Title:

Hecke algebras and the Lusztig isomorphism

Let G be a Chevalley group over a finite field with q elements and let B be a Borel subgroup of G. Let H(G,B) be the Hecke algebra of the pair (G,B). J. Tits showed that the Hecke algebra over C is isomorphic to the group algebra over C of the Weyl group. N. Iwahori conjectured that the Hecke algebra over Q of the pair (G,B) is isomorphic to the group algebra over Q of the Weyl group. Benson and Curtis proved that this conjecture is true whenever G is simple of type different from E7, E8. With the help of Springer they proved that the conjecture is no longer true when G is of type E7, E8. G. Lusztig constructed an explicit isomorphism from the Hecke algebra over Q(q*) to the group algebra over Q(q*) of the Weyl group. The main purpose of this thesis is to investigate the general properties of this isomorphism. As a consequence of our investigation we introduce a way of obtaining orthogonal primitive idempotents inside the Hecke algebra. This thesis has been divided into six chapters. In Chapter 1 we recall some auxiliarly results about the structure of Coxeter groups and their associated Hecke algebras. We also recall the KazhdanLusztig decomposition of a Coxeter group into left, right and two sided cells and we explain how the cells give rise to representations of the Coxeter groups and of the corresponding Hecke algebras. Let W be a finite indecomposable Coxeter group satisfying a certain property (property (A)) for the structure of its two sided cells. We recall an explicit isomorphism from HQ (u*) (W) to Q(u*)(W) constructed by G. Lusztig, where Q(u*) 1c the field of fractions of the polynomial ring Q[u*]. The subsequent chapters are our own work. In Chapter 2 we find an explicit formula for Lusztig's Isomorphism 1n the case where W  D2n the dihedral groups. It turns out that these groups satisfy the required property (A). Here we achieve our results using classical properties of the Chebyshev polynomials of the second kind. In Chapter 3 we investigate the centre of the Hecke algebra over the polynomial ring Q[u], following some ideas of R.W. Carter. These ideas give a natural basis for the centre of the Hecke algebras of dihedral groups and they lead to an interesting conjecture for the form of a basis of the centre of the Hecke algebra in the general case. In Chapter 4 we find the images of the central basis elements of the Hecke algebra of dihedral type determined in the previous chapter, under Lusztig's isomorphism. Here we show that the images of these elements no longer involve u*. In Chapter 5 we prove results valid for arbitrary Hecke algebras. Here we show that the images of the generators Ts of the Hecke algebra under Lusztig's isomorphism θ are given by θ (Ts) – u1/2.1 + u+1/2.s + (u*1)2Fs for some Fs E QW. We give two independent proofs of this result. The second one 1s based on some conjectures made by R.W. Carter and uses the results of A. Gyoja for the irreducible representations of Coxeter groups and Hecke algebras. We also show that if c – wεw Σ awTw is an element in the centre of the Hecke algebra with aw ε Q[u], then in most of the cases the image of c under Lusztig's isomorphism θ, belongs to Q[u](W). In Chapter 6 we deal with the construction of orthogonal primitive idempotents Inside the Hecke algebra. These ide mpotents are obtained naturally from the decomposition of a maximal commutative subalgebra Inside the Hecke algebra. We shall achieve this decomposition in some special cases. Finally we discuss some open questions which arise naturally from our work, and we make some conjectures which would allow these questions to be settled.
