Two proofs of a Jantzen Conjecture for Whittaker Modules
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911964924477440 |
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| author | Eberhardt, Jens Niklas Romanov, Anna |
| author_facet | Eberhardt, Jens Niklas Romanov, Anna |
| contents | We define a filtration of a standard Whittaker module over a complex semisimple Lie algebra and and establish its fundamental properties. Our filtration specialises to the Jantzen filtration of a Verma module for a certain choice of parameter. We prove that embeddings of standard Whittaker modules are strict with respect to our filtration, and that the filtration layers are semisimple. This provides a generalisation of the Jantzen conjectures to Whittaker modules. We prove these statements in two ways. First, we give an algebraic proof which compares Whittaker modules to Verma modules using a functor introduced by Backelin. Second, we give a geometric proof using mixed twistor $\mathcal{D}$-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16330 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two proofs of a Jantzen Conjecture for Whittaker Modules Eberhardt, Jens Niklas Romanov, Anna Representation Theory We define a filtration of a standard Whittaker module over a complex semisimple Lie algebra and and establish its fundamental properties. Our filtration specialises to the Jantzen filtration of a Verma module for a certain choice of parameter. We prove that embeddings of standard Whittaker modules are strict with respect to our filtration, and that the filtration layers are semisimple. This provides a generalisation of the Jantzen conjectures to Whittaker modules. We prove these statements in two ways. First, we give an algebraic proof which compares Whittaker modules to Verma modules using a functor introduced by Backelin. Second, we give a geometric proof using mixed twistor $\mathcal{D}$-modules. |
| title | Two proofs of a Jantzen Conjecture for Whittaker Modules |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2407.16330 |