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Algebraic families of groups and commuting involutions. (English) Zbl 1428.20046
This paper constructs real forms of one-parameter algebraic families of complex affine algebraic groups. For the notion of an algebraic family, see [“Algebraic families of Harish-Chandra pairs”, Preprint, arXiv:1610.03435; “Contractions of representations and algebraic families of Harish-Chandra modules”, Preprint, arXiv:1703.04028] by J. Bernstein, N. Higson and E. Subag.
Let \(\sigma_1\) and \(\sigma_2\) be two commuting antiholomorphic involutions of a complex affine algebraic group \(G\). Its main result states that there exist an algebraic family \(\boldsymbol{G}\) of affine algebraic groups and an antiholomorphic involution \(\boldsymbol{\sigma}\) of the family \(\boldsymbol{G}\) that interpolates between the real forms \(G^{\sigma_1}\) and \(G^{\sigma_2}\). More precisely, if \([\alpha: \beta] \in \mathbb{RP}^1\) then \[ \boldsymbol{G}^{\boldsymbol{\sigma}}|_{[\alpha:\beta]} \cong \begin{cases} G^{\sigma_1}, & \alpha\beta >0,\\ (G^{\sigma_1}\cap G^{\sigma_2}) \ltimes (\mathfrak{g}^{\sigma_1} \cap \mathfrak{g}^{-\sigma_2}), & \alpha \beta =0,\\ G^{\sigma_2}, & \alpha \beta <0. \end{cases} \]

20G20 Linear algebraic groups over the reals, the complexes, the quaternions
22E15 General properties and structure of real Lie groups
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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