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LIMIT MULTIPLICITIES FOR PRINCIPAL CONGRUENCE SUBGROUPS OF $\text{GL}(n)$ AND $\text{SL}(n)$

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  08 May 2014

Tobias Finis ,
Erez Lapid  and
Werner Müller
Tobias Finis
Affiliation:
Freie Universität Berlin, Institut für Mathematik, Arnimallee 3, D-14195 Berlin, Germany (finis@math.fu-berlin.de)
Erez Lapid
Affiliation:
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel (erez.m.lapid@gmail.com)
Werner Müller
Affiliation:
Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany (mueller@math.uni-bonn.de)
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Abstract

We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups $\text{GL}(n)$ and $\text{SL}(n)$ we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur’s trace formula obtained in [Finis, Lapid, and Müller, Ann. of Math. (2) 174(1) (2011), 173–195; Finis and Lapid, Ann. of Math. (2) 174(1) (2011), 197–223], which allows us to show that for $\text{GL}(n)$ and $\text{SL}(n)$ the contribution of the continuous spectrum is negligible in the limit.

Keywords

limit multiplicitieslattices in Lie groupstrace formula

MSC classification

Primary: 11F72: Spectral theory; Selberg trace formula
Secondary: 22E40: Discrete subgroups of Lie groups 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 3 , July 2015 , pp. 589 - 638
Copyright
© Cambridge University Press 2014 

