Unitary representations of real groups and localization theory for Hodge modules
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866912232431943680 |
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| author | Davis, Dougal Vilonen, Kari |
| author_facet | Davis, Dougal Vilonen, Kari |
| contents | We prove a conjecture of Schmid and the second named author that the unitarity of a representation of a real reductive Lie group with real infinitesimal character can be read off from a canonical filtration, the Hodge filtration. Our proof rests on three main ingredients. The first is a wall crossing theory for mixed Hodge modules: the key result is that, in certain natural families, the Hodge filtration varies semi-continuously with jumps controlled by extension functors. The second ingredient is a Hodge-theoretic refinement of Beilinson-Bernstein localization: we show that the Hodge filtration of a mixed Hodge module on the flag variety satisfies the usual cohomology vanishing and global generation properties enjoyed by the underlying $\mathcal{D}$-module. The third ingredient is an explicit calculation of the Hodge filtration on a tempered Hodge module. As byproducts of our work, we obtain a version of Saito's Kodaira vanishing for twisted mixed Hodge modules, a calculation of the Hodge filtration on a certain object in category $\mathcal{O}$, and a host of new vanishing results for coherent sheaves on flag varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_13215 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Unitary representations of real groups and localization theory for Hodge modules Davis, Dougal Vilonen, Kari Representation Theory Algebraic Geometry We prove a conjecture of Schmid and the second named author that the unitarity of a representation of a real reductive Lie group with real infinitesimal character can be read off from a canonical filtration, the Hodge filtration. Our proof rests on three main ingredients. The first is a wall crossing theory for mixed Hodge modules: the key result is that, in certain natural families, the Hodge filtration varies semi-continuously with jumps controlled by extension functors. The second ingredient is a Hodge-theoretic refinement of Beilinson-Bernstein localization: we show that the Hodge filtration of a mixed Hodge module on the flag variety satisfies the usual cohomology vanishing and global generation properties enjoyed by the underlying $\mathcal{D}$-module. The third ingredient is an explicit calculation of the Hodge filtration on a tempered Hodge module. As byproducts of our work, we obtain a version of Saito's Kodaira vanishing for twisted mixed Hodge modules, a calculation of the Hodge filtration on a certain object in category $\mathcal{O}$, and a host of new vanishing results for coherent sheaves on flag varieties. |
| title | Unitary representations of real groups and localization theory for Hodge modules |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2309.13215 |