We define the Iwahori–Hecke algebra
for an almost split
Kac–Moody group
over a local nonarchimedean field. We use the hovel
associated to
this situation, which is the analogue of the Bruhat–Tits building for a reductive group. The
fixer
of some
chamber in the standard apartment plays the role of the Iwahori subgroup. We can define
as the algebra of
some
-bi-invariant
functions on
with support consisting of a finite union of double classes. As two chambers in the hovel
are not always in a same apartment, this support has to be in some large subsemigroup
of
.
In the split case, we prove that the structure constants of
are
polynomials in the cardinality of the residue field, with integer coefficients depending on
the geometry of the standard apartment. We give a presentation of this algebra , similar
to the Bernstein–Lusztig presentation in the reductive case, and embed it in a greater
algebra
,
algebraically defined by the Bernstein–Lusztig presentation. In the affine case, this
algebra
contains the Cherednik’s double affine Hecke algebra. Actually, our results apply to
abstract “locally finite” hovels, so that we can define the Iwahori–Hecke algebra with
unequal parameters.
Keywords
hovels, Hecke algebras, Bernstein–Lusztig relations,
Kac–Moody groups, local fields