THE MOTIVE OF A REDUCTIVE GROUP

BENEDICT H. GROSS

1. Motives and L-functions

Let k be any field, k^{s} a fixed separable closure of k, and Gal(k^{s}/k) the Galois group of k^{s} over k. Let V be a finite dimensional vector space over ℚ, and let

ρ : Gal(k^{s}/k) ⟶ GL(V)  (1.1)

be a continuous Galois representation on V. By definition, ρ factors through a finite Galois extension of k.

The category of such continuous Galois representations is equivalent to the category of Artin motives, which are those motives corresponding to zero dimensional varieties over k. Hence we shall call such a (ρ, V) an Artin motive over k. As we do not need the theory of motives in general, it is best to take (1.1) as the definition of an Artin motive.

Now if k is a finite field, local field or global field, we have an L-function attached to any Artin motive (ρ, V) in the following way:

• If k is finite of order q, then let

σ : x ↦ x^{q}, x ∈ k

be a topological generator of Gal(k^{s}/k). Let F = σ^{−1} be the Frobenius element. Note that the eigenvalues of F on V are roots of unity. Now we define:

L(V, s) = det(1 − F q^{−s}|V)^{−1}  (1.2)

for s ∈ ℂ.

• If k is a non-archimedean local field, then we have the exact sequence:

1 ⟶ I ⟶ Gal(k^{s}/k) ⟶ ℤ̂ · F ⟶ 1

where I is the inertia subgroup, and ℤ̂ · F is the absolute Galois group of the residue field of k, with F chosen as above. Now we define:

L(V, s) = det(1 − F q^{−s}|V^{I})^{−1}  (1.3)

where q is the order of the residue field of k, and s ∈ ℂ. Note that this L-function has no zeros, but may have poles; for example, at s = 0 when V^{Gal(k^{s}/k)} ≠ 0.

Date: 14 July 1997.

• If k = ℝ, we let V^{+} be the (+1)-eigenspace of the action of complex conjugation on V, and V^{−} the (−1)-eigenspace. Let

Γ_{ℝ}(s) = π^{−s/2}Γ(s/2)  (1.4)

where Γ is the usual gamma function. Then we define

L(V, s) = Γ_{ℝ}(s)^{dim(V^{+})} · Γ_{ℝ}(s + 1)^{dim(V^{−})}  (1.5)

If k = ℂ, we define

L(V, s) = (Γ_{ℝ}(s) · Γ_{ℝ}(s + 1))^{dim(V)}  (1.6)

• If k is a global field and v is a place of k, then

Gal(k_{v}^{s}/k_{v}) ↪ Gal(k^{s}/k)

as the decomposition group at v. This embedding is well-defined up to conjugation, and so we can regard (ρ, V) as an Artin motive over k_{v}, by restriction. Let L_{v}(V, s) be the L-function of this Artin motive, as defined in (1.3), (1.5) and (1.6). Now we define

L(V, s) = ∏_{v} L_{v}(V, s)  (1.7)

where the product is taken over all places of k. This product converges absolutely for Re(s) > 1.

It was proved by Artin and Brauer that the above product has a meromorphic continuation, which we shall also denote by L(V, s), to all s ∈ ℂ, and satisfies a functional equation with respect to the substitution s ↦ 1−s. It is conjectured that L(V, s) is entire, except possibly at s = 0 and s = 1, and that all the zeroes of L(V, s) satisfy Re(s) = \frac{1}{2}. When k is a function field, this was proved by Weil.

If S is a finite set of places of k, containing the set S_{∞} of all archimedean places, we set

L_{S}(V, s) = ∏_{v∉S} L_{v}(V, s)  (1.8)

which is called the partial L-function. By results of Euler and Siegel, L_{S}(V, n) is a rational number if n ≤ 0 is a non-positive integer.

Example: If V = ℚ is the trivial representation, and S = S_{∞}, then L_{S}(V, s) = ζ_{k}(s), the zeta function of the global field k.

This completes our definition of the L-function attached to an Artin motive. For our purposes, we need to consider motives which are slightly more general than Artin motives, namely the Artin-Tate motives. If V is an Artin motive over k, and n ∈ ℤ, there is an Artin-Tate motive V (n), obtained from V by n Tate twists. Without giving the precise definition of V (n), let us simply define its L-function to be

L(V (n), s) = L(V, s + n)  (1.9)

If F is a Frobenius element in Gal(k^{s}/k) (with k one of the fields above), then the eigenvalues of F on V (n) are of the form ζq^{−n}, with ζ a root of unity. Hence V (n) is not a rational representation of Gal(k^{s}/k) (unless n = 0). However there is an l-adic representation V_{l}(n) with these eigenvalues for all l ≠ char(k).

