The Bernstein presentation for general connected reductive groups

Abstract

Let F be a non-Archimedean local field and let G be a connected reductive affine algebraic F-group. Let I be an Iwahori subgroup of G(F) and denote by H(G; I) the Iwahori-Hecke algebra, i.e. the convolution algebra of complex-valued functions on G(F) which are left- and right-invariant by I-translations. This article proves that the Iwahori-Hecke algebra H(G; I) has both an Iwahori-Matsumoto Presentation and a Bernstein Presentation analogous to those for affine Hecke algebras on root data found in Lusztig's "Affine Hecke algebras and their graded version", and gives a basis (in other words, an explicit Bernstein Isomorphism) for the center Z[H(G; I)] also analogous to that found in loc. cit.