Duke Mathematical Journal

Representation zeta functions of compact p-adic analytic groups and arithmetic groups

Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of p-adic analytic pro-p groups obtained from a global, “perfect” Lie lattice satisfy functional equations. In the case of “semisimple” compact p-adic analytic groups, we exhibit a link between the relevant p-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by the centralizer dimension.

Based on this algebro-geometric description, we compute explicit formulas for the representation zeta functions of principal congruence subgroups of the groups SL3(o), where o is a compact discrete valuation ring of characteristic 0, and of the groups SU3(O,o), where O is an unramified quadratic extension of o. These formulas, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type A2. Assuming a conjecture of Serre on the congruence subgroup problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type A2 defined over number fields.

Article information

Source
Duke Math. J., Volume 162, Number 1 (2013), 111-197.

Dates
First available in Project Euclid: 14 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1358172076

Digital Object Identifier
doi:10.1215/00127094-1959198

Mathematical Reviews number (MathSciNet)
MR3011874

Zentralblatt MATH identifier
1281.22005

Subjects
Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05] 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings [See also 20G05] 20F69: Asymptotic properties of groups
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 20C15: Ordinary representations and characters 20G25: Linear algebraic groups over local fields and their integers

Citation

Avni, Nir; Klopsch, Benjamin; Onn, Uri; Voll, Christopher. Representation zeta functions of compact $p$ -adic analytic groups and arithmetic groups. Duke Math. J. 162 (2013), no. 1, 111--197. doi:10.1215/00127094-1959198. https://projecteuclid.org/euclid.dmj/1358172076


