Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909198449639424 |
|---|---|
| author | Jacques, Simon |
| author_facet | Jacques, Simon |
| contents | Let G be a reductive algebraic group in classical types A, B, D and e be an element of its Lie algebra with Z its centraliser in G for the adjoint action. We suppose that e identifies with an nilpotent matrix of order two, which guarantees the number of Z-orbits in the flag variety of G is finite. For types B, D in characteristic two, we also suppose the image of e is totally isotropic. We show that any closure Y of such orbit is normal. We also prove that Y is Cohen-Macaulay with rational singularities provided that the base field is of characteristic zero, and that Cohen-Macaulayness remains in any characteristic for type A. We exhibit a birational, rational morphism onto Y involving Schubert varieties. Our work generalizes a result by N. Perrin and E. Smirnov on Springer fibers ([PS12]). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_09123 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D Jacques, Simon Algebraic Geometry Group Theory Representation Theory Let G be a reductive algebraic group in classical types A, B, D and e be an element of its Lie algebra with Z its centraliser in G for the adjoint action. We suppose that e identifies with an nilpotent matrix of order two, which guarantees the number of Z-orbits in the flag variety of G is finite. For types B, D in characteristic two, we also suppose the image of e is totally isotropic. We show that any closure Y of such orbit is normal. We also prove that Y is Cohen-Macaulay with rational singularities provided that the base field is of characteristic zero, and that Cohen-Macaulayness remains in any characteristic for type A. We exhibit a birational, rational morphism onto Y involving Schubert varieties. Our work generalizes a result by N. Perrin and E. Smirnov on Springer fibers ([PS12]). |
| title | Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D |
| topic | Algebraic Geometry Group Theory Representation Theory |
| url | https://arxiv.org/abs/2207.09123 |