Real and symmetric matrices

TL;DR

The paper introduces involutions connecting real and symmetric matrices with real eigenvalues, establishing homeomorphisms and isomorphisms between their orbit spaces and extending these results to Lie algebras and quiver varieties.

Contribution

It constructs a family of involutions interpolating conjugation and transpose, leading to new homeomorphisms and isomorphisms between real and symmetric matrix spaces and their orbits.

Findings

Established a stratified homeomorphism between real and symmetric matrices with real eigenvalues.

Proved real analytic isomorphisms between individual adjoint orbits.

Extended results to Lie algebras of classical types and quiver varieties.

Abstract

We construct a family of involutions on the space $\mathfrak{gl}_n'(\mathbb C)$ 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 of $n\times n$ real matrices with real eigenvalues and the space 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-K\"ahler 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.