Export citation

References

  • [1] N. Avni, Arithmetic groups have rational representation growth, Ann. of Math. (2) 174 (2011), 1009–1056.
  • [2] N. Avni, B. Klopsch, U. Onn, and C. Voll, On representation zeta functions of groups and a conjecture of Larsen–Lubotzky, C. R. Math. Acad. Sci. Paris 348 (2010), 363–367.
  • [3] N. Avni, B. Klopsch, U. Onn, C. Voll, “Representation zeta functions of some compact $p$-adic analytic groups” in Zeta functions in algebra and geometry, Contemp. Math. 566, Amer. Math. Soc., Providence, 2012, 295–330.
  • [4] N. Avni, B. Klopsch, U. Onn, C. Voll, Representation zeta functions for $\operatorname{SL}_{3}$ and $\operatorname{SU}_{3}$, preprint, 2012.
  • [5] H. Bass, A. Lubotzky, A. R. Magid, and S. Mozes, “The proalgebraic completion of rigid groups” in Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), Geom. Dedicata 95, Springer, Dordrecht, 2002, 19–58.
  • [6] W. Borho, Über Schichten halbeinfacher Lie-Algebren, Invent. Math. 65 (1981/82), 283–317.
  • [7] J. Brill, On the minors of a skew-symmetrical determinant, Proc. London Math. Soc. 34 (1904), 103–111.
  • [8] A. Broer, Decomposition varieties in semisimple Lie algebras, Canad. J. Math. 50 (1998), 929–971.
  • [9] C. J. Bushnell and P. C. Kutzko, The Admissible Dual of $\mathrm{GL}(N)$ via Compact Open Subgroups, Ann. of Math. Stud. 129, Princeton Univ. Press, Princeton, 1993.
  • [10] J. W. S. Cassels and A. Fröhlich, eds., Algebraic Number Theory, 2nd revised ed., London Math. Soc., London, 2010.
  • [11] J. Denef, On the degree of Igusa’s local zeta function, Amer. J. Math. 109 (1987), 991–1008.
  • [12] J. Denef, Report on Igusa’s local zeta function, Astérisque 201-203 (1991), 359–386, Séminaire Bourbaki 1990/91, no. 741.
  • [13] J. Denef and D. Meuser, A functional equation of Igusa’s local zeta function, Amer. J. Math. 113 (1991), 1135–1152.
  • [14] J. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic Pro-$p$ Groups, 2nd ed., Cambridge Stud. Adv. Math. 61, Cambridge Univ. Press, Cambridge, 1999.
  • [15] M. du Sautoy and F. Grunewald, Analytic properties of zeta functions and subgroup growth, Ann. of Math. (2) 152 (2000), 793–833.
  • [16] M. du Sautoy, F. Grunewald, “Zeta functions of groups and rings” in International Congress of Mathematicians, Vol. II, Eur. Math. Soc., Zürich, 2006, 131–149.
  • [17] V. Ennola, On the conjugacy classes of the finite unitary groups, Ann. Acad. Sci. Fenn. Ser. A I 313 (1962), 13 pp.
  • [18] J. González-Sánchez, On $p$-saturable groups, J. Algebra 315 (2007), 809–823.
  • [19] J. González-Sánchez, Kirillov’s orbit method for $p$-groups and pro-$p$ groups, Comm. Algebra 37 (2009), 4476–4488.
  • [20] J. González-Sánchez and B. Klopsch, Analytic pro-$p$ groups of small dimensions, J. Group Theory 12 (2009), 711–734.
  • [21] M. J. Greenberg, Schemata over local rings, Ann. of Math. (2) 73 (1961), 624–648.
  • [22] B. H. Gross and G. Nebe, Globally maximal arithmetic groups, J. Algebra 272 (2004), 625–642.
  • [23] P. Heymans, Pfaffians and skew-symmetric matrices, Proc. London Math. Soc. (3) 19 (1969), 730–768.
  • [24] R. E. Howe, On representations of discrete, finitely generated, torsion-free, nilpotent groups, Pacific J. Math. 73 (1977), 281–305.
  • [25] R. E. Howe, Kirillov theory for compact $p$-adic groups, Pacific J. Math. 73 (1977), 365–381.
  • [26] E. Hrushovski and B. Martin, Zeta functions from definable equivalence relations, preprint, arXiv:math/0701011v1 [math.LO].
  • [27] B. Huppert, Character Theory of Finite Groups, de Gruyter Exp. Math. 25, de Gruyter, Berlin, 1998.
  • [28] J. Igusa, An Introduction to the Theory of Local Zeta Functions, AMS/IP Stud. Adv. Math. 14, Amer. Math. Soc., Providence, 2000.
  • [29] I. M. Isaacs, Counting characters of upper triangular groups, J. Algebra 315 (2007), 698–719.
  • [30] A. Jaikin-Zapirain, Zeta function of representations of compact $p$-adic analytic groups, J. Amer. Math. Soc. 19 (2006), 91–118.
  • [31] N. Kawanaka, “Generalized Gel’fand-Graev representations and Ennola duality” in Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math. 6, North-Holland, Amsterdam, 1985, 175–206.
  • [32] H. D. Kloosterman, The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups. I, Ann. of Math. (2) 47 (1946), 317–375; II, 376–447.
  • [33] B. Klopsch, On the Lie theory of $p$-adic analytic groups, Math. Z. 249 (2005), 713–730.
  • [34] M. Larsen and A. Lubotzky, Representation growth of linear groups, J. Eur. Math. Soc. (JEMS) 10 (2008), 351–390.
  • [35] M. Lazard, Groupes analytiques $p$-adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389–603.
  • [36] M. W. Liebeck and A. Shalev, Character degrees and random walks in finite groups of Lie type, Proc. London Math. Soc. (3) 90 (2005), 61–86.
  • [37] A. Lubotzky and B. Martin, Polynomial representation growth and the congruence subgroup problem, Israel J. Math. 144 (2004), 293–316.
  • [38] A. Lubotzky and D. Segal, Subgroup Growth, Progr. Math. 212, Birkhäuser, Basel, 2003.
  • [39] F. Lübeck, Character degrees and their multiplicities for some groups of Lie type of rank $<9$, online data, http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/DegMult/index.html (accessed 10 October 2012).
  • [40] D. Meuser, The meromorphic continuation of a zeta function of Weil and Igusa type, Invent. Math. 85 (1986), 493–514.
  • [41] A. Moretó, “Characters of $p$-groups and Sylow $p$-subgroups” in Groups St. Andrews 2001 in Oxford, Vol. II, London Math. Soc. Lecture Note Ser. 305, Cambridge Univ. Press, Cambridge, 2003, 412–421.
  • [42] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure Appl. Math. 139, Academic Press, Boston, 1994.
  • [43] I. Reiner, Maximal Orders, London Math. Soc. Monogr. Ser. 5, Academic Press, London, 1975.
  • [44] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren Math. Wiss. 270, Springer, Berlin, 1985.
  • [45] J.-P. Serre, Le problème des groupes de congruence pour $\operatorname{SL}_{2}$, Ann. of Math. (2) 92 (1970), 489–527.
  • [46] J.-P. Serre, Local Fields, Grad. Texts in Math. 67, Springer, New York, 1979.
  • [47] R. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Progr. Math. 41, Birkhäuser Boston, Boston, 1996.
  • [48] R. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge Stud. Adv. Math. 49, Cambridge Univ. Press, Cambridge, 1997.
  • [49] W. Veys and W. A. Zúñiga-Galindo, Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra, Trans. Amer. Math. Soc. 360 (2008), no. 4, 2205–2227.
  • [50] C. Voll, Functional equations for zeta functions of groups and rings, Ann. of Math. (2) 172 (2010), 1181–1218.
  • [51] H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton Univ. Press, Princeton, 1997.