Spherical Hecke algebras for Kac-Moody groups over local fields. (English) Zbl 1315.20046

In this paper the authors use the hovel associated to an almost split Kac-Moody group \(G\) over a local non-archimedean field to define an analogue \(\mathcal H\) of the classical spherical Hecke algebra in this situation. The authors show that the structure constants of \(\mathcal H\) are polynomials in the cardinality of the residue field and have integer coefficients depending on the geometry of the standard apartments of the hovel. The paper establishes an analogue of Satake isomorphism from \(\mathcal H\) to the algebra of certain invariant elements in a certain completion of a Laurent polynomial ring. In particular, this shows that \(\mathcal H\) is commutative. The paper also contains a direct proof of commutativity of \(\mathcal H\) in the case of a split group \(G\). The proof of the Satake isomorphism involves parabolic retraction.


20G44 Kac-Moody groups
20C08 Hecke algebras and their representations
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
20E42 Groups with a \(BN\)-pair; buildings
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