On the spectral side of Arthur’s trace formula — absolute convergence

Abstract

We derive a refinement of the spectral expansion of Arthur’s trace formula. The expression is absolutely convergent with respect to the trace norm.

  • [MR518111] Go to document J. Arthur, "A trace formula for reductive groups. I. Terms associated to classes in $G({\bf Q})$," Duke Math. J., vol. 45, iss. 4, pp. 911-952, 1978.
    @article {MR518111, MRKEY = {0518111},
      AUTHOR = {Arthur, James},
      TITLE = {A trace formula for reductive groups. {I}. {T}erms associated to classes in {$G({\bf Q})$}},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {45},
      YEAR = {1978},
      NUMBER = {4},
      PAGES = {911--952},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {10D40 (22E55)},
      MRNUMBER = {0518111},
      MRREVIEWER = {Stephen Gelbart},
      ZBLNUMBER = {0499.10032},
      DOI = {10.1215/S0012-7094-78-04533-7},
      }
  • [MR558260] J. Arthur, "A trace formula for reductive groups. II. Applications of a truncation operator," Compositio Math., vol. 40, iss. 1, pp. 87-121, 1980.
    @article {MR558260, MRKEY = {0558260},
      AUTHOR = {Arthur, James},
      TITLE = {A trace formula for reductive groups. {II}. {A}pplications of a truncation operator},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {40},
      YEAR = {1980},
      NUMBER = {1},
      PAGES = {87--121},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {22E55 (10D40 20G05)},
      MRNUMBER = {0558260},
      MRREVIEWER = {Stephen Gelbart},
      ZBLNUMBER = {0499.10033},
     }
  • [MR625344] Go to document J. Arthur, "The trace formula in invariant form," Ann. of Math., vol. 114, iss. 1, pp. 1-74, 1981.
    @article {MR625344, MRKEY = {0625344},
      AUTHOR = {Arthur, James},
      TITLE = {The trace formula in invariant form},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {114},
      YEAR = {1981},
      NUMBER = {1},
      PAGES = {1--74},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {10D40 (22E55)},
      MRNUMBER = {0625344},
      MRREVIEWER = {Freydoon Shahidi},
      DOI = {10.2307/1971376},
      ZBLNUMBER = {0495.22006},
      }
  • [MR681737] Go to document J. Arthur, "On a family of distributions obtained from Eisenstein series. I. Application of the Paley-Wiener theorem," Amer. J. Math., vol. 104, iss. 6, pp. 1243-1288, 1982.
    @article {MR681737, MRKEY = {0681737},
      AUTHOR = {Arthur, James},
      TITLE = {On a family of distributions obtained from {E}isenstein series. {I}. {A}pplication of the {P}aley-{W}iener theorem},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {104},
      YEAR = {1982},
      NUMBER = {6},
      PAGES = {1243--1288},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {22E55 (11F20 11F70 11F72 46F99)},
      MRNUMBER = {0681737},
      DOI = {10.2307/2374061},
      ZBLNUMBER = {0541.22010},
     }
  • [MR681738] Go to document J. Arthur, "On a family of distributions obtained from Eisenstein series. II. Explicit formulas," Amer. J. Math., vol. 104, iss. 6, pp. 1289-1336, 1982.
    @article {MR681738, MRKEY = {0681738},
      AUTHOR = {Arthur, James},
      TITLE = {On a family of distributions obtained from {E}isenstein series. {II}. {E}xplicit formulas},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {104},
      YEAR = {1982},
      NUMBER = {6},
      PAGES = {1289--1336},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {22E55 (11F70 11F72 46F99)},
      MRNUMBER = {0681738},
      MRREVIEWER = {Freydoon Shahidi},
      DOI = {10.2307/2374062},
      ZBLNUMBER = {0562.22004},
     }
