The efficacy of using complexifications to understand the structure of real algebraic groups is demonstrated. In particular, the following two results are proved: (i) Let G be a connected solvable linear group whose eigenvalues are all real. If the complexification Gℂ of G is algebraic and operates algebraically on a complex variety V, and some G orbit in V is compact, then this orbit is a point. (ii) If L is a connected subgroup of a connected real linear semisimple group G such that the complexification Lℂ of L is algebraic and Lℂ contains a maximal torus of Gℂ, then L contains a maximal torus of G which complexifies to a maximal torus of Gℂ.
Funding source: KFUPM
Award Identifier / Grant number: IN131058
Funding statement: The first author thanks KFUPM for funding Research Project IN131058. The second author acknowledges the support of the J. C. Bose Fellowship.
© 2016 by De Gruyter