Real and symmetric matrices

2020/06/18 by Chen, Tsao-Hsien, Nadler, David
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2006.10279

Abstract

We construct a family of involutions on the space \mathfrakgl_n'(\mathbb 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 of n× n real matrices with real eigenvalues and the space of n× n symmetric matrices with real eigenvalues, which restricts to a real analytic isomorphism between individual GL_n(\mathbb R)-adjoint orbits and 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.

Related

• Involution matrix loci and orbit harmonics
• Involutions of the Symmetric Group and Congruence B-orbits of Anti-Symmetric Matrices
• Real adjoint orbits of special linear groups
• The orbit structure of the Gelfand-Zeitlin group on n x n matrices
• Eigenvalues of real symmetric matrices