“…, and similarly for G. Since ζ m comes from the isomorphism X * (S) ∼ = X * ( S), it induces a linear bijection X * (A G ) → X * (A G), and hence an isomorphism from (3.9) to its counterpart for G. The A G -characters of representations in Rep(G, G x,m ) are precisely the characters of (3.9), and ζ G m pushes them forward along ζ m . (c) This can be shown as in [ABPS1, Theorem 5.1.c], using [Gan1,Gan2]. We only have to note that by [AMS1, Lemma 6.6], the groups M and M have the same parameter in Irr(Z(M * ∨ sc ) W F ), via (3.4).…”