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References

Abert, M., Bergeron, N., Biringer, I., Gelander, T., Nikolov, N., Raimbault, J. and Samet, I., On the growth of $L^{2}$-invariants for sequences of lattices in Lie groups (arXiv:1210.2961).Google Scholar
Abert, M., Bergeron, N., Biringer, I., Gelander, T., Nikolov, N., Raimbault, J. and Samet, I., On the growth of Betti numbers of locally symmetric spaces, C. R. Math. Acad. Sci. Paris 349(15–16) (2011), 831–835, MR 2835886.Google Scholar
Arthur, J. G., A trace formula for reductive groups. I. Terms associated to classes in G (Q), Duke Math. J. 45(4) (1978), 911–952, MR 518111 (80d:10043).Google Scholar
Arthur, J., Eisenstein series and the trace formula, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, pp. 253–274 (Amer. Math. Soc., Providence, R.I., 1979), MR 546601 (81b:10020).Google Scholar
Arthur, J., A trace formula for reductive groups. II. Applications of a truncation operator, Compositio Math. 40(1) (1980), 87–121, MR 558260 (81b:22018).Google Scholar
Arthur, J., The trace formula in invariant form, Ann. of Math. (2) 114(1) (1981), 1–74, MR 625344 (84a:10031).Google Scholar
Arthur, J., On a family of distributions obtained from Eisenstein series. I. Application of the Paley-Wiener theorem, Amer. J. Math. 104(6) (1982), 1243–1288, MR 681737 (85k:22044).Google Scholar
Arthur, J., On a family of distributions obtained from Eisenstein series. II. Explicit formulas, Amer. J. Math. 104(6) (1982), 1289–1336, MR 681738 (85d:22033).CrossRefGoogle Scholar
Arthur, J., A measure on the unipotent variety, Canad. J. Math. 37(6) (1985), 1237–1274, MR 828844 (87m:22049).Google Scholar
Arthur, J., On a family of distributions obtained from orbits, Canad. J. Math. 38(1) (1986), 179–214, MR 835041 (87k:11058).Google Scholar
Arthur, J., Intertwining operators and residues. I. Weighted characters, J. Funct. Anal. 84(1) (1989), 19–84, MR 999488 (90j:22018).CrossRefGoogle Scholar
Arthur, J., The endoscopic classification of representations, American Mathematical Society Colloquium Publications, Volume 61 (American Mathematical Society, Providence, RI, 2013). Orthogonal and symplectic groups, MR 3135650.Google Scholar
Borwein a, P. and Erdélyi, T., Sharp extensions of Bernstein’s inequality to rational spaces, Mathematika 43(2) (1997), 413–423, MR 1433285 (97k:26014).Google Scholar
Borel, A. and Garland, H., Laplacian and the discrete spectrum of an arithmetic group, Amer. J. Math. 105(2) (1983), 309–335, MR 701563 (84k:58220).Google Scholar
Bushnell, C. J. and Henniart, G., An upper bound on conductors for pairs, J. Number Theory 65(2) (1997), 183–196, MR 1462836 (98h:11153).Google Scholar
Barbasch, D. and Moscovici, H., L 2-index and the Selberg trace formula, J. Funct. Anal. 53(2) (1983), 151–201, MR 722507 (85j:58137).Google Scholar
Borel, A. and Tits, J., Groupes réductifs, Inst. Hautes Études Sci. Publ. Math.(27) (1965), 55–150, MR 0207712 (34 #7527).Google Scholar
Borel, A. and Tits, J., Homomorphismes “abstraits” de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499–571, MR 0316587 (47 #5134).Google Scholar
Clozel, L. and Delorme, P., Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs, Invent. Math. 77(3) (1984), 427–453, MR 759263 (86b:22015).Google Scholar
Clozel, L. and Delorme, P., Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs. II, Ann. Sci. École Norm. Sup. (4) 23(2) (1990), 193–228, MR 1046496 (91g:22013).Google Scholar
Chao, K.F. and Li, W.-W., Dual R-groups of the inner forms of SL (N), Pacific J. Math. 267(1) (2014), 35–90, MR 3163476.Google Scholar
Clozel, L., On limit multiplicities of discrete series representations in spaces of automorphic forms, Invent. Math. 83(2) (1986), 265–284, MR 818353 (87g:22012).Google Scholar
Clozel, L., On the cohomology of Kottwitz’s arithmetic varieties, Duke Math. J. 72(3) (1993), 757–795, MR 1253624 (95b:11055).Google Scholar
Collingwood, D. H. and McGovern, W. M., Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series (Van Nostrand Reinhold Co., New York, 1993), MR 1251060 (94j:17001).Google Scholar
DeGeorge, D. L., On a theorem of Osborne and Warner. Multiplicities in the cuspidal spectrum, J. Funct. Anal. 48(1) (1982), 81–94, MR 671316 (84h:22027).Google Scholar
Delorme, P., Formules limites et formules asymptotiques pour les multiplicités dans L 2(G∕Γ), Duke Math. J. 53(3) (1986), 691–731, MR 860667 (87m:22040).Google Scholar