  • [MR650368] Go to document J. Arthur, "On the inner product of truncated Eisenstein series," Duke Math. J., vol. 49, iss. 1, pp. 35-70, 1982.
    @article {MR650368, MRKEY = {0650368},
      AUTHOR = {Arthur, James},
      TITLE = {On the inner product of truncated {E}isenstein series},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {49},
      YEAR = {1982},
      NUMBER = {1},
      PAGES = {35--70},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {22E55 (10D40 22E50)},
      MRNUMBER = {0650368},
      MRREVIEWER = {Martin L. Karel},
      ZBLNUMBER = {0518.22012},
      DOI = {10.1215/S0012-7094-82-04904-3},
     }
  • [MR999488] Go to document J. Arthur, "Intertwining operators and residues. I. Weighted characters," J. Funct. Anal., vol. 84, iss. 1, pp. 19-84, 1989.
    @article {MR999488, MRKEY = {0999488},
      AUTHOR = {Arthur, James},
      TITLE = {Intertwining operators and residues. {I}. {W}eighted characters},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {84},
      YEAR = {1989},
      NUMBER = {1},
      PAGES = {19--84},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {22E55 (11F70 11F72)},
      MRNUMBER = {0999488},
      MRREVIEWER = {David Joyner},
      DOI = {10.1016/0022-1236(89)90110-9},
      ZBLNUMBER = {0679.22011},
     }
  • [MR2192011] J. Arthur, "An introduction to the trace formula," in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Providence, RI: Amer. Math. Soc., 2005, vol. 4, pp. 1-263.
    @incollection {MR2192011, MRKEY = {2192011},
      AUTHOR = {Arthur, James},
      TITLE = {An introduction to the trace formula},
      BOOKTITLE = {Harmonic Analysis, the Trace Formula, and {S}himura Varieties},
      SERIES = {Clay Math. Proc.},
      VOLUME = {4},
      PAGES = {1--263},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2005},
      MRCLASS = {11F72},
      MRNUMBER = {2192011},
      MRREVIEWER = {Erez M. Lapid},
      ZBLNUMBER = {1152.11021},
      }
  • [MR1433285] Go to document P. Borwein and T. Erdélyi, "Sharp extensions of Bernstein’s inequality to rational spaces," Mathematika, vol. 43, iss. 2, pp. 413-423 (1997), 1996.
    @article {MR1433285, MRKEY = {1433285},
      AUTHOR = {Borwein, Peter and Erd{é}lyi, Tam{á}s},
      TITLE = {Sharp extensions of {B}ernstein's inequality to rational spaces},
      JOURNAL = {Mathematika},
      FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
      VOLUME = {43},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {413--423 (1997)},
      ISSN = {0025-5793},
      CODEN = {MTKAAB},
      MRCLASS = {26D15},
      MRNUMBER = {1433285},
      MRREVIEWER = {L. Leindler},
      DOI = {10.1112/S0025579300011876},
      ZBLNUMBER = {0869.41010},
     }
  • [FL2] T. Finis and E. M. Lapid, "On the Arthur-Selberg trace formula for ${G}{L}(2)$," Groups Geom. Dyn., vol. 5, pp. 367-391, 2011.
    @article{FL2,
      author = {Finis, Tobias and Lapid, Erez M.},
      TITLE={On the {A}rthur-{S}elberg trace formula for ${G}{L}(2)$},
      JOURNAL={Groups Geom. Dyn.},
      VOLUME={5},
      YEAR={2011},
      NOTE={special issue on the occasion of {F}ritz {G}runewald's 60th birthday},
      PAGES={367--391},
      }
  • [FL3] Go to document T. Finis and E. M. Lapid, "On the continuity of Arthur’s trace formula: the semisimple terms," Compos. Math., vol. 147, pp. 784-802, 2011.
    @article{FL3,
      author = {Finis, Tobias and Lapid, Erez M.},
      TITLE={On the continuity of {A}rthur's trace formula: the semisimple terms},
      JOURNAL={{C}ompos. {M}ath.},
      VOLUME={147},
      YEAR={2011},
      PAGES={784--802},
      DOI={10.1112/S0010437X11004891},
      }
