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SOME OBSERVATIONS ON THE SIEGEL FORMULA

on the choice of T nor on the linear order in Hom(T, G_m). Furthermore,
if K is a finite algebraic extension of к and if p is admissible over K, it is
admissible over k. We say that ρ is absolutely admissible if it is admissible
over any such K. We observe that, if G splits over K and if p is admissible
over K, then p is absolutely admissible. The first problem is to determine
all absolutely admissible representations.

First of all, if G* → G is an isogeny defined over k, p is admissible over
k if and only if the product of p and G* → G is admissible over k. Secondly,
if ρ is absolutely admissible, G is necessarily semisimple. Therefore, for
our purpose, we may assume that G is a connected, simply connected,
semisimple group defined over k.

Now, any rational representation p of G defined over k decomposes
uniquely (up to equivalence over k) into a sum of к-irreducible represen-
tations. Moreover, if p is admissible over k and decomposes over k into
a sum of p₁ and p₂, they are both admissible over k. Therefore, we shall
be interested in к-irreducible, absolutely admissible representations. We
observe that the trivial representation (mapping every element of G to 1)
is absolutely admissible. In the following, we shall mostly disregard this
representation. The following lemma is basic:

Lemma. If p: G → Aut(X) is a к-irreducible, absolutely admissible
representation of a connected, simply connected, semisimple group G
defined over k, there exist an absolutely simple factor G₁ of G defined
over a finite algebraic extension K of k, a G-invariant subspace X₁ of X
defined over K satisfying X = ∙Rκ∕⅛(X₁), and a К-irreducible represen-
tation p₁: G₁ →Aut(X₁) such that p decomposes into the product of the
projection G → R_κ∣_k(G₁) and the representation R_κ∣_k(Gι) → Aut(X).

Because of this lemma, we may assume that G = R_κ∕_k(G_l). Then, after
the identification of the adelizations of G, X, etc., over k and the adeliza-
tions of G₁, X₁ etc. over K, the integral I for p: G →Aut(X) relative to k
becomes equal to the integral I for p_i'∙G_l → Aut(X₁) relative to K. Con-
sequently, without losing the generality, we may assume that G is a connected,
simply connected, absolutely simple group defined over k. In this case,
we can determine, up to making them explicit for an exceptional G, all
absolutely admissible representations. This will be based on the following
classification theorem :

Theorem 1. Let G₀ denote a connected, simply connected, simple
к-split group. Then, the irreducible constituents of an admissible repre-

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