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We describe the Harder--Narasimhan filtration of the Hodge bundle forTeichm\"uller curves in the non-varying strata of quadratic differentialsappearing in [CM2]. Moreover, we show that the Hodge bundle on the non-varyingstrata away from the irregular components can split as a direct sum of linebundles. As applications, we determine all individual Lyapunov exponents ofalgebraically primitive Teichm\"uller curves in the non-varying strata andderive new results regarding the asymptotic behavior of Lyapunov exponents.

Filtration and splitting of the Hodge bundle on the non-varying strata of quadratic differentials