  • [FLAn] T. Finis and E. M. Lapid, On the spectral side of Arthur’s trace formula — combinatorial setup.
    @misc{FLAn,
      author = {Finis, Tobias and Lapid, Erez M.},
      TITLE={On the spectral side of {A}rthur's trace formula -- combinatorial setup},
      NOTE={this volume},
      }
  • [MR2550527] Go to document T. Finis, E. M. Lapid, and W. Müller, "The spectral side of Arthur’s trace formula," Proc. Natl. Acad. Sci. USA, vol. 106, iss. 37, pp. 15563-15566, 2009.
    @article {MR2550527, MRKEY = {2550527},
      AUTHOR = {Finis, Tobias and Lapid, Erez M. and M{ü}ller, Werner},
      TITLE = {The spectral side of {A}rthur's trace formula},
      JOURNAL = {Proc. Natl. Acad. Sci. USA},
      FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America},
      VOLUME = {106},
      YEAR = {2009},
      NUMBER = {37},
      PAGES = {15563--15566},
      ISSN = {1091-6490},
      CODEN = {PNASFB},
      MRCLASS = {11F72 (22E55)},
      MRNUMBER = {2550527},
      MRREVIEWER = {{\Dbar}{\cftil{o}} Ng{\d{o}}c Di{\cfudot{e}}p},
      DOI = {10.1073/pnas.0905408106},
      ZBLNUMBER = {1203.11042},
     }
  • [MR1622604] Go to document L. Ji, "The trace class conjecture for arithmetic groups," J. Differential Geom., vol. 48, iss. 1, pp. 165-203, 1998.
    @article {MR1622604, MRKEY = {1622604},
      AUTHOR = {Ji, Lizhen},
      TITLE = {The trace class conjecture for arithmetic groups},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {48},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {165--203},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {11F72 (22E40)},
      MRNUMBER = {1622604},
      MRREVIEWER = {Werner Hoffmann},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214460610},
      ZBLNUMBER = {0926.11034},
      }
  • [MR1625060] Go to document H. Jacquet, E. Lapid, and J. Rogawski, "Periods of automorphic forms," J. Amer. Math. Soc., vol. 12, iss. 1, pp. 173-240, 1999.
    @article {MR1625060, MRKEY = {1625060},
      AUTHOR = {Jacquet, Herv{é} and Lapid, Erez and Rogawski, Jonathan},
      TITLE = {Periods of automorphic forms},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {12},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {173--240},
      ISSN = {0894-0347},
      MRCLASS = {11F67 (11F70 11F72 22E55)},
      MRNUMBER = {1625060},
      MRREVIEWER = {Jiandong Guo},
      DOI = {10.1090/S0894-0347-99-00279-9},
      ZBLNUMBER = {1012.11044},
      }
  • [MR0579181] Go to document R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, New York: Springer-Verlag, 1976, vol. 544.
    @book {MR0579181, MRKEY = {0579181},
      AUTHOR = {Langlands, Robert P.},
      TITLE = {On the Functional Equations Satisfied by {E}isenstein Series},
      SERIES = {Lecture Notes in Math.},
      VOLUME={544},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1976},
      PAGES = {v+337},
      MRCLASS = {22E55 (10C15 10D20 32N05)},
      MRNUMBER = {0579181},
      MRREVIEWER = {James G. Arthur},
      ZBLNUMBER = {0332.10018},
      DOI = {1007/BFb0079929},
     }
  • [MR1159111] R. P. Langlands, "Rank-one residues of Eisenstein series," in Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday, Part II, Jerusalem, 1990, pp. 111-125.
    @inproceedings{MR1159111,
      author = {Langlands, Robert P.},
      TITLE = {Rank-one residues of {E}isenstein series},
      BOOKTITLE={Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday, Part {\rm II}},
      VENUE={Ramat Aviv, 1989},
      SERIES={Israel Math. Conf. Proc.},
      VOLUME={3},
      PUBLISHER={Weizmann},
      ADDRESS={Jerusalem},
      YEAR={1990},
      PAGES={111--125},
      MRNUMBER={1159111},
      ZBLNUMBER = {0711.11022},
