×

zbMATH — the first resource for mathematics

The base change fundamental lemma for central elements in parahoric Hecke algebras. (English) Zbl 1194.22019
Let \(E/F\) be finite unramified extension of \(p\)-adic fields, and let \(\theta\) be a generator of \(\mathrm{Gal}(E/F)\). Let \(G\) be an unramified connected reductive group over \(F\). The article under review proves the fundamental lemma for parahoric Hecke algebras in the case of stable base change from \(G(F)\) to \(G(E)\). Let \(J_F\) be a parahoric subgroup of \(G(F)\) and let \(J_E\) be the corresponding parahoric subgroup of \(G(E)\). Let \(Z(\mathcal H_{J_F}(G(F)))\) and \(Z(\mathcal H_{J_E}(G(E)))\) be the centers of the Hecke algebras corresponding to these parahorics. The author defines a “base change homomorphism”
\[ b:Z(\mathcal H_{J_E}(G(E))) \longrightarrow Z(\mathcal H_{J_F}(G(F))) \]
and proves that if \(\phi\in Z(\mathcal H_{J_E}(G(E)))\), then \(\phi\) and \(b(\phi)\) are associated in the sense that the stable orbital integral of \(b(\phi)\) over a semisimple orbit in \(G(F)\) agrees with the stable \(\theta\)-twisted orbital integral of \(\phi\) over a corresponding orbit in \(G(E)\). This generalizes results of L. Clozel [Duke Math. J. 61, No. 1, 255–302 (1990; Zbl 0731.22011)] and J.-P. Labesse [Duke Math. J. 61, No. 2, 519–530 (1990; Zbl 0731.22012)] in the case of spherical Hecke algebras. The present version of the fundamental lemma can be applied to compute the local factors of Hasse-Weil zeta functions associated to certain Shimura varieties with parahoric level structure at a given prime \(p\).
The proof that \(\phi\) and \(b(\phi)\) are associated proceeds by induction on the semisimple rank of \(G\). There are two main steps, as in the spherical setting. First, via the induction hypothesis, descent formulas, and other reasoning, the author reduces to the case of strongly regular elliptic semisimple orbits in an adjoint group \(G\). In the second part, he proves the statement in this more restricted case. This uses a global argument, akin to that of Clozel, which involves the twisted trace formula of P. Deligne, D. Kazhdan and M.-F. Vignéras [Représentations des groupes réductifs sur un corps local (Hermann, Paris), 33–117 (1984; Zbl 0583.22009)], as well as the stabilization of its geometric side due to Kottwitz.
The author also gives a short discussion of the relationship between the article under review and other recent work, including that of Ngô, Laumon, and Waldspurger.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
20G25 Linear algebraic groups over local fields and their integers
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] J. Arthur, A trace formula for reductive groups, I: Terms associated to classes in \(G(\mathbb Q)\) , Duke Math. J. 45 (1978), 911–952. · Zbl 0499.10032
[2] -, “The trace Paley-Wiener theorem for Schwartz functions” in Representation Theory and Analysis on Homogeneous Spaces (New Brunswick, N.J., 1993) , Contemp. Math. 177 , Amer. Math. Soc., Providence, 1994, 171–180. · Zbl 0854.22011
[3] J. N. Bernstein, “Le ‘centre’ de Bernstein” in Représentations des groupes réductifs sur un corps local , Travaux en Cours, Hermann, Paris, 1984, 1–32. · Zbl 0599.22016
[4] A. Borel, “Automorphic \(L\)-functions” in Automorphic Forms, Representations and \(L\)-Functions (Corvallis, Ore., 1977) , Proc. Sympos. Pure Math. 33 , Amer. Math. Soc., Providence, 1979, 27–61. · Zbl 0412.10017
[5] -, Linear Algebraic Groups , 2nd ed., Grad. Texts in Math. 126 , Springer, New York, 1991. · Zbl 0726.20030
[6] A. Borel and H. Jacquet, “Automorphic forms and automorphic representations,” with a supplement “On the notion of an automorphic representation” by R. P. Langlands, in Automorphic Forms, Representations and \(L\)-Functions (Corvallis, Ore., 1977) , Proc. Sympos. Pure Math. 33 , Amer. Math. Soc., Providence, 1979, 189–207.
