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

Abstract: 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 $\mathcal{H}(\mathbf{G}(F)//K_{\mathcal{F}}, \mathbb{Z}[q^{-1}])$ with level $K_{\mathcal{F}}$ and prove the compatibility of generalized (twisted) Bernstein and Satake homomorphisms.