a split Ricci-type solvable symmetric space is an asymptotic expansion of a tempered non-formal quantization. This note illustrates and partially reviews through an example a problematic studied by the author regarding non-formal quantization in presence of large groups of symmetries.