The idea of an Iwahori-Hecke algebra originated in Iwahori's 1964 paper 'On the structure of a Hecke ring of a Chevalley group over a finite field' [22] . The finite Chevalley groups, such as SL(n, q), are analogues over finite fields of the simple Lie groups. Such a Chevalley group G(q) has a Borel subgroup B(q) (the subgroup of triangular matrices in the case SL(n, q)), and one considers the G(q)-module V affording the representation induced from the unit representation of B(q). The Hecke

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... bra H is the endomorphism algebra of V . Iwahori showed that the dimension of H is the order of the Weyl group W of G(q) and that H has a basis T w , w ∈ W , of elements satisfying the relations T w T w = T ww if l(ww ) = l(w) + l(w ) and also the quadratic relations