      }
  • [LSh] E. Lapid, "On Arthur’s asymptotic inner product formula of truncated Eisenstein series," in On Certain ${L}$-Functions: A Volume in Honor of Freydoon Shahidi on the Occasion of his 60 th Birthday, Providence, RI: Amer. Math. Soc., 2011, vol. 13, pp. 309-331.
    @incollection{LSh,
      author={Lapid, Erez},
      TITLE={On {A}rthur's asymptotic inner product formula of truncated {E}isenstein series},
      BOOKTITLE={On {C}ertain ${L}$-Functions: A Volume in Honor of Freydoon Shahidi on the Occasion of his 60{\rm th} Birthday},
      SERIES={Clay Math. Proc.},
      VOLUME={13},
      PUBLISHER={Amer. Math. Soc.},
      ADDRESS={Providence, RI},
      PAGES={309--331},
      YEAR={2011},
      }
  • [MR2541128] Go to document E. Lapid and W. Müller, "Spectral asymptotics for arithmetic quotients of ${ SL}(n,\Bbb R)/{ SO}(n)$," Duke Math. J., vol. 149, iss. 1, pp. 117-155, 2009.
    @article {MR2541128, MRKEY = {2541128},
      AUTHOR = {Lapid, Erez and M{ü}ller, Werner},
      TITLE = {Spectral asymptotics for arithmetic quotients of {${\rm SL}(n,\Bbb R)/{\rm SO}(n)$}},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {149},
      YEAR = {2009},
      NUMBER = {1},
      PAGES = {117--155},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {11F70 (11F72 22E55)},
      MRNUMBER = {2541128},
      MRREVIEWER = {{\Dbar}{\cftil{o}} Ng{\d{o}}c Di{\cfudot{e}}p},
      DOI = {10.1215/00127094-2009-037},
      ZBLNUMBER = {pre05588173},
     }
  • [MR2053600] Go to document W. Müller and B. Speh, "Absolute convergence of the spectral side of the Arthur trace formula for ${ GL}_n$," Geom. Funct. Anal., vol. 14, iss. 1, pp. 58-93, 2004.
    @article {MR2053600, MRKEY = {2053600},
      AUTHOR = {M{ü}ller, W. and Speh, B.},
      TITLE = {Absolute convergence of the spectral side of the {A}rthur trace formula for {${\rm GL}\sb n$}},
      NOTE = {with an appendix by {E. M. L}apid},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {14},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {58--93},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {22E55 (11F70 11F72)},
      MRNUMBER = {2053600},
      MRREVIEWER = {Werner Hoffmann},
      DOI = {10.1007/s00039-004-0452-0},
      ZBLNUMBER = {1083.11031},
     }
  • [MR1025165] W. Müller, "The trace class conjecture in the theory of automorphic forms," Ann. of Math., vol. 130, pp. 473-529, 1989.
    @article {MR1025165, MRKEY = {1025165},
      AUTHOR = {M{ü}ller, W.},
      TITLE = {The trace class conjecture in the theory of automorphic forms},
      JOURNAL = {Ann. of Math.},
      VOLUME = {130},
      YEAR = {1989},
      PAGES = {473--529},
      MRCLASS = {11F72 (11F70 22E55)},
      MRNUMBER = {1025165},
      ZBLNUMBER = {0701.11019},
      }
  • [MR1616155] Go to document W. Müller, "The trace class conjecture in the theory of automorphic forms. II," Geom. Funct. Anal., vol. 8, iss. 2, pp. 315-355, 1998.
    @article {MR1616155, MRKEY = {1616155},
      AUTHOR = {M{ü}ller, W.},
      TITLE = {The trace class conjecture in the theory of automorphic forms. {II}},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {8},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {315--355},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {11F72 (22E40)},
      MRNUMBER = {1616155},
      MRREVIEWER = {Werner Hoffmann},
      DOI = {10.1007/s000390050059},
      ZBLNUMBER = {1073.11514},
     }
  • [MR1800065] Go to document W. Müller, "On the singularities of residual intertwining operators," Geom. Funct. Anal., vol. 10, iss. 5, pp. 1118-1170, 2000.
    @article {MR1800065, MRKEY = {1800065},
      AUTHOR = {M{ü}ller, W.},
