Haar Measure and the Artin Conductor

Let G be a connected reductive group, defined over a local, non-archimedean field k. The group G(k) is locally compact and unimodular. In [Gr], a Haar measure |ωG| was defined on G(k), using the theory of Bruhat and Tits. In this note, we give another construction of the measure |ωG|, using the Artin conductor of the motive M of G over k. The equivalence of the two constructions is deduced from a result of Prasad [P]. 1. The Root Datum and Motive of G In this section, k is an arbitrary field and G is a connected reductive group over k. We let k be an algebraic closure of k, ks the separable closure of k in k, and Γ = Gal(ks/k). Let T ⊂ B ⊂ G be a maximal torus, contained in a Borel subgroup, defined over ks. Let Ψ = Ψ(G,B, T ) be the based root datum defined by this choice. We recall (cf [Sp], pg3-12) that: Ψ = (X(T ),∆(T,B),X•(T ),∆•(T,B)) , (1.1) with X•(T ) and X•(T ) the character and cocharacter groups of T respectively, and ∆• and ∆• the simple roots and coroots determined by B respectively. Let W = NG(T )/T be the Weyl group of Ψ. The finite group W acts as automorphisms of X•(T ), and is generated by the reflections: sα(x) = x− 〈x, α〉α (1.2) for α ∈ ∆•. The Galois group Γ acts as automorphisms of Ψ, ie as automorphisms of the group X•(T ) preserving the finite set ∆•, as follows. If σ ∈ Γ, then we can find g ∈ G(ks) such that: Int(g)(σT ) = gσ(T )g = T, and Int(g)(σB) = gσ(B)g = B, with g well-defined up to left multiplication by T (ks). Hence it induces a well-defined automorphism ψ(σ) : X(T ) −→ X(T ) Date: 14th January 1997. 1 2 BENEDICT H. GROSS AND WEE TECK GAN preserving ∆•. Hence we get a group homomorphism ψ : Γ −→ Aut(Ψ). Via ψ, Γ acts on Aut(Ψ) by inner automorphisms. Similarly, if f : G −→ G is any automorphism of G over ks, it induces an automorphism ψ(f) of Ψ, which depends only on the image of f in the quotient group Outks(G) of outer automorphisms. The resulting map Outks(G) −→ Aut(Ψ) is an isomorphism which respects the respective Galois actions on the two groups (cf [Sp], pg10). The Galois group Γ also acts on W , via the formula: σ(sα) = sσ(α) (1.3) and the semi-direct product W ⋊ Γ acts on the rational vector space E = X(T )⊗Q (1.4) Let R = Sym•(E)W , which is a graded Q[Γ]-module. Let R+ be the ideal of elements of positive degree in R, and define: V = R+/R 2 + = ⊕d≥1Vd (1.5) This is a graded Q[Γ]-module, and Chevalley proved that dim(V ) = dim(E) (cf [Ch]). Steinberg extended the proof to show that E and V are isomorphic Γ-modules (cf [St], pg22). We sketch a proof of this result that does not involve the classification of irreducible root systems. Proposition 1.6. The Q[Γ]-modules E and V are isomorphic. Proof. By the criterion in [Se, pg 104], it suffices to show that for all σ ∈ Γ, the fixed spaces Eσ and V σ have the same dimension. For any graded Γ-module A = ⊕Am, we define the Poincare series of σ by: P (A,σ)(t) = ∑ tr(σ|Am)t. Then P (A ⊗ B) = P (A)P (B). Steinberg showed that there is an isomorphism of graded Γ-modules: S(E) ∼= S(⊕Vd)⊗A Here A is finite dimensional, with basis {bw}w∈W , and Γ-action given by: σ(bw) = bσ(w) The degree of bw is the length l(w) of w, with respect to the generators sα furnished by ∆•. This isomorphism yields the following identity of Poincare series: det(1− σt|E) = ∏ d≥1 det(1 − σt|Vd) · ∑