GKM Theory for Manifolds of Isospectral Matrices in Lie Type D

We study the manifold $Q_{\mathrm{\Gamma },\lambda }$ of isospectral real skew-symmetric matrices with a prescribed sparsity pattern determined by a graph $\mathrm{\Gamma }$. The compact torus $T^n$ acts naturally on $Q_{\mathrm{\Gamma },\lambda }$ by conjugation, and this action can be studied using GKM theory. We prove two results about this manifold and its GKM graph. The first theorem describes how the GKM graph of $Q_{\mathrm{\Gamma },\lambda }$ is obtained from the GKM graph of the corresponding manifold $M_{\mathrm{\Gamma },\lambda }$ of isospectral Hermitian matrices. The second theorem gives a criterion for equivariant formality of $Q_{\mathrm{\Gamma },\lambda }$.