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 $W(E_8)$ 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 $W(H_4)\subset W(E_8)$.