\(P = W\): a strange identity for \(\mathrm{GL}(2,\mathbb C)\)

Start with a compact Riemann surface \(X\) and a complex reductive group \(G\), like \(\mathrm{GL}(n,\mathbb C)\). According to Hitchin-Simpson's ``non abelian Hodge theory", the pair \((X,G)\) comes with two new complex manifolds: the character variety \(\mathcal M_B\) and the Higgs moduli space \(\mathcal{M}_\text{Dolbeault}\). When \(G= \mathbb C^*\), these manifolds are two instances of the usual first cohomology group of \(X\) with coefficients in the abelian \(\mathbb C^*\). For general \(G\), we do not have cohomology groups, but we can study the two manifolds \(\mathcal M_B\) and \(\mathcal M_D\). I will present some aspects of this story and discuss a new identity --\(P = W\) for \(G = \mathrm{GL}(2,\mathbb C)\)-- occurring inside the singular cohomology of \(\mathcal M_B\) and \(\mathcal M_D\), where \(P\) and \(W\) dwell. The question as to whether the analogous identity \(P = W\) holds for other \(G\)'s, e.g. \(\mathrm{GL}(3,\mathbb C)\), is open.