2013/10/29 by Jeffrey Adams, Adams, Jeffrey · 2 citations
Mathematics · #11E72 #20G10 #20G20 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR #msc:11E72 #msc:20G10 #msc:20G20
paper · pdf · doi:10.48550/arxiv.1310.7917
revision 1: highlighted definition of real forms; divided Proposition 8.2 into Prop. 8.2/Corollary 8.3; fixed several typos and 2 references revision 2: added discussion of rational Weyl group and cohomology of spin groups
openalex publication_date 2013/10/29 · arxiv created 2014/07/01 · arxiv updated 2014/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Real forms of a complex reductive group are classified in terms of Galois cohomology H1(Γ,Gad) where Gad is the adjoint group. Alternatively, the theory of the Cartan involution gives a description in terms of cohomology with respect to a holomorphic involution: H1(\mathbb Z/2\mathbb Z,Gad) where the non trivial element acts by a holomorphic involution θ. The main theorem is that in general, if θ is the Cartan involution of a real form σ, there is a canonical isomorphism H1(Γ,G)≃ H1(\mathbb Z/2\mathbb Z,G). This has applications to the structure and representation theory of real groups. We give two such applications. The first is a simple proof of Matsuki's result on conjugacy classes of tori in real groups. The second is a computation of H1(Γ,G) in general. The answer is expressed in terms of the notion of strong real forms. We include tables for all simply connected simple groups.