Zeta functions associated to admissible representations of compact $p$-adic Lie groups
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- by Steffen Kionke and Benjamin Klopsch PDF
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Abstract:
Let $G$ be a profinite group. A strongly admissible smooth representation $\varrho$ of $G$ over ${\mathbb {C}}$ decomposes as a direct sum $\varrho \cong \bigoplus _{\pi \in \text {Irr}(G)} m_\pi (\varrho ) \pi$ of irreducible representations with finite multiplicities $m_\pi (\varrho )$ such that, for every positive integer $n$, the number $r_n(\varrho )$ of irreducible constituents of dimension $n$ is finite. Examples arise naturally in the representation theory of reductive groups over nonarchimedean local fields. In this article we initiate an investigation of the Dirichlet generating function \[ \zeta _\varrho (s) = \sum \nolimits _{n=1}^\infty r_n(\varrho ) n^{-s} = \sum \nolimits _{\pi \in \text {Irr}(G)} \frac {m_\pi (\varrho )}{(\dim \pi )^s} \] associated to such a representation $\varrho$.
Our primary focus is on representations $\varrho = \text {Ind}_H^G(\sigma )$ of compact $p$-adic Lie groups $G$ that arise from finite-dimensional representations $\sigma$ of closed subgroups $H$ via the induction functor. In addition to a series of foundational results—including a description in terms of $p$-adic integrals—we establish rationality results and functional equations for zeta functions of globally defined families of induced representations of potent pro-$p$ groups. A key ingredient of our proof is Hironaka’s resolution of singularities, which yields formulae of Denef type for the relevant zeta functions.
In some detail we consider representations of open compact subgroups of reductive $p$-adic groups that are induced from parabolic subgroups. Explicit computations are carried out by means of complementing techniques: (i) geometric methods that are applicable via distance-transitive actions on spherically homogeneous rooted trees, and (ii) the $p$-adic Kirillov orbit method. Approach (i) is closely related to the notion of Gelfand pairs and works equally well in positive defining characteristic.
References
- Avraham Aizenbud and Nir Avni, Representation growth and rational singularities of the moduli space of local systems, Invent. Math. 204 (2016), no. 1, 245–316. MR 3480557, DOI 10.1007/s00222-015-0614-8
- Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, Representation zeta functions of some compact $p$-adic analytic groups, Zeta functions in algebra and geometry, Contemp. Math., vol. 566, Amer. Math. Soc., Providence, RI, 2012, pp. 295–330. MR 2858928, DOI 10.1090/conm/566/11226
- Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, Representation zeta functions of compact $p$-adic analytic groups and arithmetic groups, Duke Math. J. 162 (2013), no. 1, 111–197. MR 3011874, DOI 10.1215/00127094-1959198
- Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, Similarity classes of integral $\mathfrak {p}$-adic matrices and representation zeta functions of groups of type $\mathsf {A}_2$, Proc. Lond. Math. Soc. (3) 112 (2016), no. 2, 267–350. MR 3471251, DOI 10.1112/plms/pdv071
- Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, Arithmetic groups, base change, and representation growth, Geom. Funct. Anal. 26 (2016), no. 1, 67–135. MR 3494486, DOI 10.1007/s00039-016-0359-6
- Laurent Bartholdi and Rostislav I. Grigorchuk, On parabolic subgroups and Hecke algebras of some fractal groups, Serdica Math. J. 28 (2002), no. 1, 47–90. MR 1899368
- Hyman Bass, Alexander Lubotzky, Andy R. Magid, and Shahar Mozes, The proalgebraic completion of rigid groups, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), 2002, pp. 19–58. MR 1950883, DOI 10.1023/A:1021221727311
- Victor V. Batyrev, Birational Calabi-Yau $n$-folds have equal Betti numbers, New trends in algebraic geometry (Warwick, 1996) London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 1–11. MR 1714818, DOI 10.1017/CBO9780511721540.002
- M. Bachir Bekka and Pierre de la Harpe, Irreducibility of unitary group representations and reproducing kernels Hilbert spaces, Expo. Math. 21 (2003), no. 2, 115–149. Appendix by the authors in collaboration with Rostislav Grigorchuk. MR 1978060, DOI 10.1016/S0723-0869(03)80014-2
