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Let   F   be a non‐Archimedean local field and let   G   be a connected reductive affine algebraic   F  ‐group. Let    I  ⊂  G  (  F  )    be an Iwahori subgroup and denote by    H  (  G  ;  I  )    the Iwahori–Hecke algebra, that is, the convolution algebra of functions    G  (  F  )  →  C    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 ( J. Amer. Math. Soc . 2 (1989) 599–635), 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 .

The Bernstein Presentation for general connected reductive groups