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The companion section for classical groups

Thomas Hameister

University of Chicago, Eckhart Hall, 5734 University Ave, Chicago, IL 60637, USA

E-mail Address: thameister@uchicago.edu

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and

Bao Châu Ngô

University of Chicago, Eckhart Hall, 5734 University Ave, Chicago, IL 60637, USA

E-mail Address: ngo@uchicago.edu

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https://doi.org/10.1142/S0129167X24410106Cited by:1 (Source: Crossref)

This article is part of the issue:

Special Volume Dedicated to Prof. Oscar Garcia-Prada’s 60th Birthday

Abstract

We use the companion matrix construction for ${GL}_{n}$ to build canonical sections of the Chevalley map $[𝔤 / G] \to 𝔤 / / G$ for classical groups $G$ as well as the group $G_{2}$ . To do so, we construct canonical tensors on the associated spectral covers. As an application, we make explicit lattice descriptions of affine Springer fibers and Hitchin fibers for classical groups and $G_{2}$ .

Communicated by Luis Álvarez-Cónsul and Steven Bradlow

Keywords:

AMSC: 57M25, 57M27

References

1. J. D. Adler, J. Fintzen and S. Varma , On Kostant sections and topological nilpotence, J. London Math. Soc. 97(2) (2018) 325–351. Crossref, Google Scholar
2. A. Bouthier and K. Česnavičius , Torsors on loop groups and the Hitchin fibration, Ann. Sci. Éc. Norm. Supér. 55(3) (2022) 791–864. Crossref, Google Scholar
3. F. Engel , Ein neues, dem linearen Komplexe analoges Gebilde. (1900). Google Scholar
4. N. Hitchin , Langlands duality and $G_{2}$ spectral curves, Quart. J. Math. 58(3) (2007) 319–344. Crossref, Google Scholar
5. N. Hitchin , Stable bundles and integrable systems, Duke Math. J. 54(1) (1987) 91–114. Crossref, Web of Science, Google Scholar
6. N. Hitchin , The geometry of three-forms in six dimensions, J. Diff. Geom. 55(3) (2000) 547–576. Crossref, Web of Science, Google Scholar
7. D. Kazhdan and G. Lusztig , Fixed point varieties on affine flag manifolds, Israel J. Math. 62(2) (1988) 129–168. Crossref, Google Scholar
8. B. Kostant , Lie group representations on polynomial rings, Amer. J. Math. 85(3) (1963) 327–404. Crossref, Web of Science, Google Scholar
9. B. C. Ngo , Le lemme fondamental pour les algebres de Lie, Publ. Math. l’IHÉS 111(1) (2010) 1–169. Crossref, Google Scholar
10. J.-P. Serre , Local Fields, Vol. 67 (Springer Science & Business Media, 2013). Google Scholar
11. Z. Yun, Lectures on Springer theories and orbital integrals, arXiv preprint arXiv:1602.01451 (2016). Google Scholar