Real and symmetric matrices

Abstract

We construct a family of involutions on the space ${\mathfrak{g}\mathfrak{l}}_n^{\prime }(\mathbb{C})$ of $n\times n$ matrices with real eigenvalues interpolating the complex conjugation and the transpose. We deduce from it a stratified homeomorphism between the space ${\mathfrak{g}\mathfrak{l}}_n^{\prime }(\mathbb{R})$ of $n\times n$ real matrices with real eigenvalues and the space ${\mathfrak{p}}_n^{\prime }(\mathbb{C})$ of $n\times n$ symmetric matrices with real eigenvalues, which restricts to a real analytic isomorphism between individual ${\mathrm{GL}}_n(\mathbb{R})$-adjoint orbits and ${\mathrm{O}}_n(\mathbb{C})$-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.