  Ben-Zvi--Sakellaridis--Venkatesh described a conjectural extension of the
geometric Satake equivalence to spherical varieties, whose spectral
decomposition is described by Hamiltonian varieties. The goal of this article
is to study their conjecture, especially in the case of spherical varieties of
relative rank 1, using tools from homotopy theory. Our discussion relates their
conjecture to classical topics in homotopy theory such as the EHP sequence and
Hopf fibrations, as well as more modern topics such as Hochschild (co)homology.
We will also study an analogue of the derived geometric Satake equivalence and
of the Ben-Zvi--Sakellaridis--Venkatesh conjecture with coefficients in
connective complex K-theory. In this generalized setting, the dual group (a la
Langlands, Gaitsgory--Nadler, Sakellaridis--Venkatesh, Knop--Schalke) remains
unchanged, but the specific dual "representation" of the dual group changes. On
the spectral/Langlands dual side, we expect that the appropriate replacement of
Hamiltonian varieties are given by what we term "ku-Hamiltonian varieties";
this is a notion interpolating between Hamiltonian and quasi-Hamiltonian
varieties (a la Alekseev--Malkin--Meinrenken). Finally, we suggest possible
generalizations to more exotic cohomology theories such as complex cobordism.
