REAL AND SYMMETRIC MATRICES

We construct a family of involutions on the space gl0 n.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 gl0 n.R/ of n × n real matrices with real eigenvalues and the space p0 n.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.