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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2310.20006 |
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| _version_ | 1866912473797361664 |
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| author | Chen, Tsao-Hsien Yi, Lingfei |
| author_facet | Chen, Tsao-Hsien Yi, Lingfei |
| contents | We study the singularities of closures of Iwahori orbits on loop spaces of symmetric varieties extending the celebrated work of Lusztig-Vogan to the affine setting. We show that the IC-complexes of orbit closures (with possible non-trivial coefficients) are pointwise pure and satisfy a parity vanishing property. We apply those geometric results to study the affine Lusztig-Vogan modules and obtain fundational results about them including the positivity properties of the affine Kazhdan-Lusztig-Vogan polynomials. Along the way, we construct conical transversal slices inside loop spaces of symmetric varieties generalizing the work of Mars-Springer in the finite dimensional setting. Our results answer a question of Lusztig.
We deduce results for singularities of spherical orbit closures and provide applications to relative Langlands duality including the positivity for the relative Kostka-Foulkes polynomials and the formality conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20006 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Singularities of orbit closures in loop spaces of symmetric varieties Chen, Tsao-Hsien Yi, Lingfei Representation Theory Algebraic Geometry We study the singularities of closures of Iwahori orbits on loop spaces of symmetric varieties extending the celebrated work of Lusztig-Vogan to the affine setting. We show that the IC-complexes of orbit closures (with possible non-trivial coefficients) are pointwise pure and satisfy a parity vanishing property. We apply those geometric results to study the affine Lusztig-Vogan modules and obtain fundational results about them including the positivity properties of the affine Kazhdan-Lusztig-Vogan polynomials. Along the way, we construct conical transversal slices inside loop spaces of symmetric varieties generalizing the work of Mars-Springer in the finite dimensional setting. Our results answer a question of Lusztig. We deduce results for singularities of spherical orbit closures and provide applications to relative Langlands duality including the positivity for the relative Kostka-Foulkes polynomials and the formality conjecture. |
| title | Singularities of orbit closures in loop spaces of symmetric varieties |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2310.20006 |