The name is absent

by R_κ∣_k the functor called the “restriction of the field of definition from K
to k.” We refer to Weil [7] for the details.

Let G denote a connected, reductive group, X a vector space (of finite
dimension), and p: G → Aut(X) a rational representation of G in X. We
assume that G, X_i and p are defined over k. The basic reference on al-
gebraic groups is Borel [7]. Let <S^(X_a) denote the Scbwartz-Bruhat space
of the locally compact, abelian group X_a . Then, denoting by g a typical
element of the unimodular group G_a and by dμ(g) its Haar measure, we
shall consider the following integral:

/(Φ)=f   ( ∑ Φ(pω-ξ))∙⅛ω,

Jg_λ∣G_i< ∖ieXk           1

in which Φ is taken from 3^(X_x). We shall be interested in such a repre-
sentation p for which 7(Φ) is absolutely convergent, i.e., the integrand is
integrable on G_a∣G_1i in the usual sense for every Φ in SA(X_λ). A criterion
for this has been formulated by Weil in the language of the reduction theory
[5, ρ. 20]. In the following, we shall recall this criterion.

Fix a maximal /с-split torus Tof G. If we apply the functor R_t/Q to T_i
we get a torus T' defined over Q. Let T" denote the unique, maximal ɑ-split
torus of T'. Then, the connected component Θ of (T'')_fi can be considered
as a subgroup of T₄. We see that Θ is isomorphic to R^r _i in which r is the
/с-rank of G, i.e., the dimension of T. We shall denote a typical element
of Θ by 0 and its Haar measure by dθ. On the other hand, denoting by G_m the
universally split torus of dimension 1, we convert the free abelian group
Hom(T_jG_m) into a linearly ordered abelian group. Then we define Θ ⁺
as the set of those 0 satisfying ∣α(0)∣₄ ≤ 1 for every positive root a of G
relative to T. The absolute value with the subscript A denotes the idele
norm. Let λ denote a weight of p relative to T and m_λ its multiplicity in p.
Then, the aforesaid criterion is that the following integral:

[_θ+ (∏ sup(l, IΛ(0) [7^,'^λ)) (∏₀1 ^α(⁰)lə ∙M

be convergent, in which the second product is extended over the set of
positive roots α of G relative to T counted with their multiplicities (in the
adjoint representation of G).

It is worthwhile to observe that the adjoint representation of G does not
satisfy this criterion unless G_A/G_k is compact. In this case, the criterion
is satisfied for every p. In the general case, we say that p is admissible over
к if it satisfies this criterion. The admissibility over к does not depend

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