Algebra & Number Theory

$\ell$-modular representations of unramified $p$-adic U(2,1)

Robert Kurinczuk

Full-text: Open access


We construct all irreducible cuspidal -modular representations of a unitary group in three variables attached to an unramified extension of local fields of odd residual characteristic p with p. We describe the -modular principal series and show that the supercuspidal support of an irreducible -modular representation is unique up to conjugacy.

Article information

Algebra Number Theory, Volume 8, Number 8 (2014), 1801-1838.

Received: 21 October 2013
Revised: 24 June 2014
Accepted: 8 September 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]

representations of p-adic groups modular representations


Kurinczuk, Robert. $\ell$-modular representations of unramified $p$-adic U(2,1). Algebra Number Theory 8 (2014), no. 8, 1801--1838. doi:10.2140/ant.2014.8.1801.

Export citation


  • L. Blasco, “Description du dual admissible de ${\rm U}(2,1)(F)$ par la théorie des types de C. Bushnell et P. Kutzko”, Manuscripta Math. 107:2 (2002), 151–186.
  • C. Blondel, “Quelques propriétés des paires couvrantes”, Math. Ann. 331:2 (2005), 243–257.
  • C. Blondel, “Représentation de Weil et $\beta$-extensions”, Ann. Inst. Fourier $($Grenoble$)$ 62:4 (2012), 1319–1366.
  • C. Bonnafé and R. Rouquier, “Catégories dérivées et variétés de Deligne–Lusztig”, Publ. Math. Inst. Hautes Études Sci. 97 (2003), 1–59. 1054.20024
  • C. J. Bushnell and P. C. Kutzko, The admissible dual of ${\rm GL}(N)$ via compact open subgroups, Annals of Mathematics Studies 129, Princeton University Press, 1993.
  • C. J. Bushnell and P. C. Kutzko, “The admissible dual of ${\rm SL}(N)$, I”, Ann. Sci. École Norm. Sup. $(4)$ 26:2 (1993), 261–280.
  • C. J. Bushnell and P. C. Kutzko, “Smooth representations of reductive $p$-adic groups: Structure theory via types”, Proc. London Math. Soc. $(3)$ 77:3 (1998), 582–634.
  • J.-F. Dat, “$v$-tempered representations of $p$-adic groups, I: $l$-adic case”, Duke Math. J. 126:3 (2005), 397–469.
  • J.-F. Dat, “Finitude pour les représentations lisses de groupes $p$-adiques”, J. Inst. Math. Jussieu 8:2 (2009), 261–333.
  • F. Digne and J. Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts 21, Cambridge University Press, 1991.
  • V. Ennola, “On the characters of the finite unitary groups”, Ann. Acad. Sci. Fenn. Ser. A I No. 323 (1963), 35.
  • M. Geck, “Irreducible Brauer characters of the $3$-dimensional special unitary groups in nondefining characteristic”, Comm. Algebra 18:2 (1990), 563–584.
  • M. Geck, G. Hiss, and G. Malle, “Towards a classification of the irreducible representations in non-describing characteristic of a finite group of Lie type”, Math. Z. 221:3 (1996), 353–386.
  • G. Henniart and V. Sécherre, “Types et contragrédientes”, Canad. J. Math. 66:6 (2014), 1287–1304.
  • G. Hiss, “Hermitian function fields, classical unitals, and representations of 3-dimensional unitary groups”, Indag. Math. $($N.S.$)$ 15:2 (2004), 223–243.
  • R. B. Howlett and G. I. Lehrer, “Induced cuspidal representations and generalised Hecke rings”, Invent. Math. 58:1 (1980), 37–64.
  • D. Keys, “Principal series representations of special unitary groups over local fields”, Compositio Math. 51:1 (1984), 115–130.
  • J.-L. Kim, “Supercuspidal representations: An exhaustion theorem”, J. Amer. Math. Soc. 20:2 (2007), 273–320.
  • R. Kurinczuk and S. Stevens, “$\ell$-modular cuspidal representations of classical $p$-adic groups”, preprint, 2014.
  • A. Mínguez and V. Sécherre, “Représentations lisses modulo $\ell$ de $\textrm{GL}\sb m(D)$”, Duke Math. J. 163:4 (2014), 795–887.
  • A. Mínguez and V. Sécherre, “Types modulo $\ell$ pour les formes intérieures de ${\rm GL}(n)$ sur un corps local non archimédien”, Proc. London Math. Soc. $(3)$ 109:4 (2014), 823–891.
  • L. Morris, “Tamely ramified intertwining algebras”, Invent. Math. 114:1 (1993), 1–54.
  • L. Morris, “Level zero $\bf G$-types”, Compositio Math. 118:2 (1999), 135–157. 2000g:22029
  • T. Okuyama and K. Waki, “Decomposition numbers of ${\rm SU}(3,q\sp 2)$”, J. Algebra 255:2 (2002), 258–270.
  • P. Schneider and E.-W. Zink, “$K$-types for the tempered components of a $p$-adic general linear group”, J. Reine Angew. Math. 517 (1999), 161–208.
  • V. Sécherre and S. Stevens, “Représentations lisses de ${\rm GL}\sb m(D)$, IV: Représentations supercuspidales”, J. Inst. Math. Jussieu 7:3 (2008), 527–574.
  • S. Stevens, “Semisimple characters for $p$-adic classical groups”, Duke Math. J. 127:1 (2005), 123–173.
  • S. Stevens, “The supercuspidal representations of $p$-adic classical groups”, Invent. Math. 172:2 (2008), 289–352.
  • M.-F. Vignéras, Représentations $l$-modulaires d'un groupe réductif $p$-adique avec $l\ne p$, Progress in Mathematics 137, Birkhäuser, Boston, 1996.
  • M.-F. Vignéras, “Induced $R$-representations of $p$-adic reductive groups”, Selecta Math. $($N.S.$)$ 4:4 (1998), 549–623.
  • M.-F. Vignéras, “Irreducible modular representations of a reductive $p$-adic group and simple modules for Hecke algebras”, pp. 117–133 in European Congress of Mathematics, I (Barcelona, 2000), edited by C. Casacuberta et al., Progr. Math. 201, Birkhäuser, Basel, 2001.
  • M.-F. Vignéras, “On highest Whittaker models and integral structures”, pp. 773–801 in Contributions to automorphic forms, geometry, and number theory, edited by H. Hida et al., Johns Hopkins Univ. Press, Baltimore, MD, 2004.
  • J.-K. Yu, “Construction of tame supercuspidal representations”, J. Amer. Math. Soc. 14:3 (2001), 579–622.