On the parameterized Tate construction

Abstract We introduce and study a genuine equivariant refinement of the Tate construction associated to an extension of a finite group by a compact Lie group , which we call the parameterized Tate construction . Our main theorem establishes the coincidence of three conceptually distinct approaches to its construction when is also finite: one via recollement theory for the ‐free ‐family, another via parameterized ambidexterity for ‐local systems, and the last via parameterized assembly maps. We also show that uniquely admits the structure of a lax ‐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 ‐spectrum from its geometric fixed points; our method of proof further yields a formula for the geometric fixed points of an ‐complete ‐spectrum for any ‐family .