Semisimplicity and Rigidity of the Kontsevich--Zorich Cocycle for Products of Strata

Abstract

We study the Kontsevich--Zorich cocycle, its fixed tensor constructions, and their finite direct sums over products of strata of Abelian differentials. The acting groups are closed subgroups $G\subset(\mathrm{SL}_2\mathbb R)^n$ that, up to factor permutations and coordinatewise conjugation by rotations, are products of diagonal $\mathrm{SL}_2\mathbb R$ blocks and a finite central group. For ergodic invariant probability measures, we prove Deligne-type semisimplicity: the bundles admit Hodge-compatible decompositions into $G$-invariant pieces $V_i\otimes_{A_i}W_i$, where $A_i$ are division algebras, $V_i$ carry pure variations of Hodge structure, and the multiplicity spaces $W_i$ carry finite direct sums of pure Hodge structures. Every invariant subbundle arises from $A_i$-submodules of the $W_i$. If block marginal measures have smooth local supports and are locally equivalent to smooth volume, Hodge-orthogonal projections onto invariant subbundles are locally polynomial in period coordinates on the normalized support. The proof obtains a local product-of-quadrics model and joint polynomiality from polynomiality along stable and unstable leaves. On the cone obtained by scaling each diagonal block together, projection coefficients admit homogeneous degree-zero expressions with integral powers of reciprocal block areas. For the full product group $(\mathrm{SL}_2\mathbb R)^n$, single-stratum measure classification supplies the support hypothesis, yielding rigidity for every ergodic invariant probability measure. Under the transported support hypothesis, arbitrary coordinatewise $\mathrm{SL}_2\mathbb R$-conjugates also admit polynomial equivariant projectors, though not necessarily Hodge-orthogonal. Genus-two examples show that nonrotational conjugation can destroy Deligne-torus invariance, and independent factor scalings can require square roots of area functions.