This chapter is an introduction to the fundamental paper of Kazhdan–Lusztig [1]. We begin with some generalities about Hecke algebras, which arise in the study by Iwahori [1] and Iwahori–Matsumoto [1] of certain groups of Lie type. The underlying idea is to replace the problem of decomposing an induced representation by the equivalent problem of determining irreducible representations of the associated algebra of intertwining operators (the Hecke algebra); see Curtis [1]–[3]. Here the Hecke algebra is a sort of deformation of the group algebra of the related Weyl group or affine Weyl group. The later work shares this philosophy, but is appreciably more subtle. In any case, what we do is hard to motivate strictly in terms of Coxeter groups.

The treatment in 7.1–7.3 follows Couillens [1] (expanding Bourbaki [1], IV, §2, Exercise 23), but the remainder of the chapter is drawn almost entirely from Kazhdan–Lusztig [1], with added references to work influenced by theirs. Throughout the chapter (W, S) is an arbitrary Coxeter system. Both letters s and t may be used for elements of S, while u, v, w, x, y, z will be used to denote arbitrary elements of W.

Generic algebras

We begin with a very general construction of associative algebras over a commutative ring A (with 1). Such an algebra will have a free A-basis parametrized by the elements of W, together with a multiplication law which reflects in a certain way the multiplication in W.