Normalizers of maximal tori and real forms of Lie groups

European Journal of Mathematics - 2022
A. Gerasimov1, Д. В. Лебедев1, Sergey Oblezin2
1Laboratory for Quantum Field Theory and Information, Institute for Information Transmission Problems, RAS, Moscow, Russia, 127994
2School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

Tóm tắt

AbstractGiven a complex connected reductive Lie group G with a maximal torus $$H\subset G$$ H ⊂ G , Tits defined an extension $$W_G^{\mathrm{T}}$$ W G T of the corresponding Weyl group $$W_G$$ W G . The extended group is supplied with an embedding into the normalizer $$N_G(H)$$ N G ( H ) such that $$W_G^{\mathrm{T}}$$ W G T together with H generate $$N_G(H)$$ N G ( H ) . In this paper we propose an interpretation of the Tits classical construction in terms of the maximal split real form $$G(\mathbb {R})\subset G$$ G ( R ) ⊂ G , which leads to a simple topological description of $$W^{\mathrm{T}}_G$$ W G T . We also consider a variation of the Tits construction associated with compact real form U of G. In this case we define an extension $$W_G^U$$ W G U of the Weyl group $$W_G$$ W G , naturally embedded into the group extension $$\widetilde{U}:=U\,{\rtimes }\, \Gamma $$ U ~ : = U ⋊ Γ of the compact real form U by the Galois group $$\Gamma ={\mathrm{Gal}}(\mathbb {C}/\mathbb {R})$$ Γ = Gal ( C / R ) . Generators of $$W^U_G$$ W G U are squared to identity as in the Weyl group $$W_G$$ W G . However, the non-trivial action of $$\Gamma $$ Γ by outer automorphisms requires $$W^U_G$$ W G U to be a non-trivial extension of $$W_G$$ W G . This gives a specific presentation of the maximal torus normalizer of the group extension $${\widetilde{U}}$$ U ~ . Finally, we describe explicitly the adjoint action of $$W_G^{\mathrm{T}}$$ W G T and $$W^U_G$$ W G U on the Lie algebra of G.

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Tài liệu tham khảo

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European Journal of Mathematics

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