Real and symmetric matrices

We construct a family of involutions on the space gln′(C) of n×n matrices with real eigenvalues interpolating the complex conjugation and the transpose. We deduce from it a stratified homeomorphism between the space gln′(R) of n×n real matrices with real eigenvalues and the space pn′(C) of n×n symmetric matrices with real eigenvalues, which restricts to a real analytic isomorphism between individual GLn(R)-adjoint orbits and On(C)-adjoint orbits. We also establish similar results in more general settings of Lie algebras of classical types and quiver varieties. To this end, we prove a general result about involutions on hyper-Kähler quotients of linear spaces. We provide applications to the (generalized) Kostant–Sekiguchi correspondence, singularities of real and symmetric adjoint orbit closures, and Springer theory for real groups and symmetric spaces.