Hyperbolic Localization and Second Adjointness

Following an idea of Drinfeld, we present an abstract proof of Braden's
result on hyperbolic localization, relating sheaves on a space with
$\mathbb G_m$-action to their restriction to the attractor, repeller and
fixed point locus. Our argument is purely formal and works in an
abstract setting, and it roughly reduces the general statement to the
base case of $\mathbb G_m$ acting on $\mathbb A^1$.  This allows us to
extend the result to stacks and to arbitrary 6-functor formalisms.
Applying the result to the classifying stack of a reductive group, we
obtain new results on second adjointness for smooth representations in
natural characteristic, and we expect a similar version for locally
analytic representations as well as other applications to various
geometric incarnations of second adjointness in the Langlands program.
The main technical difficulty lies in handling the higher category
theory involved in the argument, and we have found efficient ways to
deal with "enriched (infty,2)-categories" and lax functors between them.
This is joint work with Claudius Heyer.