Real and symmetric matrices

We construct a family of involutions on the space $\mathfrak{gl}_n'(\mathbb C)$ $n\times n$ matrices with real eigenvalues interpolating complex conjugation and transpose. deduce from it stratified homeomorphism between symmetric eigenvalues, which restricts to analytic isomorphism individual $\mathrm{GL}_n(\mathbb R)$-adjoint orbits $\mathrm{O}_n(\mathbb C)$-adjoint orbits. also establish similar results in more general settings Lie algebras classical types quiver varieties. To this end, we prove result about hyper-K\"ahler quotients linear spaces. provide applications (generalized) Kostant-Sekiguchi correspondence, singularities adjoint orbit closures, Springer theory for groups