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A twisted topological trace formula for Hecke operators and liftings from symplectic to general linear groups

  • Uwe Weselmann (a1)

Abstract

For the locally symmetric space X attached to an arithmetic subgroup of an algebraic group G of ℚ-rank r, we construct a compact manifold by gluing together 2r copies of the Borel–Serre compactification of X. We apply the classical Lefschetz fixed point formula to and get formulas for the traces of Hecke operators ℋ acting on the cohomology of X. We allow twistings of ℋ by outer automorphisms η of G. We stabilize this topological trace formula and compare it with the corresponding formula for an endoscopic group of the pair (G,η) . As an application, we deduce a weak lifting theorem for the lifting of automorphic representations from Siegel modular groups to general linear groups.

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References

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[Art88]Arthur, J., The invariant trace formula II. Global theory, J. Amer. Math. Soc. 1 (1988), 501554.
[Bal01]Ballmann, J., Berechnung der Kottwitz–Shelstad-Transferfaktoren für unverzweigte Tori in nicht zusammenhängenden reduktiven Gruppen, Dissertation, University of Mannheim (2001), http://bibserv7.bib.uni-mannheim.de/madoc/volltexte/2002/38/.
[BGS85]Ballmann, W., Gromov, M. and Schroeder, V., Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61 (Birkhäuser, Boston–Basel–Stuttgart, 1985).
[BWW02]Ballmann, J., Weissauer, R. and Weselmann, U., Remarks on the fundamental lemma for stable twisted endoscopy of classical groups, Manuskripte der Forschergruppe Arithmetik, vol. 7 (Universities of Mannheim and Heidelberg, 2002).
[Bew85]Bewersdorff, J., Eine Lefschetzsche Fixpunktformel für Hecke-Operatoren, Bonner Mathematische Schriften, vol. 164 (University of Bonn, 1985).
[Bor91]Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (Springer, Berlin–Heidelberg–New York, 1991).
[BS73]Borel, A. and Serre, J. P., Corners and arithmetic groups, Comment. Math. Helv. 73 (1973), 436491.
[Bor98]Borovoi, M. V., Abelian Galois cohomology of reductive groups, Mem. Amer. Math. Soc. 626 (1998).
[Fli96]Flicker, Y. F., On the symmetric square. Unit elements, Pacific J. Math. 175 (1996), 507526.
[Fli99]Flicker, Y. F., Matching of orbital integrals on GL(4) and GSp(2), Mem. Amer. Math. Soc. 137 (1999).
[Fli05]Flicker, Y. F., Automorphic forms and Shimura varieties of PGSp(2) (World Scientific, Singapore, 2005).
[Fra98]Franke, J., Harmonic analysis in weighted L 2-spaces, Ann. Sci. Éc. Norm. Supér. 31 (1998), 181279.
[God62/63]Godement, R., Domaines fondamentaux des groupes arithmétiques, Sémin. Bourbaki (1962/63), exp. 257.
[GHM94]Goresky, M., Harder, G. and MacPherson, R., Weighted cohomology, Invent. Math. 116 (1994), 139213.
[GKM97]Goresky, M., Kottwitz, R. and MacPherson, R., Discrete series characters and the Lefschetz formula for Hecke operators, Duke Math. J. 89 (1997), 477554.
[GKM98]Goresky, M., Kottwitz, R. and MacPherson, R., Correction to ‘Discrete series characters and the Lefschetz formula for Hecke operators’, Duke Math. J. 92 (1998), 665666.
[GM92]Goresky, M. and MacPherson, R., Lefschetz numbers of Hecke correspondences, in The zeta functions of Picard modular surfaces, eds Langlands, R. P. and Ramakrishnan, D. (Centre de Recherche Mathématiques, Montréal, 1992), 465478.
[GM93]Goresky, M. and MacPherson, R., Local contribution to the Lefschetz fixed point formula, Invent. Math. 111 (1993), 133.
[GM03]Goresky, M. and MacPherson, R., The topological trace formula, J. Reine Angew. Math. 560 (2003), 77150.
[GT99]Goresky, M. and Tai, Y.-S., Toroidal and reductive Borel–Serre compactifications of locally symmetric spaces, Amer. J. Math. 121 (1999), 10951151.
