The spectral action for small primes, and good filtration subgroups

In the original version of my work with Fargues, we constructed the spectral action with Z_\ell-coefficients, but omitting some small primes \ell. In the revised version, we removed this restriction on \ell, by giving an improved construction of the spectral action that also applies in the setting of Betti geometric Langlands. I will explain this argument, and a related (new) result on preservation of good filtrations for restriction of representations along an embedding of reductive groups H\subset G, when H is the group of fixed points of a semisimple automorphism of G.