Home Mathematics Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of SLn(ℤp)
Article
Licensed
Unlicensed Requires Authentication

Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of SLn(ℤp)

  • Jishnu Ray EMAIL logo
Published/Copyright: November 8, 2019
Forum Mathematicum
From the journal Forum Mathematicum Volume 32 Issue 2

Abstract

Iwasawa algebras of compact p-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of p-adic Lie groups. We previously determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over ℤp which were uniform pro-p groups in the sense of Dixon, du Sautoy, Mann and Segal. In this paper, for prime p>n+1, we determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-p Iwahori subgroup of GLn⁢(ℤp) which is not, in general, a uniform pro-p group.

Keywords: Iwasawa algebra
MSC 2010: 11R23; 22E35; 22E50

Communicated by Freydoon Shahidi

References

[1] L. Clozel, Presentation of an Iwasawa algebra: The case of Γ1⁢SL⁢(2,ℤp), Doc. Math. 16 (2011), 545–559. 10.4171/dm/341Search in Google Scholar

[2] L. Clozel, Globally analytic p-adic representations of the pro-p-Iwahori subgroup of GL⁢(2) and base change, I: Iwasawa algebras and a base change map, Bull. Iranian Math. Soc. 43 (2017), no. 4, 55–76. Search in Google Scholar

[3] C. Cornut and J. Ray, Generators of the pro-p Iwahori and Galois representations, Int. J. Number Theory 14 (2018), no. 1, 37–53. 10.1142/S1793042118500045Search in Google Scholar

[4] M. Demazure and A. Grothendieck, Schémas en groupes (SGA 3). Tome III. Structure des schémas en groupes réductifs, Doc. Math. (Paris) 8, Société Mathématique de France, Paris, 2011. Search in Google Scholar

[5] J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic pro-p Groups, 2nd ed., Cambridge Stud. Adv. Math. 61, Cambridge University, Cambridge, 1999. 10.1017/CBO9780511470882Search in Google Scholar

[6] M. Emerton, Locally analytic vectors in representations of locally p-adic analytic groups, Mem. Amer. Math. Soc. 248 (2017), no. 1175, 1–158. 10.1090/memo/1175Search in Google Scholar

[7] P. Garrett, Buildings and Classical Groups, Chapman & Hall, London, 1997. 10.1007/978-94-011-5340-9Search in Google Scholar

[8] K. Iwasawa, On Γ-extensions of algebraic number fields, Bull. Amer. Math. Soc. 65 (1959), 183–226. 10.1090/S0002-9904-1959-10317-7Search in Google Scholar

[9] M. Lazard, Groupes analytiques p-adiques, Inst. Hautes Études Sci. Publ. Math. (1965), no. 26, 389–603. Search in Google Scholar

[10] R. Ollivier, Modules simples en caractéristique p de l’algèbre de Hecke du pro-p-Iwahori de GL3⁢(F), J. Algebra 304 (2006), no. 1, 1–38. 10.1016/j.jalgebra.2006.03.046Search in Google Scholar

[11] R. Ollivier, Le foncteur des invariants sous l’action du pro-p-Iwahori de GL2⁢(F), J. Reine Angew. Math. 635 (2009), 149–185. 10.1515/CRELLE.2009.078Search in Google Scholar

[12] R. Ollivier and P. Schneider, The modular pro-p Iwahori–Hecke Ext-algebra, Representations of Reductive Groups, Proc. Sympos. Pure Math. 101, American Mathematical Society, Providence (2019), 255–308. 10.1090/pspum/101/11Search in Google Scholar

[13] J. Ray, Explicit ring-theoretic presentation of Iwasawa algebras, C. R. Math. Acad. Sci. Paris 356 (2018), no. 11–12, 1075–1080. 10.1016/j.crma.2018.09.002Search in Google Scholar

[14] J. Ray, Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over ℤp, J. Algebra 511 (2018), 405–419. 10.1016/j.jalgebra.2018.05.039Search in Google Scholar

[15] P. Schneider, p-adic Lie Groups, Grundlehren Math. Wiss. 344, Springer, Heidelberg, 2011. 10.1007/978-3-642-21147-8Search in Google Scholar

[16] P. Schneider and U. Stuhler, Representation theory and sheaves on the Bruhat–Tits building, Publ. Math. Inst. Hautes Études Sci. 85 (1997), 97–191. 10.1007/BF02699536Search in Google Scholar

