Cyclotomic structures on root lattices

The centralizer C(w) of an element w in a Weyl group W plays an important role in the structure and representation theory of split reductive groups G over finite and p-adic fields k, where W is the absolute Weyl group of G. If k is finite, this is well-known: the element w determines a maximal ktorus Tw ⊂ G and C(w) may be identified with the k-rational points in the Weyl group W (Tw, G) of Tw in G. For every character χ of Tw(k), the DeligneLusztig construction [12] gives a virtual representation R Tw(χ) of G(k) whose self-intertwining number is the order of the stabilizer of χ in C(w). If k is p-adic then w determines an unramified maximal k-torus Tw, which now can be embedded as a maximal torus in G in several ways. Assume G is simply-connected. Each class in the Galois cohomology group H(k, Tw) determines an embedding Tw ↪→ G and two classes in H(k, Tw) give G(k)-conjugate embeddings iff they are conjugate under the natural action of C(w) on H(k, Tw). Let C(w, ρ) be the stabilizer of the class ρ ∈ H(k, Tw). If Tw ∼ → T ρ w ⊂ G is an embedding belonging to the class ρ ∈ H(Tw, G), then C(w) is isomorphic to the big Weyl group of k-rational elements in W (T ρ w, G) and C(w, ρ) is isomorphic to the small Weyl group of elements in W (T ρ w, G) which have representatives in G(k). On the representation theory side, suppose w is elliptic (i.e. Tw is anisotropic) and χ is a sufficiently regular character of Tw(k). Then, in accordance with the