      TITLE = {On the singularities of residual intertwining operators},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {10},
      YEAR = {2000},
      NUMBER = {5},
      PAGES = {1118--1170},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {22E55 (11F72)},
      MRNUMBER = {1800065},
      MRREVIEWER = {David Manderscheid},
      DOI = {10.1007/PL00001649},
      ZBLNUMBER = {0964.22016},
      }
  • [MR1935546] Go to document W. Müller, "On the spectral side of the Arthur trace formula," Geom. Funct. Anal., vol. 12, iss. 4, pp. 669-722, 2002.
    @article {MR1935546, MRKEY = {1935546},
      AUTHOR = {M{ü}ller, W.},
      TITLE = {On the spectral side of the {A}rthur trace formula},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {12},
      YEAR = {2002},
      NUMBER = {4},
      PAGES = {669--722},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {11F72 (11F70 22E55)},
      MRNUMBER = {1935546},
      MRREVIEWER = {Werner Hoffmann},
      DOI = {10.1007/s00039-002-8263-7},
      ZBLNUMBER = {1169.11313},
     }
  • [MR1217488] P. Orlik and H. Terao, Arrangements of Hyperplanes, New York: Springer-Verlag, 1992, vol. 300.
    @book {MR1217488, MRKEY = {1217488},
      AUTHOR = {Orlik, Peter and Terao, Hiroaki},
      TITLE = {Arrangements of Hyperplanes},
      SERIES = {Grundl. Math. Wissen.},
      VOLUME = {300},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1992},
      PAGES = {xviii+325},
      ISBN = {3-540-55259-6},
      MRCLASS = {52B30 (14F35 20F36 20F55 32S25 57N65)},
      MRNUMBER = {1217488},
      MRREVIEWER = {Michel Yves Jambu},
      ZBLNUMBER = {0757.55001},
      }
  • [MR2487491] A. Postnikov, "Permutohedra, associahedra, and beyond," Int. Math. Res. Not., vol. 2009, iss. 6, pp. 1026-1106, 2009.
    @article {MR2487491, MRKEY = {2487491},
      AUTHOR = {Postnikov, Alexander},
      TITLE = {Permutohedra, associahedra, and beyond},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2009},
      NUMBER = {6},
      PAGES = {1026--1106},
      ISSN = {1073-7928},
      MRCLASS = {05E30},
      MRNUMBER = {2487491},
      VOLUME = {2009},
      ZBLNUMBER = {1162.52007},
      }
  • [MR0088511] A. Selberg, "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series," J. Indian Math. Soc., vol. 20, pp. 47-87, 1956.
    @article {MR0088511, MRKEY = {0088511},
      AUTHOR = {Selberg, A.},
      TITLE = {Harmonic analysis and discontinuous groups in weakly symmetric {R}iemannian spaces with applications to {D}irichlet series},
      JOURNAL = {J. Indian Math. Soc.},
      VOLUME = {20},
      YEAR = {1956},
      PAGES = {47--87},
      MRCLASS = {10.1X},
      MRNUMBER = {0088511},
      MRREVIEWER = {F. V. Atkinson},
      ZBLNUMBER = {0072.08201},
      }
  • [MR1311028] G. M. Ziegler, Lectures on Polytopes, New York: Springer-Verlag, 1995, vol. 152.
    @book {MR1311028, MRKEY = {1311028},
      AUTHOR = {Ziegler, G{ü}nter M.},
      TITLE = {Lectures on Polytopes},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {152},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1995},
      PAGES = {x+370},
      ISBN = {0-387-94365-X},
      MRCLASS = {52Bxx},
      MRNUMBER = {1311028},
      MRREVIEWER = {Margaret M. Bayer},
      ZBLNUMBER = {0823.52002},
      }

Authors

Tobias Finis

Lehrstuhl für Algebra und Zahlentheorie
Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf
Universitätstrasse. 1
40225 Düsseldorf, Germany

Erez Lapid

Institute of Mathematics
Hebrew University of Jerusalem
Jerusalem 91904
Israel

Werner Müller

Mathematisches Institut
Rheinische
Friedrich-Wilhelms-Universität Bonn
53012 Bonn
Germany