Semilinear automorphisms of reductive algebraic groups

2019/01/16 by Thierry Stulemeijer, Stulemeijer, Thierry
Mathematics · Physics and Astronomy · #16K20 #20G05 #20G10 #20G15 #20G25 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1901.05368

openalex publication_date 2019/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a connected reductive algebraic group over a field k . We study the group of semilinear automorphisms Aut( G→ Spec k) consisting of algebraic automorphisms of G over automorphisms of k . We focus on the exact sequence 1→ Aut G→ Aut ( G→ Spec k)→ Aut_G(k)→ 1 . When G is quasi-split, we show that Aut_G(k) is isomorphic to Aut_R(G)(k), where R(G) denotes the scheme of based root datum of G. Furthermore, the exact sequence 1→ Aut G→ Aut ( G→ Spec k)→ Aut_G(k)→ 1 splits if and only if the exact sequence 1→ Aut R(G) → Aut (R(G) → Spec k)→ Aut_R(G)(k)→ 1 splits. As a corollary, we get many examples of algebraic groups G over k whose group of abstract automorphisms does not decompose as the semidirect product of Aut G with Aut_G(k) . We also study the same questions for inner forms of SL_n over a local field.

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