[7] M. V. Borovoi, Abelian Galois cohomology of reductive algebraic groups , Mem. Amer. Soc. Math. 626 (1998), no. 626. · Zbl 0918.20037
[8] N. Bourbaki, Éléments de Mathématiques: Groupes et Algèbres de Lie, Chapitres 4, 5, et 6 , Masson, Paris, 1981. · Zbl 0483.22001
[9] F. Bruhat and J. Tits, Groupes réductifs sur un corps local , Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251. · Zbl 0254.14017
[10] -, Groupes réductifs sur un corps local, II , Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376.
[11] W. Casselman, Characters and Jacquet modules , Math. Ann. 230 (1977), 101–105. · Zbl 0337.22019
[12] -, The unramified principal series of \(\mathfrak p\)-adic groups, I: The spherical function , Compositio Math. 40 (1980), 387–406. · Zbl 0472.22004
[13] -, Introduction to the theory of admissible representations of \(p\)-adic reductive groups , unpublished notes.
[14] N. Chriss and K. Khuri-Makdisi, On the Iwahori-Hecke algebra of a \(p\)-adic group , Internat. Math. Res. Notices 1998 , no. 2, 85–100. · Zbl 0894.22013
[15] L. Clozel, “Théorème d’Atiyah-Bott pour les variétés \(\mathfrak p\)-adiques et caractères des groupes réductifs” in Harmonic Analysis on Lie Groups and Symmetric Spaces (Kleebach, Germany, 1983), Mém. Soc. Math. France (N.S.) 15 (1984), 39–64. · Zbl 0555.22003
[16] -, Sur une conjecture de Howe, I , Compositio Math. 56 (1985), 87–110. · Zbl 0599.22015
[17] -, Orbital integrals on \(p\)-adic groups: A proof of the Howe conjecture , Ann. of Math. (2) 129 (1989), 237–251. JSTOR: · Zbl 0675.22007
[18] -, The fundamental lemma for stable base change , Duke Math. J. 61 (1990), 255–302. · Zbl 0731.22011
[19] -, On the cohomology of Kottwitz’ arithmetic varieties , Duke Math J. 72 (1993), 757–795. · Zbl 0974.11019
[20] P. Deligne, D. Kazhdan, and M.-F. Vigneras, “Représentations des algèbres centrales simples p-adiques” in Représentations des groupes réductifs sur un corps local , Travaux en cours, Hermann, Paris, 1984, 33–117. · Zbl 0583.22009
[21] M. J. Greenberg, Schemata over local rings, II, Ann. of Math. (2) 78 (1963) 256–266. JSTOR: · Zbl 0126.16704
[22] T. J. Haines, “Introduction to Shimura varieties with bad reduction of parahoric type” in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Math. Proc. 4 , Amer. Math. Soc., Providence, 2005, 583–642. · Zbl 1148.11028
[23] -, Intertwiners for unramified groups , unpublished article.
[24] T. J. Haines, R. Kottwitz, and A. Prasad, Iwahori-Hecke algebras ,\arxivmath.RT/0309168v3[math.RT]
[25] T. J. Haines and B. C. Ngô, Nearby cycles for local models of some Shimura varieties , Compositio Math. 133 (2002), 117–150. · Zbl 1009.11042
[26] T. J. Haines and M. Rapoport, “On parahoric subgroups,” appendix to G. Pappas and M. Rapoport, Twisted loop groups and their affine flag varieties , Adv. Math. 219 (2008), 188–198. · Zbl 1159.22010
[27] T. Hales, Unipotent representations and unipotent classes in SL \((n)\), Amer. J. Math. 115 (1993), 1347–1383. JSTOR: · Zbl 0810.22008
[28] -, On the fundamental lemma for standard endoscopy: Reduction to unit elements , Canad. J. Math. 47 (1995), 974–994. · Zbl 0840.22032
[29] D. Keys, Reducibility of unramified unitary principal series representations of \(p\)-adic groups and class- 1 representations, Math. Ann. 260 (1982), 397–402. · Zbl 0488.22026
[30] R. Kottwitz, Orbital integrals on \(\mathrm GL_3\) , Amer. J. Math. 102 (1980), 327–384. JSTOR: · Zbl 0437.22011
[31] -, Rational conjugacy classes in reductive groups , Duke Math. J. 49 (1982), 785–806. · Zbl 0506.20017
[32] -, Sign changes in harmonic analysis on reductive groups , Trans. Amer. Math. Soc. 278 (1983), 289–297. · Zbl 0538.22010
[33] -, Base change for unit elements of Hecke algebras , Compositio Math. 60 (1986), 237–250.
