Semisimplicity conjecture for the complex Hodge bundle

Let ${\mathcal L}_1$ be a suborbifold in the moduli space of unit area Abelian differentials, or in the moduli space of unit area meromorphic quadratic differentials with at most simple poles. Suppose that ${\mathcal L}_1$ carries a Borel probability measure invariant under the natural action of $\operatorname{SL}(2,{\mathbb R})$ (respectively, $\operatorname{PSL}(2,{\mathbb R})$) and ergodic for the Teichmüller geodesic flow. Let $H^1_{\mathbb C}$ be the complex Hodge bundle over ${\mathcal L}_1$, and let $r$ denote the total number of zero entries in its Lyapunov spectrum. For a suitable finite, possibly ramified, cover $\hat{\mathcal L}_1$ of ${\mathcal L}_1$, consider the induced Hodge bundle and the pseudo-Hermitian intersection form on it. Semisimplicity conjecture. The induced Hodge bundle decomposes as a direct sum of irreducible $\operatorname{SL}(2,{\mathbb R})$-invariant (respectively, irreducible $\operatorname{PSL}(2,{\mathbb R})$-invariant), continuous, split subbundles. If $(p_i,q_i)$ is the signature of the restriction of the pseudo-Hermitian intersection form to the $i$th summand, then

$$\sum_i |p_i-q_i|=r.$$

Moreover, the cover can be chosen so that the nonzero part of the Lyapunov spectrum of each summand is simple. This conjectural decomposition extends the semisimplicity principle of Deligne to invariant suborbifolds, where the required algebraic-geometric hypotheses for a direct application of Deligne's theorem are not currently available. The conjecture also predicts how the signatures of the summands account for all zero Lyapunov exponents.