The name is absent

sentation of G₀ over к are “fundamental” in the sense of É. Cartan.
Furthermore, modulo the “duality” or the “triality,” the list of all
admissible representations over к is as follows:

PPi + 4P_n (p + q ≤ n), Pi + P2, P2 + P_n> P₂ S
pp_l (p ≤ n — 1) in general, and also

Pi + P_n> ²p_π, P_n (ⁿ = 2,3),

²Pi + P₄, Pi + P₄> P₄ (« = 4), p₅ (n = 5);

pp_l (p ≤ n), ρ_l + p₂, ρ₂ in general, and also

Рз (n = 3);

PPi (p ≤ n — 2) in general, and also

Pι+ Рз + Pm Pi + Рз (« = 4),
²Pi + p₄, Pi + p₄, p₄ (n = 5),
Pi + Ps> Ps (« = 6);

type E₆          p₁+p_s, 2p_l, p₁;

type E₁           p_l (of degree 56);

type F₄.           2pι, p₁ (of degree 26);

type G₂           2p₁, p₁ (of degree 7).

The numbering of the fundamental representations is the same as in
Chevalley [2] except for F₄, where it is reversed. If G is a connected, simply
connected, semisimple group defined over к, we represent к-simple factors
of G as R_κ∣_k(G_l), in which G₁ are absolutely simple. We pick any rational
representation of G₁ defined over K which is equivalent (over Ω) to the one
in the list. This determines a rational representation of -Rκ∕∣_c(G₁), hence
of G, defined over k. Take the sum of such representations for all ∕<-simple
factors of G. In this manner, we get absolutely admissible representations
of G defined over k. If we apply this consideration to /с-forms of G₀ of
type A-D or to a special к-form of G₀ of type E₆, we will get the represen-
tations that have appeared in the works of Siegel [6], Weil [<8], and Mars
[<3]. We refer to Serre [5] for the concept of к-forms. We observe that,
among the remaining admissible representations for G₀, there are essen-
tially two infinite sequences, one for type A and another for type C.

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