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In this paper, we provide an $L^2$ fine resolution of the prolongation of anilpotent harmonic bundle in the sense of Simpson-Mochizuki (an analyticanalogue of the Kashiwara-Malgrange filtrations). This is the logarithmicanalogue of Cattani-Kaplan-Schmid's and Kashiwara-Kawai's results on the $L^2$interpretation of the intersection complex. As an application, we give an$L^2$-theoretic proof to the Nadel-Kawamata-Viehweg vanishing theorem withcoefficients in a nilpotent Higgs bundle.

$L^2$ representation of Simpson-Mochizuki's prolongation of Higgs  bundles and the Kawamata-Viehweg vanishing theorem for semistable parabolic  Higgs bundles