Decompositions of Kac-Moody groups

2017/08/18 by Max Horn, Horn, Max
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1708.05566

openalex publication_date 2017/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a split (minimal) Kac-Moody group over ℝ or ℂ with maximal torus T, and let θ be a Cartan-Chevalley involution of G, twisted by complex conjugation, and satisfying that θ(T)=T. Furthermore, let K be the subgroup fixed by θ, and τ:G→ G, g↦ gθ(g)^-1. Let A:=τ(T). In this note, we show resp. revisit that G admits a (refined) Iwasawa decompositions G=UAK. We also show that if G is of non-spherical type, then it never admits a polar decomposition G=τ(G)K nor a Cartan decompositions G=KAK. This has implications for the geometrical structure of the Kac-Moody symmetric space G/K ≅ τ(G).

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