de George, D. L. and Wallach, N. R., Limit formulas for multiplicities in L 2(Γ∖G), Ann. of Math. (2) 107(1) (1978), 133–150, MR 0492077 (58 #11231).Google Scholar
Deitmar, A. and Hoffman, W., Spectral estimates for towers of noncompact quotients, Canad. J. Math. 51(2) (1999), 266–293, MR 1697144 (2000f:11061).Google Scholar
Deitmar, A. and Hoffmann, W., On limit multiplicities for spaces of automorphic forms, Canad. J. Math. 51(5) (1999), 952–976, MR 1718672 (2001f:11080).Google Scholar
Dixmier, J., Les $C^{\ast }$-algèbres et leurs représentations, Deuxième édition. Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969. MR 0246136 (39 #7442).Google Scholar
Duistermaat, J. J., Kolk, J. A. C. and Varadarajan, V. S., Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math. 52(1) (1979), 27–93, MR 532745 (82a:58050a).CrossRefGoogle Scholar
DeGeorge, D. L. and Wallach, N. R., Limit formulas for multiplicities in L 2(Γ∖G). II. The tempered spectrum, Ann. of Math. (2) 109(3) (1979), 477–495, MR 534759 (81g:22010).Google Scholar
Finis, T. and Lapid, E., On the spectral side of Arthur’s trace formula—combinatorial setup, Ann. of Math. (2) 174(1) (2011), 197–223, MR 2811598.Google Scholar
Finis, T., Lapid, E. and Müller, W., On the spectral side of Arthur’s trace formula—absolute convergence, Ann. of Math. (2) 174(1) (2011), 173–195, MR 2811597.Google Scholar
Finis, T., Lapid, E. and Müller, W., On the degrees of matrix coefficients of intertwining operators, Pacific J. Math. 260(2) (2012), 433–456, MR 3001800.Google Scholar
Gelbart, S. and Jacquet, H., Forms of GL(2) from the analytic point of view, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977, Part 1, Proc. Sympos. Pure Math., XXXIII, pp. 213–251 (Amer. Math. Soc., Providence, R.I., 1979), MR 546600 (81e:10024).Google Scholar
Gelbart, S. and Shahidi, F., Boundedness of automorphic L-functions in vertical strips, J. Amer. Math. Soc. 14(1) (2001), 79–107, (electronic), MR 1800349 (2003a:11056).CrossRefGoogle Scholar
Hiraga, K. and Saito, H., On L-packets for inner forms of SL n, Mem. Amer. Math. Soc. 215 (2012), 1013, vi+97, MR 2918491.Google Scholar
Iwaniec, H., Nonholomorphic modular forms and their applications, in Modular forms (Durham, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., pp. 157–196 (Horwood, Chichester, 1984), MR 803367 (87f:11030).Google Scholar
Iwaniec, H., Small eigenvalues of Laplacian for Γ0(N), Acta Arith. 56(1) (1990), 65–82, MR 1067982 (92h:11045).Google Scholar
Jacquet, H., Principal L-functions of the linear group, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Part 2, Proc. Sympos. Pure Math., XXXIII, pp. 63–86 (Amer. Math. Soc., Providence, R.I., 1979), MR 546609 (81f:22029).Google Scholar
Jacquet, H., A correction to conducteur des représentations du groupe linéaire [mr620708], Pacific J. Math. 260(2) (2012), 515–525, MR 3001803.Google Scholar
Jacquet, H. and Langlands, R. P., Automorphic forms on GL(2), Lecture Notes in Mathematics, 114, (Springer-Verlag, Berlin, 1970), MR 0401654 (53 #5481).Google Scholar
Jacquet, H., Piatetski-Shapiro, I. I. and Shalika, J., Conducteur des représentations du groupe linéaire, Math. Ann. 256(2) (1981), 199–214, MR 620708 (83c:22025).Google Scholar
Kim, H. H. and Shahidi, F., Symmetric cube L-functions for GL2 are entire, Ann. of Math. (2) 150(2) (1999), 645–662, MR 1726704 (2000k:11065).CrossRefGoogle Scholar
Knapp, A. W. and Zuckerman, G. J., Classification of irreducible tempered representations of semisimple groups, Ann. of Math. (2) 116(2) (1982), 389–455, MR 672840 (84h:22034a).CrossRefGoogle Scholar
Knapp, A. W. and Zuckerman, G. J., Classification of irreducible tempered representations of semisimple groups. II, Ann. of Math. (2) 116(3) (1982), 457–501, MR 678478 (84h:22034b).Google Scholar
Langlands, R. P., Euler Products (Yale University Press, New Haven, Conn., 1971). A James K. Whittemore Lecture in Mathematics given at Yale University, 1967, Yale Mathematical Monographs, 1, MR 0419366 (54 #7387).Google Scholar
Lang, S., Algebraic Number Theory, 2nd ed., Graduate Texts in Mathematics, Volume 110, (Springer-Verlag, New York, 1994), MR 1282723 (95f:11085).Google Scholar
Lubotzky, A., Subgroup growth and congruence subgroups, Invent. Math. 119(2) (1995), 267–295, MR 1312501 (95m:20054).CrossRefGoogle Scholar
