Abstract
We introduce a notion of \(\xi \)-stability on the affine grassmannian \({\fancyscript{X}}\) for the classical groups, this is the local version of the \(\xi \)-stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient \({\fancyscript{X}}^{\xi }/T\) of the stable part \({\fancyscript{X}}^{\xi }\) by the maximal torus \(T\) exists as an ind-\(k\)-scheme, and we introduce a reduction process analogous to the Harder–Narasimhan reduction for vector bundles over an algebraic curve. For the group \({\mathrm {SL}}_{d}\), we calculate the Poincaré series of the quotient \({\fancyscript{X}}^{\xi }/T\).
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Acknowledgments
This article is based on part of the author’s thesis at Université Paris-Sud 11 at Orsay. We want to thank Gérard Laumon for having posed this question, and for his encouragements during the preparation of this work. We also want to thank an anonymous referee for his careful reeding and helpful suggestions.
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Chen, Z. The \(\xi \)-stability on the affine grassmannian. Math. Z. 280, 1163–1184 (2015). https://doi.org/10.1007/s00209-015-1471-2
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DOI: https://doi.org/10.1007/s00209-015-1471-2