[17] P. Schneider and J. Teitelbaum, Banach space representations and Iwasawa theory, Israel J. Math. 127 (2002), 359–380. 10.1007/BF02784538Search in Google Scholar

[18] P. Schneider and J. Teitelbaum, Algebras of p-adic distributions and admissible representations, Invent. Math. 153 (2003), no. 1, 145–196. 10.1007/s00222-002-0284-1Search in Google Scholar

[19] R. Steinberg, Lectures on Chevalley Groups, Yale University, New Haven, 1968. Search in Google Scholar

[20] J. Tits, Reductive groups over local fields, Automorphic Forms, Representations and L-Functions. Part I (Corvallis 1977), Proc. Sympos. Pure Math. 23, American Mathematical Society, Providence (1979), 29–69. 10.1090/pspum/033.1/546588Search in Google Scholar

[21] M.-F. Vigneras, The pro-p-Iwahori Hecke algebra of a reductive p-adic group III (spherical Hecke algebras and supersingular modules), J. Inst. Math. Jussieu 16 (2017), no. 3, 571–608. 10.1017/S1474748015000146Search in Google Scholar

[22] Rabindranath Tagore, Gitabitan Part I, Viswa-Bharati Granthana Vibhaga, (1979) Aug-Sep, page 50, song 109. Search in Google Scholar

[23] Rabindranath Tagore, Prano Bhoriye Trisha Horiye, song written (3rd June 1912) during a voyage on the Red Sea enroute to Europe. English translation: http://www.geetabitan.com/lyrics/rs-p/pran-bhoriye-trisa-hariye-english-translation.html. Search in Google Scholar

Received: 2019-09-19
Published Online: 2019-11-08
Published in Print: 2020-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Operational rules and generalized special polynomials
  3. Linearity of some low-complexity mapping class groups
  4. Ultramatricial algebras over commutative chain semirings and application to MV-algebras
  5. On a group analogue of the Heyde theorem
  6. Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of SLn(ℤp)
  7. Generalized Riesz potentials of functions in Morrey spaces L(1,ϕ;κ)(G) over non-doubling measure spaces
  8. On rough singular integrals along real-analytic submanifolds
  9. Dimension-free estimates for the vector-valued variational operators
  10. Upper bound of multiplicity in prime characteristic
  11. Asymptotic behaviour for elliptic operators with second-order discontinuous coefficients
  12. Irreducible and permutative representations of ultragraph Leavitt path algebras
  13. Twisted tensor products of φ-coordinated modules for nonlocal vertex algebras
  14. Explicit decomposition theorem for special Schubert varieties
  15. On the Hirzebruch–Kobayashi–Ono proportionality principle and the non-existence of compact solvable Clifford–Klein forms of certain homogeneous spaces
  16. The Lie group of vertical bisections of a regular Lie groupoid
  17. Chain conditions for graph C*-algebras
  18. Congruences in Hermitian Jacobi and Hermitian modular forms
  19. The Jensen–Pólya program for various L-functions
https://doi.org/10.1515/forum-2019-0260
Keywords for this article
Iwasawa algebra

Articles in the same Issue

  1. Frontmatter
  2. Operational rules and generalized special polynomials
  3. Linearity of some low-complexity mapping class groups
  4. Ultramatricial algebras over commutative chain semirings and application to MV-algebras
  5. On a group analogue of the Heyde theorem
  6. Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of SLn(ℤp)
  7. Generalized Riesz potentials of functions in Morrey spaces L(1,ϕ;κ)(G) over non-doubling measure spaces
  8. On rough singular integrals along real-analytic submanifolds
  9. Dimension-free estimates for the vector-valued variational operators
  10. Upper bound of multiplicity in prime characteristic
  11. Asymptotic behaviour for elliptic operators with second-order discontinuous coefficients
  12. Irreducible and permutative representations of ultragraph Leavitt path algebras
  13. Twisted tensor products of φ-coordinated modules for nonlocal vertex algebras
  14. Explicit decomposition theorem for special Schubert varieties
  15. On the Hirzebruch–Kobayashi–Ono proportionality principle and the non-existence of compact solvable Clifford–Klein forms of certain homogeneous spaces
  16. The Lie group of vertical bisections of a regular Lie groupoid
  17. Chain conditions for graph C*-algebras
  18. Congruences in Hermitian Jacobi and Hermitian modular forms
  19. The Jensen–Pólya program for various L-functions
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 3.2.2026 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2019-0260/pdf
Scroll to top button