Finite Coxeter groups are finite groups of real matrices that are generated by reflections. They include the Weyl groups, which are fundamental in the classification of simple complex Lie algebras as well as simple algebraic groups. Iwahori– Hecke algebras associated to Weyl groups appear naturally as endomorphism algebras of induced representations in the study of finite reductive groups. They can also be defined independently as deformations of group algebras of finite Coxeter groups, where the deformation depends on an indeterminate q and a weight function L. For q = 1, we recover the group algebra. For a finite Coxeter group W, we will denote by 𝓗 (W, L) the associated Iwahori–Hecke algebra.

When q is an indeterminate, the Iwahori–Hecke algebra 𝓗 (W, L) is semi-simple. By Tits’s deformation theorem, there exists a bijection between the set of irreducible representations of𝓗 (W, L) and the set Irr(W) of irreducible representations of W. Using this bijection, Lusztig attaches to every irreducible representation of W an integer depending on L, thus defining the famous a-function. The a-function is used in his definition of families of characters, a partition of Irr(W) which plays a key role in the organisation of families of unipotent characters in the case of finite reductive groups.