Elliptic Centralizers in Weyl Groups and Their Coinvariant Representations

The centralizer C(w) of an elliptic element w in a Weyl group has a natural symplectic representation on the group of w-coinvariants in the root lattice. We give the basic properties of this representation, along with applications to p-adic groups—classifying maximal tori and computing inducing data in L-packets—as well as to elucidating the structure of the centralizer C(w) itself. We give the structure of each elliptic centralizer in W (E8) in terms of its coinvariant representation, and we refine Springer’s theory for elliptic regular elements to give explicit complex reflections generating C(w). The case where w has order three is examined in detail, with connections to mathematics of the nineteenth century. A variation of the methods recovers the subgroup W (H4) ⊂ W (E8).