[34] -, Stable trace formula: Elliptic singular terms , Math. Ann. 275 (1986), 365–399. · Zbl 0577.10028
[35] -, Tamagawa numbers , Ann. of Math. (2) 127 (1988), 629–646. JSTOR: · Zbl 0678.22012
[36] -, Points on some Shimura varieties over finite fields , J. Amer. Math. Soc. 5 (1992), 373–444. JSTOR: · Zbl 0796.14014
[37] -, Isocrystals with additional structure, II , Compositio Math. 109 (1997), 255–339. · Zbl 0966.20022
[38] R. Kottwitz and J. Rogawski, The distributions in the invariant trace formula are supported on characters , Canad. J. Math. 52 (2000), 804–814. · Zbl 0991.22014
[39] J.-P. Labesse, Fonctions élémentaires et lemme fondamental pour le changement de base stable , Duke Math. J. 61 (1990), 519–530. · Zbl 0731.22012
[40] -, Cohomologie, stabilisation et changement de base , Appendix A by L. Clozel, Appendix B by L. Breen, Astérisque 257 , Soc. Math. France, Montrouge, 1999. · Zbl 1024.11034
[41] R. P. Langlands, Base Change for GL \((2)\), Ann. of Math. Stud. 96 , Princeton Univ. Press, Princeton, 1980. · Zbl 0444.22007
[42] B. C. Ngô, Le lemme fondamental pour les algebres de Lie , preprint,\arxiv0801.0446v3[math.AG]
[43] M. Rapoport, A guide to the reduction modulo \(p\) of Shimura varieties: Automorphic forms, I , Astérisque 298 (2005), 271–318. · Zbl 1084.11029
[44] M. Rapoport and T. Zink, Period spaces for \(p\)-divisible groups , Ann. of Math. Stud. 141 , Princeton Univ. Press, Princeton, 1996. · Zbl 0873.14039
[45] J. D. Rogawski, Trace Paley-Wiener theorem in the twisted case , Trans. Amer. Math. Soc. 309 (1988), 215–229. JSTOR: · Zbl 0663.22011
[46] J.-P. Serre, Abelian \(l\)-Adic Representation and Elliptic Curves , with the collaboration of W. Kuyk and J. Labute, 2nd ed., Adv. Book Classics, Addison-Wesley, Redwood City, Calif., 1989.
[47] A. J. Silberger, Introduction to harmonic analysis on reductive p-adic groups , based on lectures by Harish-Chandra at the Institute for Advanced Studies, 1971–1973., Math. Notes 23 , Princeton Univ. Press, Princeton, 1979. · Zbl 0458.22006
[48] T. A. Springer, Linear Algebraic Groups , 2nd ed., Progr. Math. 9 , Birkhäuser, Boston, 1998. · Zbl 0927.20024
[49] R. Steinberg, Endomorphisms of linear algebraic groups , Mem. Amer. Math. Soc. 80 , Amer. Math. Soc., Providence, 1968. · Zbl 0164.02902
[50] J. Tits, “Reductive groups over local fields” in Automorphic Forms, Representations and L-Functions, Part I (Corvallis, Ore., 1977) , Proc. Sympos. Pure Math. 33 , Amer. Math. Soc., Providence, 1979, 29–69. · Zbl 0415.20035
[51] J.-L. Waldspurger, Le lemme fondamental implique le transfert , Compositio Math. 105 (1997), 153–236. · Zbl 0871.22005
[52] -, Endoscopie et changement de caractéristique , J. Inst. Math. Jussieu 5 (2006), 423–525. · Zbl 1102.22010
[53] -, L’endoscopie tordue n’est pas si tordue , Mem. Amer. Math. Soc. 194 (2008), no. 908. · Zbl 1146.22016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.