[Hal94]Hales, Th., The twisted endoscopy of GL(4) and GL(5): transfer of Shalika germs, Duke Math. J. 76 (1994), 595632.
[Hal95]Hales, Th., On the fundamental lemma for standard endoscopy: reduction to the unit element, Canad. J. Math. 47 (1995), 974994.
[Har71]Harder, G., A Gauss–Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Éc. Norm. Supér. (4) (1971), 409455.
[Har93]Harder, G., Eisensteinkohomologie und die Konstruktion gemischter Motive, Lecture Notes in Mathematics, vol. 1562 (Springer, Berlin, 1993).
[Har95]Harder, G., Five notes on the topological trace formula and its applications (1995), unpublished manuscript, Bonn.
[Hel62]Helgason, , Differential geometry and symmetric spaces, Pure and Applied Mathematics, vol. 12 (Academic Press, New York, 1962).
[Hum72]Humphreys, J. E., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9 (Springer, New York, 1972).
[Kos61]Kostant, B., Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. (2) 74 (1961), 329387.
[Kot83]Kottwitz, R., Sign changes in harmonic analysis on reductive groups, Trans. Amer. Math. Soc. 278 (1983), 289297.
[Kot88]Kottwitz, R., Tamagawa numbers, Ann. of Math. (2) 127 (1988), 629646.
[KS99]Kottwitz, R. and Shelstad, D., Foundations of twisted endoscopy, Astérisque 255 (1999).
[KS72]Kuga, M. and Sampson, J. H., A coincidence formula for locally symmetric spaces, Amer. J. Math. 94 (1972), 486500.
[LS78]Lee, R. and Szczarba, R. H., On the torsion in K 4(ℤ) and K 5(ℤ), Duke Math. J. 45 (1978), 101129.
[Leu96]Leuzinger, E., On the Gauss–Bonnet formula for locally symmetric spaces of noncompact type, Enseign. Math. II. Sér. 42 (1996), 201214.
[Ngo10]Ngô, B. C., Le lemme fondamental pour les algêbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1271.
[Osh78]Oshima, T., A realization of Riemannian symmetric spaces, J. Math. Soc. Japan 30 (1978), 117132.
[Roh90]Rohlfs, J., Lefschetz numbers for arithmetic groups, in Cohomology of arithmetic groups and automorphic forms: Proceedings of a Conference held in Luminy/Marseille, France, May 22–27 1989, Lecture Notes in Mathematics, vol. 1447 (Springer, New York, 1990), 303313.
[RS93]Rohlfs, J. and Speh, B., Lefschetz numbers and twisted stabilized orbital integrals, Math. Ann. 296 (1993), 191214.
[San81]Sansuc, J.-J., Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 1280.
[She79]Shelstad, D., Characters and inner forms of a quasi-split group over ℝ, Compositio Math. 39 (1979), 1145.
[Ste68]Steinberg, R., Endomorphisms of algebraic groups, Mem. Amer. Math. Soc. 80 (1968).
[Wal97]Waldspurger, J.-L., Le lemme fondamental implique le transfert, Compositio Math. 105 (1997), 153236.
[Wal06]Waldspurger, J.-L., Endoscopie et changement de caractéristique, J. Inst. Math. Jussieu 5 (2006), 423525.
[Wal08]Waldspurger, J.-L., L’endoscopie tordue n’est pas si tordue, Mem. Amer. Math. Soc. 908 (2008).
[Wei06]Weissauer, R., Spectral approximation of twisted local κ-orbital integrals, Preprint (2006).
[Wei08]Weissauer, R., A remark on the existence of Whittaker models for L-packets of automorphic representations of GSp(4), in Modular forms on Schiermonnikoog. Based on the conference on Modular Forms, Schiermonnikoog, Netherlands, October 2006, eds Edixhoven, B.et al. (Cambridge University Press, Cambridge, 2008), 285310.
[Wei09]Weissauer, R., Endoscopy for GSp(4) and the cohomology of Siegel modular threefolds, Lecture Notes in Mathematics, vol. 1968 (Springer, Heidelberg–Berlin, 2009).
[Wen01]Wendt, R., Weyl’s character formula for non-connected Lie groups and orbital theory for twisted affine Lie algebras, J. Funct. Anal. 180 (2001), 3165.
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