Margulis, G. A., Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Results in Mathematics and Related Areas (3), Volume 17 (Springer-Verlag, Berlin, 1991), MR 1090825 (92h:22021).Google Scholar
Matringe, N., Essential Whittaker functions for GL (n), Doc. Math. 18 (2013), 1191–1214, MR 3138844.CrossRefGoogle Scholar
Mœglin, C., Fonctions L de paires pour les groupes classiques, Ramanujan J. 28(2) (2012), 187–209, MR 2925174.Google Scholar
Müller, W. and Speh, B., Absolute convergence of the spectral side of the Arthur trace formula for GLn, Geom. Funct. Anal. 14(1) (2004), 58–93, With an appendix by E. M. Lapid, MR 2053600 (2005m:22021).Google Scholar
Müller, W., The trace class conjecture in the theory of automorphic forms, Ann. of Math. (2) 130(3) (1989), 473–529, MR 1025165 (90m:11083).CrossRefGoogle Scholar
Müller, W., The trace class conjecture in the theory of automorphic forms. II, Geom. Funct. Anal. 8(2) (1998), 315–355, MR 1616155 (99b:11056).Google Scholar
Müller, W., On the spectral side of the Arthur trace formula, Geom. Funct. Anal. 12(4) (2002), 669–722, MR 1935546 (2003k:11086).Google Scholar
Müller, W., Weyl’s law for the cuspidal spectrum of SLn, Ann. of Math. (2) 165(1) (2007), 275–333, MR 2276771 (2008a:11065).Google Scholar
Newman, M., A complete description of the normal subgroups of genus one of the modular group, Amer. J. Math. 86 (1964), 17–24, MR 0163966 (29 #1265).Google Scholar
Scott Osborne, M. and Warner, G., The theory of Eisenstein systems, Pure and Applied Mathematics, Volume 99 (Academic Press Inc. (Harcourt Brace Jovanovich Publishers), New York, 1981), MR 643242 (83j:10034).Google Scholar
Platonov, V. and Rapinchuk, A., Algebraic groups and number theory, Pure and Applied Mathematics, 139 (Academic Press Inc., Boston, MA, 1994). Translated from the 1991 Russian original by Rachel Rowen, MR 1278263 (95b:11039).Google Scholar
Phillips, R. and Sarnak, P., The spectrum of Fermat curves, Geom. Funct. Anal. 1(1) (1991), 80–146, MR 1091611 (92a:11061).Google Scholar
Ranga Rao, R., Orbital integrals in reductive groups, Ann. of Math. (2) 96 (1972), 505–510, MR 0320232 (47 #8771).Google Scholar
Rohlfs, J. and Speh, B., On limit multiplicities of representations with cohomology in the cuspidal spectrum, Duke Math. J. 55(1) (1987), 199–211, MR 883670 (88k:22010).Google Scholar
Sarnak, P., A note on the spectrum of cusp forms for congruence subgroups, preprint (1982).Google Scholar
Sauvageot, F., Principe de densité pour les groupes réductifs, Compositio Math. 108(2) (1997), 151–184, MR 1468833 (98i:22025).Google Scholar
Savin, G., Limit multiplicities of cusp forms, Invent. Math. 95(1) (1989), 149–159, MR 969416 (90c:22035).CrossRefGoogle Scholar
Shin, S. W., Automorphic Plancherel density theorem, Israel J. Math. 192(1) (2012), 83–120, MR 3004076.Google Scholar
Shin, S. W. and Templier, N., Sato-Tate theorem for families and low-lying zeros of automorphic $L$-functions, arXiv:1208.1945.Google Scholar
Vogan, D. A. Jr, The algebraic structure of the representation of semisimple Lie groups. I, Ann. of Math. (2) 109(1) (1979), 1–60, MR 519352 (81j:22020).CrossRefGoogle Scholar
Vogan, D. A. Jr, Representations of real reductive Lie groups, Progress in Mathematics, 15 (Birkhäuser, Boston, Mass, 1981), MR 632407 (83c:22022).Google Scholar
Wallach, N. R., The spectrum of compact quotients of semisimple Lie groups, in Proceedings of the International Congress of Mathematicians (Helsinki, 1978 (Helsinki), pp. 715–719 (Acad. Sci. Fennica, 1980), MR 562677 (81f:22021).Google Scholar
Wallach, N. R., On the constant term of a square integrable automorphic form, in Operator algebras and group representations, Volume II (Neptun, 1980, Monogr. Stud. Math. 18, pp. 227–237 (Pitman, Boston, MA, 1984), MR 733320 (86i:22029).Google Scholar
Wallach, N. R., Limit multiplicities in L 2(Γ∖G), in Cohomology of arithmetic groups and automorphic forms (Luminy-Marseille, 1989, Lecture Notes in Math., 1447, pp. 31–56 (Springer, Berlin, 1990), MR 1082961 (92f:22020).Google Scholar
Waldspurger, J.-L., La formule de Plancherel pour les groupes p-adiques (d’après Harish-Chandra), J. Inst. Math. Jussieu 2(2) (2003), 235–333, MR 1989693 (2004d:22009).CrossRefGoogle Scholar
Wang, W., Dimension of a minimal nilpotent orbit, Proc. Amer. Math. Soc. 127(3) (1999), 935–936, MR 1610801 (99f:17010).Google Scholar