For certain characters of the compact torus of a reductive $p$-adic group, which we call strongly parabolic characters, we prove Satake-type isomorphisms. Our results generalize those of Satake, Howe, Bushnell and Kutzko, and Roche.
We define the $p$-adic trace of certain rank-one local systems on the multiplicative group over $p$-adic numbers, using Sekiguchi and Suwas unification of Kummer and Artin-Schrier-Witt theories. Our main observation is that, for every non-negative in
teger $n$, the $p$-adic trace defines an isomorphism of abelian groups between local systems whose order divides $(p-1)p^n$ and $ell$-adic characters of the multiplicative group of $p$-adic integers of depth less than or equal to $n$.
