On the parametrized Tate construction

We introduce and study a genuine equivariant refinement of the Tate construction associated to an extension $\widehat{G}$ of a finite group $G$ by a compact Lie group $K$, which we call the parametrized Tate construction $(-)^{t_G K}$. Our main theorem establishes the coincidence of three conceptually distinct approaches to its construction when $K$ is also finite: one via recollement theory for the $K$-free $\widehat{G}$-family, another via parametrized ambidexterity for $G$-local systems, and the last via parametrized assembly maps. We also show that $(-)^{t_G K}$ uniquely admits the structure of a lax $G$-symmetric monoidal functor, thereby refining a theorem of Nikolaus and Scholze. Along the way, we apply a theorem of the second author to reprove a result of Ayala--Mazel-Gee--Rozenblyum on reconstructing a genuine $G$-spectrum from its geometric fixed points; our method of proof further yields a formula for the geometric fixed points of an $\mathcal{F}$-complete $G$-spectrum for any $G$-family $\mathcal{F}$.