# The recognition theorem for graded Lie algebras in prime by Georgia Benkart

By Georgia Benkart

Quantity 197, quantity 920 (second of five numbers).

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**Example text**

Hence if P ⊆ t, then −γ P contains a root vector, eγ say. In this case e−γ ∈ [g−γ 0 , [g0 , eγ ]] is in P as well. So it can be assumed further that γ > 0. As the Lie algebra n+ is nilpotent, the ad n+ -module generated by eγ contains a root vector eδ (with δ > 0) such that [n+ , eδ ] = 0. 12. 63 IN THE DEPTH-ONE CASE 21 root α > 0. As t is abelian, this shows that P is contained in the center Z(g0 ) of g0 . Since dim Z(g0 ) ≤ 1, it must be that P = Z(g0 ) in this case. If P ⊆ t, we set e = eδ .

Thus, any rational G-module L(λ) can be regarded, canonically, as a restricted Lie(G)-module. 13. (Compare [Ja, Part II, Prop. 10 and Prop. ) Any irreducible restricted Lie(G)-module is isomorphic to exactly one module L(λ) with λ ∈ X1 (T ). Conversely, any G-module L(λ) with λ ∈ X1 (T ) is irreducible as a Lie(G)-module. The module V (λ) := H 0 (−w0 λ)∗ which is dual to H 0 (−w0 λ) is called the Weyl module corresponding to λ. There is an antiautomorphism τ of order 2 on G which acts as the identity on T and sends the root subgroup Uα to U−α , for each α ∈ Φ (as in [Ja, Part II, Sec.

For k = 0, 1, . . , p − 1 we set xk = [(ad eα )k (w), v] and observe that xk ∈ gη+kα 0 where η = β − λ + ν. Since e commutes with eα and (ad eα )p [w, e] = χ(eα )p [w, e], a nonzero multiple of [w, e], it must be that [u, xk ] = [(ad eα )k (w), [u, v]] = (ad eα )k [w, e] = 0 for all k ≥ 0. But then gη+kα = 0 for k = 0, 1, . . , p − 1. If η is not a 0 multiple of α, this is impossible, since root strings in classical Lie algebras have length at most 4. If η is a multiple of α, then all multiples kα for k = 1, .