The Satake isomorphism for special maximal parahoric Hecke algebras

Let $G$ denote a connected reductive group over a nonarchimedean local field $F$. Let $K$ denote a special maximal parahoric subgroup of $G(F)$. We establish a Satake isomorphism for the Hecke algebra $\mathcal{H}_K$ of $K$-bi-invariant compactly supported functions on $G(F)$. The key ingredient is a Cartan decomposition describing the double coset space $K\backslash G(F)/K$. As an application we define a transfer homomorphism $t: \mathcal{H}_{K^*}(G^*) \rightarrow \mathcal{H}_K(G)$ where $G^*$ is the quasi-split inner form of $G$. We also describe how our results relate to the treatment of Cartier [Car], where $K$ is replaced by a special maximal compact open subgroup $\widetilde{K} \subset G(F)$ and where a Satake isomorphism is established for the Hecke algebra $\mathcal{H}_{\widetilde{K}}$.