Abstract
Let G be a split Kac-Moody group over a non-Archimedean local field. We define a completion of the Iwahori-Hecke algebra of G, then we compute its center and prove that it is isomorphic (via the Satake isomorphism) to the spherical Hecke algebra of G. This is thus similar to the situation for reductive groups. Our main tool is the masure ℐ associated to this setting, which plays here the same role as Bruhat-Tits buildings do for reductive groups. In a second part, we associate a Hecke algebra to each spherical face F of type 0, extending a construction that was only known, in the Kac-Moody setting, for the spherical subgroup and for the Iwahori subgroup.
Highlights
Let G0 be a split reductive group over a non-Archimedean local field K and set G0 = G0(K )
An important tool to study complex representations of G0 are Hecke algebras attached to each open compact subgroup of G0: if K is such a subgroup, the Hecke algebra HK associated to K is the convolution algebra of complex-valued K-biinvariant functions on G0 with compact support
Ks is a maximal open compact subgroup of G and Hs = HKs is a commutative algebra called the spherical Hecke algebra of G0. This algebra can be explicitly described through the Satake isomorphism: if W v denotes the Weyl group of G0 and Q∨ is the coweight lattice of G0, Hs is isomorphic to the subalgebra C[Q∨]W v of W v−invariant elements in the group algebra of (Q∨, +)
Summary
Two choices of K are of particular interest: the first one is when K = Ks = G(O) with O being the ring of integers of K In this case, Ks is a maximal open compact subgroup of G and Hs = HKs is a commutative algebra called the spherical Hecke algebra of G0. When G is reductive, any face F between {0} and C0+ corresponds to an open compact subgroup of G (namely the parahoric subgroup associated with F ) contained in its fixer KF , one can use it to attach a Hecke algebra to F. — As explained, this paper is written in a more general framework, as we only need I to be an abstract masure and G to be a strongly transitive group of (positive, type-preserving) automorphisms of I This applies to almost split (and split) Kac-Moody groups over local fields. We thank the referee for their valuable comments and suggestions, and for his/her interesting questions
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