A tale of parahoric--Hecke algebras, Bernstein and Satake homomorphisms

Let $\mathbf{G}$ be a connected reductive group over a {non-archimedean local field} $F$. Let $K_{\mathcal{F}}$ be the parahoric subgroup attached to a facet $\mathcal{F}$ in the Bruhat–Tits building of $\mathbf{G}$. The ultimate goal of the present paper is to describe the center of the parahoric–Hecke algebra $\mathcal{H}(\mathbf{G}(F)\mathrm{/}\mathrm{/}{K}_{\mathcal{F}},\mathbb{Z}[q^{-1}])$ with level $K_{\mathcal{F}}$ and prove the compatibility of generalized (twisted) Bernstein and Satake homomorphisms.