The classical $J$-homomorphism is a map from the orthogonal group $O(n)$ to the iterated loop space $\Omega^nS^n$ of sphere $S^n$. After $K(1)$-localization, the stable $J$-homomorphism can be interpreted as a profinite transfer map. More precisely, it is a transfer map $\Sigma^{-1}KO^\wedge_2 \to S_{K(1)}$ from the $C_2$-homotopy fixed points (with a twist) to the $\mathbb{Z}_2^\times$-homotopy fixed points of the $2$-complete complex topological $K$-theory.
In joint work in progress with Guchuan Li, we extend this idea to define and study profinite transfers between homotopy fixed points of the Morava E-theory by closed subgroups of the Morava stabilizer group in $K(n)$-local homotopy theory. We introduce two definitions of the profinite transfer maps. One working definition is as duals to the profinite restriction maps in the appropriate category. At large primes, we show that the image of the transfer map $\Sigma^{-n^2}E_n \to S_{K(n)}$ on homotopy groups is filtration $n^2$-line in the homotopy fixed point spectral sequence. A second definition of the profinite transfer maps is based on the 6-functor formalism for smooth representations of $p$-adic Lie groups by Heyer–Mann. We prove that the two definitions are equivalent.
