Regular automorphisms and Calogero–Moser families

• Cédric Bonnafé ^[1]
  1. [1] IMAG, Universit´e de Montpellier, CNRS, Montpellier, France
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 68, Nº. 2, 2025, págs. 519-533
• Idioma: inglés
• DOI: 10.33044/revuma.3143
• Enlaces
  • Texto completo
• Resumen

  • We study the subvariety of fixed points of an automorphism of a Calogero–Moser space induced by a regular element of finite order of the normalizer of the associated complex reflection group W. We determine some of (and conjecturally all) the C×-fixed points of its unique irreducible component of maximal dimension in terms of the character table of W. This is inspired by the mysterious relations between the geometry of Calogero–Moser spaces and unipotent representations of finite reductive groups, which is the theme of another paper [Pure Appl. Math. Q. 21 no. 1 (2025), 131–200].

• Referencias bibliográficas
  • G. Bellamy, Generalized Calogero–Moser spaces and rational Cherednik algebras, Ph.D. thesis, University of Edinburgh, 2010. Available at http://hdl.handle.net/1842/4733.
  • C. Bonnafe´, On the Calogero–Moser space associated with dihedral groups, Ann. Math. Blaise Pascal 25 no. 2 (2018), 265–298. DOI MR Zbl
  • C. Bonnafe´, Automorphisms and symplectic leaves of Calogero–Moser spaces, J. Aust. Math. Soc. 115 no. 1 (2023), 26–57. DOI MR Zbl
  • C. Bonnafe´, Calogero–Moser spaces vs unipotent representations, Pure Appl. Math. Q. 21 no. 1 (2025), 131–200. DOI MR Zbl
  • C. Bonnafe´ and R. Maksimau, Fixed points in smooth Calogero–Moser spaces, Ann. Inst. Fourier (Grenoble) 71 no. 2 (2021), 643–678. DOI MR...
  • C. Bonnafe´ and R. Rouquier, Cherednik algebras and Calogero–Moser cells, [v1] 2017, [v3] 2022. arXiv:1708.09764v3 [math.RT].
  • C. Bonnafe´ and U. Thiel, Computational aspects of Calogero–Moser spaces, Selecta Math. (N.S.) 29 no. 5 (2023), Paper No. 79. DOI MR Zbl
  • M. Broue´, Introduction to complex reflection groups and their braid groups, Lecture Notes in Math. 1988, Springer, Berlin, 2010. DOI MR Zbl
  • M. Broue´, G. Malle, and J. Michel, Generic blocks of finite reductive groups, in Repr´esentations unipotentes g´en´eriques et blocs des groupes...
  • M. Broue´, G. Malle, and J. Michel, Split spetses for primitive reflection groups, Ast´erisque no. 359, Soci´et´e Math´ematique de France,...
  • M. Broue´ and J. Michel, Sur certains ´el´ements r´eguliers des groupes de Weyl et les vari´et´es de Deligne–Lusztig associ´ees, in Finite...
  • P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 no. 1 (1976), 103–161. DOI MR Zbl
  • P. Etingof and V. Ginzburg, Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphism, Invent. Math....
  • M. Geck and N. Jacon, Representations of Hecke algebras at roots of unity, Algebra and Applications 15, Springer, London, 2011. DOI MR Zbl
  • M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori–Hecke algebras, London Math. Soc. Monogr. New Ser. 21, Oxford University...
  • I. Gordon, Baby Verma modules for rational Cherednik algebras, Bull. London Math. Soc. 35 no. 3 (2003), 321–336. DOI MR Zbl
  • I. G. Gordon and M. Martino, Calogero–Moser space, restricted rational Cherednik algebras and two-sided cells, Math. Res. Lett. 16 no. 2 (2009),...
  • I. M. Isaacs, Character theory of finite groups, Pure Appl. Math. 69, Academic Press, New York-London, 1976. MR Zbl
  • A. Lacabanne, On a conjecture about cellular characters for the complex reflection group G(d, 1, n), Ann. Math. Blaise Pascal 27 no. 1 (2020),...
  • G. Lusztig, Characters of reductive groups over a finite field, Ann. of Math. Stud. 107, Princeton University Press, Princeton, NJ, 1984....
  • T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159– 198. DOI MR Zbl