The Hodge filtration of a monodromic mixed Hodge module and the irregular Hodge filtration

For an algebraic vector bundle E over a smooth algebraic variety X, a monodromic D-module on E is decomposed into a direct sum of some O-modules on X. We show that the Hodge filtration of a monodromic mixed Hodge module is decomposed with respect to the decomposition of the underlying D-module. By using this result, we endow the Fourier–Laplace transform M ∧ of the underlying D-module M of a monodromic mixed Hodge module with a mixed Hodge module structure. Moreover, we describe the irregular Hodge filtration on M ∧ concretely and show that it coincides with the Hodge filtration at all integer indices.