A Higgs bundle on a Hermitian symmetric space (Q2454831)

The author considers an irreducible Hermitian symmetric space \(G/K\) of non-compact type, a torsion free cocompact lattice \(\Gamma \subset G\), and the double quotient \(M = \Gamma \backslash G/K\). Then \(M\) is a compact connected Kähler manifold, which would be a Riemann surface in the case when \(G/K\) is the upper half plane \(\mathbb H\) (or equivalently the unit disk). The quotient map \(\phi \colon G/K \to M\) makes \(G/K\) a principal \(\Gamma\)-bundle over \(M\), denoted \(E_\Gamma\), and there is a unique flat connection \(\nabla^\Gamma\) on \(E_\Gamma\). We get a principal \(G_\mathbb C\)-bundle \(E^0_{G_\mathbb C}\) by taking the quotient of \(G_\mathbb C \times E_\Gamma\) by the \(\Gamma\)-action \((g, z) \mapsto (g \gamma, \gamma^{-1} z)\) (\(\gamma \in \Gamma\)). (\(G_\mathbb C\) denotes the complexified Lie group of \(G\)). From \(\nabla^\Gamma\) we get a flat connection \(\nabla_\Gamma^{G_\mathbb C}\) on \(E^0_{G_\mathbb C}\). The authors construct a \(G_\mathbb C\)-Higgs bundle \((E_{G_\mathbb C}, \theta)\) over \(M\) corresponding to \((E^0_{G_\mathbb C}, \nabla_\Gamma^{G_\mathbb C})\) and show that the Higgs bundle is rigid if \(\dim_\mathbb C M > 1\). For the Riemann surface case of \(\dim_\mathbb C M = 1\), the construction is due to \textit{N. J. Hitchin} [Proc. Lond. Math. Soc. (3) 55, 59--126 (1987; Zbl 0634.53045)].