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Functions on Nilpotent Orbit Covers and Birational Geometry

  • William Graham
  • Scott Joseph Larson
  • Alberto San Miguel Malaney
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Abstract

We use an analogue of the Springer resolution to describe the G-module structure on the ring of regular functions on the universal cover π’ͺΜƒ of any nilpotent orbit for G = SLn. Building on previous work on the extended Springer resolution, we construct a variety β„³Μƒ that is finite over the cotangent bundle of a partial flag variety G/P, and proper and birational over the affinization β„³ of π’ͺΜƒ. We use techniques in birational geometry to show that β„³Μƒ has rational singularities, which provides the cohomology vanishing needed to describe the ring of functions on π’ͺΜƒ as an induced representation from a Levi subgroup of G. Our results also yield a description of the structure of R(π’ͺΜƒ) as a graded G-module. We describe the minimal embedding of β„³, study the lifting of characters of the component group of π’ͺΜƒ to parabolics and Levi subgroups, and make a more general vanishing conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{graham2026functions,
  title = {{Functions on Nilpotent Orbit Covers and Birational Geometry}},
  author = {William Graham and Scott Joseph Larson and Alberto San Miguel Malaney},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.17272},
  eprint = {2609.17272},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17272},
}
RIS
TY  - UNPB
TI  - Functions on Nilpotent Orbit Covers and Birational Geometry
AU  - William Graham
AU  - Scott Joseph Larson
AU  - Alberto San Miguel Malaney
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.17272
UR  - https://arxiv.org/abs/2609.17272
AB  - We use an analogue of the Springer resolution to describe the G-module structure on the ring of regular functions on the universal cover π’ͺΜƒ of any nilpotent orbit for G = SL_n. Building on previous work on the extended Springer resolution, we construct a variety β„³Μƒ that is finite over the cotangent bundle of a partial flag variety G/P, and proper and birational over the affinization β„³ of π’ͺΜƒ. We use techniques in birational geometry to show that β„³Μƒ has rational singularities, which provides the cohomology vanishing needed to describe the ring of functions on π’ͺΜƒ as an induced representation from a Levi subgroup of G. Our results also yield a description of the structure of R(π’ͺΜƒ) as a graded G-module. We describe the minimal embedding of β„³, study the lifting of characters of the component group of π’ͺΜƒ to parabolics and Levi subgroups, and make a more general vanishing conjecture.
ER  -