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The null-cone and cohomology of vector bundles on flag varieties


Authors: Kari Vilonen and Ting Xue
Journal: Represent. Theory 20 (2016), 482-498
MSC (2010): Primary 14L99, 20G99
DOI: https://doi.org/10.1090/ert/491
Published electronically: December 23, 2016
MathSciNet review: 3589334
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Abstract: We study the null-cone of a semi-simple algebraic group acting on a number of copies of its Lie algebra via the diagonal adjoint action. We show that the null-cone has rational singularities in the case of $ SL_3$. We observe by example that the null-cone is not normal in general and that the normalization of the null-cone does not have rational singularities in general. This is achieved by computing cohomology of certain vector bundles on flag varieties.


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Additional Information

Kari Vilonen
Affiliation: School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia – and – Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
Email: kari.vilonen@unimelb.edu.au, kari.vilonen@helsinki.fi

Ting Xue
Affiliation: School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia – and – Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
Email: ting.xue@unimelb.edu.au

DOI: https://doi.org/10.1090/ert/491
Received by editor(s): February 3, 2016
Received by editor(s) in revised form: March 10, 2016, and October 14, 2016
Published electronically: December 23, 2016
Additional Notes: The first author was supported in part by NSF grants DMS-1402928 and DMS-1069316, the Academy of Finland, and the ARC grant DP150103525.
The second author was supported in part by the ARC grants DP150103525, DE160100975, and the Academy of Finland.
Article copyright: © Copyright 2016 American Mathematical Society