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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.19383 |
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| _version_ | 1866913127644266496 |
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| author | Lőrincz, András C. Yang, Ruijie |
| author_facet | Lőrincz, András C. Yang, Ruijie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19383 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Filtrations of D-modules along semi-invariant functions Lőrincz, András C. Yang, Ruijie Algebraic Geometry Commutative Algebra Representation Theory 32S35, 14F10, 11S90, 13A50 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. |
| title | Filtrations of D-modules along semi-invariant functions |
| topic | Algebraic Geometry Commutative Algebra Representation Theory 32S35, 14F10, 11S90, 13A50 |
| url | https://arxiv.org/abs/2504.19383 |