- Corinne Blondel, Quelques propriétés des paires couvrantes, Math. Ann. 331 (2005), no. 2, 243–257 (French). MR 2115455, DOI 10.1007/s00208-004-0579-1
- Daniel Bump, Lie groups, Graduate Texts in Mathematics, vol. 225, Springer-Verlag, New York, 2004. MR 2062813, DOI 10.1007/978-1-4757-4094-3
- Colin J. Bushnell and Guy Henniart, The local Langlands conjecture for $\rm GL(2)$, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Springer-Verlag, Berlin, 2006. MR 2234120, DOI 10.1007/3-540-31511-X
- Peter S. Campbell and Monica Nevins, Branching rules for unramified principal series representations of $\textrm {GL}(3)$ over a $p$-adic field, J. Algebra 321 (2009), no. 9, 2422–2444. MR 2504482, DOI 10.1016/j.jalgebra.2009.01.013
- William Casselman, The restriction of a representation of $\textrm {GL}_{2}(k)$ to $\textrm {GL}_{2}({\mathfrak {o}})$, Math. Ann. 206 (1973), 311–318. MR 338274, DOI 10.1007/BF01355984
- Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1013113
- V. I. Danilov, Cohomology of algebraic varieties [ MR1060324 (91f:14016)], Algebraic geometry, II, Encyclopaedia Math. Sci., vol. 35, Springer, Berlin, 1996, pp. 1–125, 255–262. MR 1392958, DOI 10.1007/978-3-642-60925-1_{1}
- J. Denef, On the degree of Igusa’s local zeta function, Amer. J. Math. 109 (1987), no. 6, 991–1008. MR 919001, DOI 10.2307/2374583
- Jan Denef and Diane Meuser, A functional equation of Igusa’s local zeta function, Amer. J. Math. 113 (1991), no. 6, 1135–1152. MR 1137535, DOI 10.2307/2374901
- J. D. Dixon, The structure of linear groups, Van Nostrand Reinhold Co., London, 1971.
- J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$ groups, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999. MR 1720368, DOI 10.1017/CBO9780511470882
- Duong H. Dung and Christopher Voll, Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6327–6349. MR 3660223, DOI 10.1090/tran/6879
- Jon González-Sánchez, On $p$-saturable groups, J. Algebra 315 (2007), no. 2, 809–823. MR 2351895, DOI 10.1016/j.jalgebra.2007.02.005
- Jon González-Sánchez, Kirillov’s orbit method for $p$-groups and pro-$p$ groups, Comm. Algebra 37 (2009), no. 12, 4476–4488. MR 2588861, DOI 10.1080/00927870802545679
- J. González-Sánchez and A. Jaikin-Zapirain, On the structure of normal subgroups of potent $p$-groups, J. Algebra 276 (2004), no. 1, 193–209. MR 2054394, DOI 10.1016/j.jalgebra.2003.12.006
- Jon González-Sánchez, Andrei Jaikin-Zapirain, and Benjamin Klopsch, The representation zeta function of a FAb compact $p$-adic Lie group vanishes at $-2$, Bull. Lond. Math. Soc. 46 (2014), no. 2, 239–244. MR 3194743, DOI 10.1112/blms/bdt090
- Jon González-Sánchez and Benjamin Klopsch, Analytic pro-$p$ groups of small dimensions, J. Group Theory 12 (2009), no. 5, 711–734. MR 2554763, DOI 10.1515/JGT.2009.006
- Benedict H. Gross, Some applications of Gel′fand pairs to number theory, Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 2, 277–301. MR 1074028, DOI 10.1090/S0273-0979-1991-16017-9
- G. Harder, Eisenstein cohomology of arithmetic groups. The case $\textrm {GL}_2$, Invent. Math. 89 (1987), no. 1, 37–118. MR 892187, DOI 10.1007/BF01404673
- Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- Guy Henniart, Une preuve simple des conjectures de Langlands pour $\textrm {GL}(n)$ sur un corps $p$-adique, Invent. Math. 139 (2000), no. 2, 439–455 (French, with English summary). MR 1738446, DOI 10.1007/s002220050012
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
- Ehud Hrushovski, Ben Martin, and Silvain Rideau, Definable equivalence relations and zeta functions of groups, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 10, 2467–2537. With an appendix by Raf Cluckers. MR 3852185, DOI 10.4171/JEMS/817
- I. Martin Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR 1280461
- A. Jaikin-Zapirain, Zeta function of representations of compact $p$-adic analytic groups, J. Amer. Math. Soc. 19 (2006), no. 1, 91–118. MR 2169043, DOI 10.1090/S0894-0347-05-00501-1
- S. Kionke, Groups acting on rooted trees and their representations on the boundary, J. Algebra (2019), DOI 10.1016/j.jalgebra.2019.03.023.
