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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 aSatake isomorphism for the Hecke algebra H of K-bi-invariant compactlysupported functions on G(F). The key ingredient is a Cartan decompositiondescribing the double coset space K\G(F)/K. We also describe how our resultsrelate to the treatment of Cartier, where K is replaced by a special maximalcompact open subgroup K' of G(F) and where a Satake isomorphism is establishedfor the Hecke algebra of K'-bi-invariant compactly supported functions on G(F).

The Satake isomorphism for special maximal parahoric Hecke algebras