Abstract irreducible representations of reductive algebraic groups with Borel-stable line

• Xiaoyu Chen ^[1]
  1. [1] Shanghai Normal University

     Shanghai Normal University

     China

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 5, 2024
• Idioma: inglés
• DOI: 10.1007/s00029-024-00991-x
• Enlaces
  • Texto completo
• Resumen

  • Let p be a prime number and k=F¯p, the algebraic closure of the finite field Fp of p elements. Let G be a connected reductive group defined over Fp and B be a Borel subgroup of G (not necessarily defined over Fp). Let kH be the group algebra of a group H over k. We show that for each (one-dimensional) character θ of B (not necessarily rational), there is a unique (up to isomorphism) irreducible kG-module L(θ) containing θ as a kB-submodule, and moreover, L(θ) is isomorphic to a parabolic induction from a finite-dimensional irreducible kL-module for some Levi subgroup L of G. Thus, we have classified and constructed all (abstract) irreducible kG-modules with B-stable line (i.e. an one-dimensional kB-submodule). As a byproduct, we give a new proof of a result of Borel and Tits on the classification of finite-dimensional irreducible kG-modules.

• Referencias bibliográficas
  • 1. Andersen, H.H., Jantzen, J.C., Soergel, W.: Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic...
  • 2. Bonnafé, C., Rouquier, R.: Catégroies dérivées et variétés de Deligne–Lusztig. Publ. Math. IHES 57, 1–57 (2003)
  • 3. Bonnafé, C., Dat, J.F., Rouquier, R.: Derived categories and Deligne–Lusztig varieties II. Ann. Math. 185, 609–670 (2017)
  • 4. Borel, A., Tis, J.: Homomorphismes abstraits de groupes algebriques simples. Ann. Math. 97, 499–571 (1973)
  • 5. Carter, R.W.: Finite groups of Lie type: conjugacy classes and complex characters. Pure and applied mathematics. John Wiley and Sons, New...
  • 6. Carter, W., Lusztig, G.: Modular representations of finite groups of Lie type. Proc. Lond Math. Soc. 32, 347–384 (1976)
  • 7. Chen, X.: On the principal series representations of semisimple groups with Frobenius maps. arXiv:1702.05686v2
  • 8. Chen, X.: Some non quasi-finite irreducible representations of semisimple groups with Frobenius
  • maps. arxiv:1705.04845v1 9. Chen, X., Dong, J.: The permutation module on flag varieties in cross characteristic. Math. Z. 293, 475–484 (2019)
  • 10. Chen, X., Dong, J.: The decomposition of permutation module for infinite Chevalley groups. Sci. China Math. 64, 921–930 (2021)
  • 11. Chen, X., Dong, J.: Abstract induced modules for reductive groups with Frobenius maps. Int. Math. Res. Not. IMRN 5, 3308–3348 (2022)
  • 12. Deligne, P., Lusztig, G.: Representations of reductive groups over finite fields. Ann. Math. 103, 103–161 (1976)
  • 13. Fiebig, P.: Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture. J. Amer. Math. Soc. 24, 133–181 (2011)
  • 14. Haboush, W.J.: Central differential operators on split semi-simple groups over fields of positive characteristic, pp. 35–85 in: M.-P....
  • 15. Jacobson, N.: Basic algebra II. W. H. Freeman and Company, New York (1985)
  • 16. Jantzen, J.C.: Representations of algebraic groups. Mathematical surveys and monograph, vol. 107, 2nd ed., American Mathematical Society,...
  • 17. Jantzen, J.C.: Filtrierungen der Darstellungen in der Hauptserie endlicher Chevalley–Gruppen. Proc. Lond. Math. Soc. 49, 445–482 (1984)
  • 18. Lusztig, G.: Hecke algebras and Jantzen’s generic decomposition patterns. Adv. Math. 37, 121–164 (1980)
  • 19. Lusztig, G.: Some problems in the representation theory of finite Chevalley groups, The Santa Cruz Conference on Finite Groups (Univ....
  • 20. Pillen, C.: Loewy series for principal series representations of finite Chevalley groups. J. Algebra 189, 101–124 (1997)
  • 21. Rudakov, A.N.: On representations of classical semisimple Lie algebras of characteristic p. Math. USSR Izvestiya 4, 741–749 (1970)
  • 22. Serre, J.P.: Linear representations of finite groups, Graduate Texts in Mathematics, vol. 42, SpringerVerlag., New York-Heidelberg, (1977)
  • 23. Serre, J.P.: A course in arithmetic, Graduate Texts in Mathematics, vol. 7, Springer-Verlag., New York-Heidelberg, (1973)
  • 24. Springer, T.A.: Reductive Groups, Proceedings of symposia in pure mathematics, vol. 33 (1979), part 1, pp. 3–27
  • 25. Steinberg, R.: Prime power representations of finite linear groups II. Can. J. Math. 9, 347–351 (1957)
  • 26. Steinberg, R.: Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, 80 (1968)
  • 27. Steinberg, R.: Representations of algebraic groups. Nagoya Math. J. 22, 33–56 (1963)
  • 28. Sawada, H.: A characterization of the modular representations of finite groups with split (B, N) pairs. Math. Z. 155, 29–41 (1977)
  • 29. Williamson, G.: Schubert calculus and torsion explosion. J. Amer. Math. Soc. 30, 1023–1046 (2017)
  • 30. Xi, Nanhua: Some infinite-dimensional representations of reductive groups with Frobenius maps. Sci. China Math. 57, 1109–1120 (2014)
  • 31. Yang, R.: Irreducibility of infinite dimensional Steinberg modules of reductive groups with Frobenius maps. J. Algebra 533, 17–24 (2019)
  • 32. Yoshida, Y.: A generalization of Pillen’s theorem for principal series modules II. J. Algebra 429, 177–191 (2015)