Filtrations of D-modules along semi-invariant functions

Given a smooth algebraic variety X with an action of a connected reductive linear algebraic group G , and an equivariant D -module M , we study the G -decompositions of the associated V -, Hodge, and weight filtrations. If M is the localization of a D -module S underlying a pure twistor D -module (e.g. when S is simple and regular holonomic) along a semi-invariant function, we determine the weight level of any element in an irreducible isotypic component of M in terms of multiplicities of roots of b -functions. If S underlies a pure Hodge module, we show that the Hodge level is governed by the degrees of another class of polynomials, also expressible in terms of b -functions. As an application, if X is an affine spherical variety, we describe these filtrations representation-theoretically in terms of roots of b -functions, and compute all higher multiplier and Hodge ideals associated with semi-invariant functions. Explicit examples include the spaces of general, skew-symmetric, and symmetric matrices, as well as the Freudenthal cubic on the fundamental representation of E 6 .