Today I want to talk about a paper by Dick Gross, where he describes how to attach a motive of Artin-Tate type to a connected reductive group over a field . The motive and its -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 is a finite field, then the twisted dual motive yields a formula for the order of , as shown by Steinberg.

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

If is a number field, let be the adèles of and let be a finite set of places of such that contains all of the infinite places and all of the finite places where over is not quasi-split. We can define a measure on the , a locally compact group, in terms of its local components by setting on for all , and , the Euler-Poincaré measure on for all .

Let be the value at of the meromorphic continuation of the Euler product

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

where is the Tamagawa number of and

where is a maximal torus which is anisotropic over . For finite places , we have .

Let denote the separable closure of and let . Let denote a connected reductive group over with center and derived subgroup . Let denote the connected component of . will be a torus. To define the motive of , we first define the motive in the case of quasi-split groups.

For a quasi-split group , let be a maximal split torus in and let be the centralizer of in . Since is quasi-split, will be a maximal torus. Let denote the Weyl group of in over the separable closure of . Then the Galois group acts on . The -vector space

admits an action of .

By Chevalley, the algebra of -invariants in the symmetric algebra on is isomorphic to a symmetric algebra on a graded -vector space . If and is the ideal of elements of degree at least 1 in , then the graded vector space is given by

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

and

Let be the Tate motive of rank 1 and weight over . If is an Artin motive over , given by a rational representation of , then is an Artin-Tate motive of weight

The *motive* of is defined as

The *rank* of is the rank of over . Each representation is self-dual and so the twisted dual of is the motive

The weights of the motive are all and the weights of its twisted dual are all .

In the case of split over , then , each is the trivial representation of , and the motive is a Tate motive.

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

Examples:

If , then

If or , then

If is a finite field with elements, then is generated by the geometric Frobenius element , which has eigenvalue on the Tate motive .

Steinberg found the twisted dual of and the following formula:

In the case that is a local non-Archimedean field of characteristic 0. Let be its ring of integers with uniformizer and let be the cardinality of the residue field . Let be the maximal unramified extension of contained in the separable closure of and let be the inertia subgroup which fixes . We define the Artin-Tate motive

over .

Let be a connected reductive group over with motive . We have the -functions

and

The -function of can be infinite, positive, or negative.

Now, let be the Haar measure on which corresponds to a differential with good reduction mod and also to the canonical absolute value with .

Let be the quasi-split inner form of over . Then, following Kottwitz, we attach the sign

to .

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

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

in the space of invariant measures on .

**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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