Excursus on modulation spaces via metaplectic operators and related time-frequency representations

We provide a comprehensive overview of the theoretical framework surrounding modulation spaces and their characterizations, particularly focusing on the role of metaplectic operators and time-frequency representations. We highlight the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. In particular, this work provides new characterizations via the sub- manifold of shift-invertible symplectic matrices. Similar results hold for the Wiener amalgam spaces.