Today I want to talk about a paper by Dick Gross, where he describes how to attach a motive M of Artin-Tate type to a connected reductive group G over a field k. The motive M and its L-function are particularly useful in computing some adèlic integrals that occur in the trace formula. Moreover, the motive contains some other encoded information about the group. For example, if k=\mathbb{F}_q is a finite field, then the twisted dual motive M^\vee(1) yields a formula for the order of G(\mathbb{F}_q), as shown by Steinberg.

If k is a local field with characteristic 0, then the L-function L(M) is finite if and only if Serre’s Euler-Poincaré measure \mu_G on G(k) is non-zero. Also, there is a local functional equation relating the L-function of M to that of its twisted dual motive.

If k is a number field, let \mathbb{A}:=\mathbb{A}_k be the adèles of k and let S be a finite set of places of k such that S contains all of the infinite places and all of the finite places where G over k_v is not quasi-split. We can define a measure \mu_S on the G(\mathbb{A}), a locally compact group, in terms of its local components by setting \mu_v=L_v(M^\vee(1))\cdot|\omega_{G_v}| on G(k_v) for all v\notin S, and \mu_v=\mu_{G_v}, the Euler-Poincaré measure on G(k_v) for all v\in S.

Let L_S(M) be the value at s=0 of the meromorphic continuation of the Euler product

\prod_{v\notin S} L_v(M,s).

This product converges in some right half plane and, by some results of Siegel, L_S(M) is a rational number, which is non-zero if and only if the measure \mu_S is non-zero on the adèlic points of G. In this case,

\int_{G(k)\backslash G(\mathbb{A}}\mu_S=L_S(M)\cdot \tau(G)\slash\prod_{v\in S}c(G_v)

where \tau(G) is the Tamagawa number of G and

c(G_v)=\dfrac{\# H^1(k_v,G)}{\#({\rm ker}:H^1(k_v,T)\longrightarrow H^1(k_v,G))},

where T is a maximal torus which is anisotropic over k. For finite places v, we have c(G_v)=\# H^1(k_v,G).

Let k^s denote the separable closure of k and let \Gamma={\rm Gal}(k^s\slash k). Let G denote a connected reductive group over k with center Z and derived subgroup G^{\rm der}. Let C denote the connected component of Z. C will be a torus. To define the motive of G, we first define the motive in the case of quasi-split groups.

For a quasi-split group G, let S be a maximal split torus in G and let T be the centralizer of S in G. Since G is quasi-split, T will be a maximal torus. Let W=N_{G(k^s)}(T(k^s)\slash T(k^s) denote the Weyl group of T in G over the separable closure k^s of k. Then the Galois group \Gamma acts on W. The \mathbb{Q}-vector space

E=X^\bullet(T)\otimes\mathbb{Q}={\rm Hom}_{k^s}(T,\mathbb{G}_m)\otimes\mathbb{Q}

admits an action of W\rtimes\Gamma.

By Chevalley, the algebra of W-invariants in the symmetric algebra on E is isomorphic to a symmetric algebra on a graded \mathbb{Q}-vector space V. If R={\rm Sym}^\bullet(E)^W and R_+ is the ideal of elements of degree at least 1 in R, then the graded vector space V=\oplus_{d\geq 1}V_d is given by

V=R_+\slash R_+^@

This V is isomorphic to E as a representation of \Gamma. The grading of V and the fact that each V_d is also a representation of \Gamma can be exploited to yield some useful information about G and its Weyl group. For example, one has

{\rm dim}\, G=\sum_{d\geq 1}(2d-1){\rm dim}\, V_d

and

\# W=\prod_{d\geq 1}d^{{\rm dim}\, V_d}

Let \mathbb{Q}(1)=H_1(\mathbb{G}_m) be the Tate motive of rank 1 and weight -2 over k. If N is an Artin motive over k, given by a rational representation of \Gamma, then N(n)=N\otimes \mathbb{Q}(1)^{\otimes n} is an Artin-Tate motive of weight -2n

The motive M of G is defined as

M=M_G=\oplus_{d\geq 1} V_d(1-d)

The rank of M is the rank of G over k^s. Each representation V_d is self-dual and so the twisted dual of M is the motive

M^\vee(1)=\oplus_{d\geq 1}V_d(d)

The weights of the motive M are all \geq 0 and the weights of its twisted dual M^\vee(1) are all \leq -2.

In the case of G split over k, then T=S, each V_d is the trivial representation of \Gamma, and the motive M is a Tate motive.

Finally, for any connected reductive group G, we define its motive as the motive of the quasi-split inner form of G.

Examples:

If G={\rm GL}_n, then
M=\mathbb{Q}+\mathbb{Q}(-1)+\mathbb{Q}(-2)+\dots+\mathbb{Q}(1-n).

If G={\rm Sp}_{2n} or G={\rm SO}_{2n+1}, then
M=\mathbb{Q}(-1)+\mathbb{Q}(-3)+\dots+\mathbb{Q}(1-2n).

If k=\mathbb{F}_q is a finite field with q elements, then \Gamma is generated by the geometric Frobenius element F, which has eigenvalue q^{-1} on the Tate motive \mathbb{Q}(1).

Steinberg found the twisted dual M^\vee(1) of M and the following formula:
\# G(\mathbb{F}_q)/q^{{\rm dim}\, G} = {\rm det}(1-F|M^\vee(1))=\prod_{d\geq 1} {\rm det}(1-F|V_d(d))=\prod_{d\geq 1}{\rm det}(1-Fq^{-d}|V_d).

In the case that k is a local non-Archimedean field of characteristic 0. Let \mathcal{O} be its ring of integers with uniformizer \varpi and let q be the cardinality of the residue field \mathcal{O}\slash\varpi\mathcal{O}. Let k^{\rm unr} be the maximal unramified extension of k contained in the separable closure k^s of k and let I be the inertia subgroup which fixes k^{\rm unr}. We define the Artin-Tate motive
M^I=\oplus_{d\geq 1}V_d^I(1-d)
over \mathcal{O}\slash\varpi\mathcal{O}.

Let G be a connected reductive group over k with motive M. We have the L-functions
L(M)={\rm det}(1-F|M^I)^{-1}
and
L(M^\vee(1))={\rm det}(1-F|M^\vee(1)^I)^{-1}
The L-function of M can be infinite, positive, or negative.

Now, let |\omega_G| be the Haar measure on G(k) which corresponds to a differential \omega_G with good reduction mod \varpi and also to the canonical absolute value |\cdot|_v:k^\times\rightarrow\mathbb{R}_+^* with |\varpi|_v=q^{-1}.

Let H be the quasi-split inner form of G over k. Then, following Kottwitz, we attach the sign
e(G)=(-1)^{{\rm rank}(G/k)-{\rm rank}(H/k)}
to G.

The local function equation relating the L-functions of the motive M with its twisted dual is given by Gross:

Theorem (Gross) Assume that the connected center C of G is anisotropic. Then

L(M)\cdot\mu_G\cdot e(G)\cdot\#H^1(k,G)=L(M^\vee(1))\cdot|\omega_G|

in the space of invariant measures on G(k).

References:

B. Gross, On the motive of a reductive group, Invent. Math. 130 (1997), no. 2, 287–313.

R. Kottwitz, Tamagawa numbers, Ann. of Math. 127 (1988), 629–646.

G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. IHES 69 (1989), 91–114.

R. Steinberg, Endomorphisms of linear algebraic groups, Memoir AMS 80 (1968).

J. Tate, Les conjectures de Stark sur les fonctions L d’Artin en s=0, Birkhäuser Progress in Math. 47 (1984).

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