# Torsors, reductive group schemes and extended affine Lie by Philippe Gille, Arturo Pianzola

By Philippe Gille, Arturo Pianzola

The authors provide a close description of the torsors that correspond to multiloop algebras. those algebras are twisted varieties of basic Lie algebras prolonged over Laurent polynomial earrings. They play a very important function within the building of prolonged Affine Lie Algebras (which are greater nullity analogues of the affine Kac-Moody Lie algebras). The torsor technique that the authors take attracts seriously from the speculation of reductive crew schemes built by way of M. Demazure and A. Grothendieck. It additionally permits the authors to discover a bridge among multiloop algebras and the paintings of F. Bruhat and J. titties on reductive teams over entire neighborhood fields

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By induction on n, we get that inside (η Y )(Fn−1 ) we have Y geo Y(k)η = ∅ as desired. (d) =⇒ (a): Since ηgeo (k) = ∅. γ y = y ∀γ ∈ Γ ˜ n,m , (η Y)(Rn ) = y ∈ Y(R ˜ n,m ) yields the inclusion the inclusion Y(k) ⊂ Y(R (Yη geo )(k) ⊂ (η Y)(Rn ). geo Thus if (Yη )(k) = ∅, then (η Y)(Rn ) = ∅. 5]. The only non trivial implication is (c) =⇒ (a). Let X = (G/S) ×k Rn . By (1), we have η Y(Kn ) = ∅. In other words, the Kn -homogeneous space η X under η G has a Kn rational point on the compactiﬁcation η Y.

11. If ψ is the trivial map the Theorem reduces to the “Main result (B)” of Borel and Mostow [BM] for g. 10) allows for a slightly more direct proof of this result. 12 to establish the following result which will play a crucial role in the the proof of the existence of maximal tori on twisted groups corresponding to loop torsors. 12. Let k/k be a ﬁnite Galois extension with Galois group Γ. ˜ Let H be a group, and assume we are given a Let G be a reductive group over k. group homomorphism ρ : H → AutΓ (G) for which we can ﬁnd a family of subgroups (Hi )0≤i≤s of H as in the Theorem, that is ker(t ◦ ρ) = Hs ⊃ Hs−1 ⊃ ...

13. 4). Then G admits a maximal torus. 3 We could not ﬁnd a reference for this basic fact in the literature. 3] we have NG (T) = NG (t). Since the natural homomorphism NG (T)/T → Aut(t) is injective we obtain T = ZG (t). 30 4. SEMILINEAR CONSIDERATIONS Proof. We try to recreate the situation of the semilinear Borel-Mostow theorem. We can assume that G is split after base extension to the Galois covering ˜ ±1/m , . . , t±1/m ˜ is a ﬁnite Galois ˜ = k[t R ] where m is a positive integers and k/k n 1 extension of ﬁelds containing all primitive m-th roots of unity of k.