2. The Motive of G

Let G be a connected reductive group over k. In this section, we show how to attach an Artin-Tate motive M to G. For more details, see [Gr].

First, assume that G is quasi-split, and fix T ⊂ B, a maximal torus contained in a Borel subgroup, both defined over k. Then

E = Hom_{k^{s}}(T, 𝔾_{⋗}) ⊗ ℚ  (2.1)

is a representation of W ⋊ Gal(k^{s}/k), where W = N_{G}(T)/T is the Weyl group of T in G. Let

Sym^{•}(E)^{W} := R = ℚ ⊕ ℝ_{+}  (2.2)

where R_{+} denotes the set of invariants of W of positive degree. Also, let

R_{+}/R_{+}^{2} := V = ⊕_{d≥1}V_{d}  (2.3)

Then V is a graded representation of Gal(k^{s}/k). It is a theorem of Steinberg that V, without its grading, is isomorphic to E as a Gal(k^{s}/k)-module. Now we make the following definition:

Definition: The motive of G is

M = ⊕_{d}V_{d}(1 − d)  (2.4)

If G is not quasi-split, we define the motive of G to be the motive of its quasi-split inner form. Let M^{∨} denote the dual motive to M, then we have

M^{∨}(1) = ⊕_{d}V_{d}(d)  (2.5)

If k is finite, local or global, we have the L-function values:

L(M) := L(M, 0) = ∏_{d≥1} L(V_{d}, 1 − d)  (2.6)

and

L(M^{∨}(1)) := L(M^{∨}(1), 0) = ∏_{d≥1} L(V_{d}, d)  (2.7)

Similarly, we may define L_{S}(M) in the global case; when S contains S_{∞}, this is either a rational number or infinity.

Examples: (i) If G = T is a torus, then V = E = V_{1}, and M is an Artin motive. In this case, L(M) = L(V, 0).

(ii) If G = GL_{n}, then E = ℚ^{⋉} as a Galois representation, and W ≅ S_{n} acts by permuting the n factors. Thus V_{1} = V_{2} = ... = V_{n} = ℚ, and

M = ⊕_{i=1}^{n}ℚ(⊮ − ℶ).

(iii) If G = U_{n}, a quasi-split unitary group associated to the separable quadratic extension K/k, and ε : Gal(K/k) ⟶ ±1 is the non-trivial character of Gal(K/k), then

M = ℚ(ε) ⊕ ℚ(−⊮) ⊕ ℚ(ε)(−⊭) ⊕ ... ⊕ ℚ(ε^{⋉})(⊮ − ⋉)

Now we give an alternative definition of M, using the L-group of G [Gr2]. Let Ĝ be the dual group of G, which is defined and split over ℚ. Then, corresponding to our fixed choice of T ⊂ B, we have a pinning of Ĝ:

T̂ ⊂ B̂ ⊂ Ĝ

eα̂ : 𝔾_{⅁} ≅ 𝕌̂α̂  (2.8)

for all simple roots α̂ (with respect to T̂ ⊂ B̂). There is an action of Gal(k^{s}/k) on Ĝ preserving the pinning. Under this setting, the W ⋊Gal(k^{s}/k)- module E discussed above is simply Lie(T̂), which is the centralizer in 𝔤̂ = Lie(Ĝ) of a regular semi-simple element in Lie(T̂). Now let:

N = ⊕_{α̂>0}Lie(eα̂)(1) ∈ 𝔤̂  (2.9)

Then N is a regular nilpotent element fixed by Gal(k^{s}/k). Hence, Gal(k^{s}/k) acts on V′, the centralizer of N in 𝔤̂. Moreover this Galois representation is graded. Indeed, the line ℚℕ is normalized by a one-dimensional torus: if

2ρ = ∑_{α>0} α : 𝔾_{⋗} ⟶ 𝕋̂

then 2ρ(𝔾_{⋗}) normalizes ℚℕ. Let V′_{d} be the subspace of V′ on which 𝔾_{⋗} acts via:

λ ↦ λ^{2(d−1)}, λ ∈ ℚ^{×}

Then we have:

V′ = ⊕_{d}V′_{d} ≅ V = ⊕_{d}V_{d}

as graded Galois modules. Hence we can use V′ to define the motive M of G.

3. Haar Measures

In this section, we give some applications of the motive of G and show how it arises in the harmonic analysis on G(k). In particular, we relate various natural Haar measures on G(k) by using the L-function of M.

