The reflection representations of some Chevalley groups

Abstract

Let G be a finite group with a BN-pair (B, N), and let its associated Weyl group W be generated by set of involutions S. Then there is a l-l correspondence between the irreducible (complex) respresentations of W and the irreducible constituents of the induced representation 1 g (see [3]). In particular, there is an irreducible constituent of 1: corresponding to PE Irr( W), the natural faithful representation of W as a group of orthogonal transformations on V, an IS)-dimensional Euclidean space, such that p(s) is a reflection for each s E S. Let this constituent of 1 g be denoted by rc(G). This is the reflection representation of G. The above correspondence is via the irreducible representations of the Hecke algebra Xc(G, B) which is the endomorphism algebra of 1 g. For the case G= G(q) is a finite Chevalley group over GF(q), the reflection representation of Xc(G, B) was constructed by Kilmoyer [S]. In this paper a module R(G) is constructed which affords x(G) when G = G(q) is a finite Chevalley group belonging to a system of type A, D, or E (see [3, Section 51).