- Benjamin Klopsch, On the Lie theory of $p$-adic analytic groups, Math. Z. 249 (2005), no. 4, 713–730. MR 2126210, DOI 10.1007/s00209-004-0717-1
- Benjamin Klopsch, Representation growth and representation zeta functions of groups, Note Mat. 33 (2013), no. 1, 107–120. MR 3071315, DOI 10.1285/i15900932v33n1p107
- János Kollár, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR 2289519
- Yasushi Komori, Kohji Matsumoto, and Hirofumi Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras II, J. Math. Soc. Japan 62 (2010), no. 2, 355–394. MR 2662849
- Michael Larsen and Alexander Lubotzky, Representation growth of linear groups, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 351–390. MR 2390327, DOI 10.4171/JEMS/113
- Alexander Lubotzky and Benjamin Martin, Polynomial representation growth and the congruence subgroup problem, Israel J. Math. 144 (2004), 293–316. MR 2121543, DOI 10.1007/BF02916715
- S. Mandelbrojt, Dirichlet series, D. Reidel Publishing Co., Dordrecht, 1972. Principles and methods. MR 0435370
- Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
- Monica Nevins, On branching rules of depth-zero representations, J. Algebra 408 (2014), 1–27. MR 3197168, DOI 10.1016/j.jalgebra.2014.03.016
- Uri Onn and Pooja Singla, On the unramified principal series of $\mathrm {GL}(3)$ over non-Archimedean local fields, J. Algebra 397 (2014), 1–17. MR 3119211, DOI 10.1016/j.jalgebra.2013.08.022
- I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original; With a foreword by M. J. Taylor. MR 1972204
- David Renard, Représentations des groupes réductifs $p$-adiques, Cours Spécialisés [Specialized Courses], vol. 17, Société Mathématique de France, Paris, 2010 (French). MR 2567785
- Tobias Rossmann, Topological representation zeta functions of unipotent groups, J. Algebra 448 (2016), 210–237. MR 3438311, DOI 10.1016/j.jalgebra.2015.09.050
- Tobias Rossmann, Computing local zeta functions of groups, algebras, and modules, Trans. Amer. Math. Soc. 370 (2018), no. 7, 4841–4879. MR 3812098, DOI 10.1090/tran/7361
- Peter Sarnak and Scot Adams, Betti numbers of congruence groups, Israel J. Math. 88 (1994), no. 1-3, 31–72. With an appendix by Ze’ev Rudnick. MR 1303490, DOI 10.1007/BF02937506
- J.-P. Serre, Local fields, Springer-Verlag, New York–Heidelberg, 1979.
- J.-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 2002.
- Richard P. Stanley, Combinatorial reciprocity theorems, Advances in Math. 14 (1974), 194–253. MR 411982, DOI 10.1016/0001-8708(74)90030-9
- R. P. Stanley, Combinatorics and commutative algebra, 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 1996.
- A. Stasinski and C. Voll, Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type $B$, Amer. J. Math. 136 (2014), no. 2, 501–550. MR 3188068, DOI 10.1353/ajm.2014.0010
- Marie-France Vignéras, Représentations $l$-modulaires d’un groupe réductif $p$-adique avec $l\ne p$, Progress in Mathematics, vol. 137, Birkhäuser Boston, Inc., Boston, MA, 1996 (French, with English summary). MR 1395151
- Christopher Voll, Functional equations for zeta functions of groups and rings, Ann. of Math. (2) 172 (2010), no. 2, 1181–1218. MR 2680489, DOI 10.4007/annals.2010.172.1185
- Christopher Voll, Zeta functions of groups and rings—recent developments, Groups St Andrews 2013, London Math. Soc. Lecture Note Ser., vol. 422, Cambridge Univ. Press, Cambridge, 2015, pp. 469–492. MR 3495675
- André Weil, Adeles and algebraic groups, Progress in Mathematics, vol. 23, Birkhäuser, Boston, Mass., 1982. With appendices by M. Demazure and Takashi Ono. MR 670072, DOI 10.1007/978-1-4684-9156-2
- Edward Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), no. 1, 153–209. MR 1133264, DOI 10.1007/BF02100009
- M. Zordan, Poincaré series of Lie lattices and representation zeta functions of arithmetic groups, arXiv:1704.04165 (2017).
Additional Information
- Steffen Kionke
- Affiliation: Karlsruhe Institute of Technology, Institute for Algebra and Geometry, Englerstrasse 2, 76131 Karlsruhe, Germany
- MR Author ID: 1070549
- Email: steffen.kionke@kit.edu
- Benjamin Klopsch
- Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstrasse 1, 40225 Düsseldorf, Germany
- MR Author ID: 658412
- Email: klopsch@math.uni-duesseldorf.de
- Received by editor(s): August 22, 2017
- Received by editor(s) in revised form: August 5, 2018, and January 14, 2019
- Published electronically: April 3, 2019
- Additional Notes: We acknowledge support by DFG grant KL 2162/1-1.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 7677-7733
- MSC (2010): Primary 20E18; Secondary 20C15, 20G25, 22E50, 11M41
- DOI: https://doi.org/10.1090/tran/7834
- MathSciNet review: 4029678