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arXiv · 2504.19383

Filtrations of D-modules along semi-invariant functions

András C. Lőrincz·Ruijie Yang
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Abstract

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) 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. Examples include the spaces of general, skew-symmetric, and symmetric matrices, as well as the Freudenthal cubic on the fundamental representation of E_6.

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András C. Lőrincz, Ruijie Yang. 2026-05-14. Filtrations of D-modules along semi-invariant functions. https://arxiv.org/abs/2504.19383

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