Twisted ambidexterity in equivariant homotopy theory

We develop the concept of twisted ambidexterity in a parametrized presentably symmetric monoidal $\mathrm{\infty }$-category, which generalizes the notion of ambidexterity by Hopkins and Lurie and the Wirthmüller isomorphisms in equivariant stable homotopy theory, and is closely related to Costenoble-Waner duality. Our main result establishes the parametrized $\mathrm{\infty }$-category of genuine $G$-spectra for a compact Lie group $G$ as the universal example of a presentably symmetric monoidal $\mathrm{\infty }$-category parametrized over $G$-spaces which is both stable and satisfies twisted ambidexterity for compact $G$-spaces. We further extend this result to the settings of orbispectra and proper genuine $G$-spectra for a Lie group $G$ which is not necessarily compact.