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).