We define the Iwahori-Hecke algebra for an almost split Kac-Moody group overa local non-archimedean field. We use the hovel associated to this situation,which is the analogue of the Bruhat-Tits building for a reductive group. Thefixer K of some chamber in the standard apartment plays the role of the Iwahorisubgroup. We can define the Iwahori-Hecke algebra as the algebra of someK-bi-invariant functions on the group with support consisting of a finite unionof double classes. As two chambers in the hovel are not always in a sameapartment, this support has to be in some large subsemigroup of the Kac-Moodygroup. In the split case, we prove that the structure constants of themultiplication in this algebra are polynomials in the cardinality of theresidue field, with integer coefficients depending on the geometry of thestandard apartment. We give a presentation of this algebra, similar to theBernstein-Lusztig presentation in the reductive case, and embed it in a greateralgebra, algebraically defined by the Bernstein-Lusztig presentation. In theaffine case, this algebra contains the Cherednik's double affine Hecke algebra.Actually, our results apply to abstract "locally finite" hovels, so that we candefine the Iwahori-Hecke algebra with unequal parameters.
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