First let k be a finite field with q elements. Then Steinberg showed that:

\frac{#G(k)}{q^{dim(G)}} = ∏_{d} det(1 − F q^{−d}|V_{d}) = L(M^{∨}(1))^{−1}  (3.1)

We can interpret this formula as follows. The finite group G(k) can be regarded as a discrete group or as a compact group. As a discrete group, the most natural invariant measure on G(k) is μ_{G,disc}, which gives each element of G(k) measure 1, but as a compact group, the natural measure is μ_{G,cpct}, which gives the whole group measure 1. Then (3.1) amounts to giving the ratio of the two measures:

\frac{μ_{G,disc}}{μ_{G,cpct}} = #G(k) = \frac{q^{dim(G)}}{L(M^{∨}(1))}  (3.2)

We shall discuss similar results when k is local or global. The local results are reinterpretations of results of Serre, Steinberg, Kottwitz and Tate, whereas the global results are inspired by the work of Harder and Prasad.

Now assume that k is a local field of characteristic zero. Then G(k) is locally compact and hence the Haar measure exists. There is a natural invariant measure μ_{G} (not necessarily positive) on G(k), called the Euler-Poincaré measure, which was introduced and studied by Serre [Se]. The measure μ_{G} can be characterized as follows. Let Γ ⊂ G(k) be any discrete, torsion-free, co-compact subgroup. Such groups exist by results of Borel and Harder. Then μ_{G} is characterized by:

∫_{Γ\G(k)} μ_{G} = χ(Γ) := ∑_{i}(−1)^{i}dimH^{i}(Γ, ℚ)  (3.3)

for all Γ. Note that μ_{G} can be zero. Indeed we have the following theorem:

Theorem 3.4. The following are equivalent:

(i) μ_{G} ≠ 0.

(ii) G has a maximal anisotropic torus over k.

(iii) G has representations in the discrete series.

(iv) L(M) ≠ ∞.

If the above conditions hold, and k is non-archimedean, then μ_{G} = (−1)^{rank(G)}μ_{St}, where μ_{St} is the Plancherel measure of the Steinberg representation of G(k).

We note that the equivalence of (i) and (ii) is due to Serre [Se]. The theorem suggests that we consider the measure L(M) · μ_{G}, when they are both defined. This measure is actually positive when G is quasi-split.

Another natural Haar measure on G(k) is |ω_{G}|, which, for k non-archimedean, is defined by an invariant differential of top degree using Bruhat-Tits theory [Gr]. For our purposes, it suffices to say that if G is unramified, and G(A) is a hyperspecial maximal compact subgroup of G(k), then ω_{G} is an invariant differential of top degree on G(A), which has good reduction modulo the maximal ideal. We refer the reader to [Gr] for the definition in the archimedean case. Now the following theorem relates the measures μ_{G} and |ω_{G}|, and can be regarded as a functional equation for the L-function of M.

Theorem 3.5.

L(M) · μ_{G} · c(G) · e(G) = L(M^{∨}(1)) · |ω_{G}|

where e(G) = ±1 ∈ Br_{2}(k) is the Kottwitz invariant [K], and

c(G) = \begin{cases}#H^{1}(k, G), if k is non-archimedean; \\ \frac{#H^{1}(k,T)}{#ker(H^{1}(k,T)→H^{1}(k,G))}, if k = ℝ. \end{cases}

Now we come to the case when k is a number field, with adele ring 𝔸. Let S ⊃ S_{∞} be a finite set of places of k such that for all v ∉ S, G_{v} := G ⊗ k_{v} is unramified. Also assume that #S ≥ 2. For v ∉ S, let μ_{v} be the Haar measure on G(k_{v}) giving G(A_{v}), a hyperspecial maximal compact subgroup, volume 1. For v ∈ S, let μ_{v} be the Euler-Poincaré measure. Then define:

μ_{S} = ∏_{v} μ_{v}  (3.6)

which is an invariant measure on G(𝔸). Also let |ω| be the Tamagawa measure of G(𝔸). Then the global analogue of Theorem (3.5) is:

Theorem 3.7.

μ_{S} = \frac{L_{S}(M)}{∏_{v∈S} c(G_{v})} · |ω|

Note that μ_{S}/|ω| is a rational number.

Example: If G = SL_{2}, and S = S_{∞}, then M = ℚ(−⊮), and μ_{S} = ζ_{k}(−1)|ω|.

References

[Gr] B.H. Gross, On the Motive of a Reductive Group, to appear in Invent. Math. (1997).

[Gr2] B.H. Gross, On the Motive of G and the Principal Homomorphism SL_{2} ⟶ Ĝ, Asian J. Math. 1(1997), Pg. 208-213.

[K] R. Kottwitz, Sign Changes in Harmonic Analysis on Reductive Groups, Trans. of AMS 278(1983), Pg.289-297.

[Se] J.P. Serre, Cohomologie des Groupes Discrets, Serre’s Collected Works Vol 2.