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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Geometric realization of Whittaker functions and the Langlands conjecture
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by E. Frenkel, D. Gaitsgory, D. Kazhdan and K. Vilonen
J. Amer. Math. Soc. 11 (1998), 451-484


We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group $GL_n(\mathbb A)$ associated to an irreducible $\ell$–adic local system of rank $n$ on an algebraic curve $X$ over a finite field. The existence of such a function is predicted by the Langlands conjecture. The first construction, which was proposed by Shalika and Piatetski-Shapiro following Weil and Jacquet-Langlands ($n=2$), is based on considering the Whittaker function. The second construction, which was proposed recently by Laumon following Drinfeld ($n=2$) and Deligne ($n=1$), is geometric: the automorphic function is obtained via Grothendieck’s “faisceaux-fonctions” correspondence from a complex of sheaves on an algebraic stack. Our proof of their equivalence is based on a local result about the spherical Hecke algebra, which we prove for an arbitrary reductive group. We also discuss a geometric interpretation of this result.
  • L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
  • Arnaud Beauville and Yves Laszlo, Un lemme de descente, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 335–340 (French, with English and French summaries). MR 1320381
  • [3] J. Bernstein, A. Zelevinsky, Representations of the group $GL(n,F)$ where $F$ is a local non-Archimedean field, Russ. Math. Surv. 31 (1976) 1-68.
  • Ranee Kathryn Brylinski, Limits of weight spaces, Lusztig’s $q$-analogs, and fiberings of adjoint orbits, J. Amer. Math. Soc. 2 (1989), no. 3, 517–533. MR 984511, DOI 10.1090/S0894-0347-1989-0984511-X
  • W. Casselman and J. Shalika, The unramified principal series of $p$-adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no. 2, 207–231. MR 581582
  • Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520, DOI 10.1007/BF02684780
  • V. G. Drinfel′d, Two-dimensional $l$-adic representations of the fundamental group of a curve over a finite field and automorphic forms on $\textrm {GL}(2)$, Amer. J. Math. 105 (1983), no. 1, 85–114. MR 692107, DOI 10.2307/2374382
  • V. G. Drinfel′d and Carlos Simpson, $B$-structures on $G$-bundles and local triviality, Math. Res. Lett. 2 (1995), no. 6, 823–829. MR 1362973, DOI 10.4310/MRL.1995.v2.n6.a13
  • I. M. Gel′fand and D. A. Kajdan, Representations of the group $\textrm {GL}(n,K)$ where $K$ is a local field, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 95–118. MR 0404534
  • [10] V. Ginzburg, Perverse sheaves on a loop group and Langlands duality, Preprint alg-geom/9511007 (1995).
  • [11] B.H. Gross, On the Satake isomorphism, Preprint (1996).
  • Benedict H. Gross and Dipendra Prasad, On the decomposition of a representation of $\textrm {SO}_n$ when restricted to $\textrm {SO}_{n-1}$, Canad. J. Math. 44 (1992), no. 5, 974–1002. MR 1186476, DOI 10.4153/CJM-1992-060-8
  • [13] A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique IV: les schémas de Hilbert, Séminaire Bourbaki 221 (1960/61), Benjamin (1966).
  • Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
  • H. Jacquet and R. P. Langlands, Automorphic forms on $\textrm {GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654, DOI 10.1007/BFb0058988
  • Hervé Jacquet, Ilja Iosifovitch Piatetski-Shapiro, and Joseph Shalika, Automorphic forms on $\textrm {GL}(3)$. I, Ann. of Math. (2) 109 (1979), no. 1, 169–212. MR 519356, DOI 10.2307/1971270
  • Shin-ichi Kato, Spherical functions and a $q$-analogue of Kostant’s weight multiplicity formula, Invent. Math. 66 (1982), no. 3, 461–468. MR 662602, DOI 10.1007/BF01389223
  • Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI 10.2307/2373130
  • R. P. Langlands, Problems in the theory of automorphic forms, Lectures in Modern Analysis and Applications, III, Lecture Notes in Mathematics, Vol. 170, Springer, Berlin, 1970, pp. 18–61. MR 0302614
  • [20] Y. Laszlo, Ch. Sorger, The line bundles on the moduli of parabolic $G$-bundles over curves and their sections, Ann. Sci. École Norm. Sup. 30 (1997) 499–525.
  • G. Laumon, Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131–210 (French). MR 908218, DOI 10.1007/BF02698937
  • Gérard Laumon, Correspondance de Langlands géométrique pour les corps de fonctions, Duke Math. J. 54 (1987), no. 2, 309–359 (French). MR 899400, DOI 10.1215/S0012-7094-87-05418-4
  • G. Laumon, Faisceaux automorphes liés aux séries d’Eisenstein, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 227–281 (French). MR 1044822
  • [24] G. Laumon, L. Moret-Bailly, Champs algébriques, Preprint 92-42, Université Paris Sud, 1992.
  • [25] G. Laumon, Faisceaux automorphes pour $GL_n$: la première construction de Drinfeld, Preprint alg-geom/9511004 (1995).
  • G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), no. 2, 169–178. MR 641425, DOI 10.1016/0001-8708(81)90038-4
  • George Lusztig, Singularities, character formulas, and a $q$-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208–229. MR 737932
  • I. G. Macdonald, Spherical functions on a group of $p$-adic type, Publications of the Ramanujan Institute, No. 2, University of Madras, Centre for Advanced Study in Mathematics, Ramanujan Institute, Madras, 1971. MR 0435301
  • [29] I. Mirković, K. Vilonen Perverse sheaves on loop Grassmannians and Langlands duality, Preprint alg-geom/9703010, to appear in the proceedings of a conference on the geometric Langlands correspondence at Luminy, June 1995.
  • I. I. Pjateckij-Šapiro, Euler subgroups, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 597–620. MR 0406935
  • Ichirô Satake, Theory of spherical functions on reductive algebraic groups over ${\mathfrak {p}}$-adic fields, Inst. Hautes Études Sci. Publ. Math. 18 (1963), 5–69. MR 195863, DOI 10.1007/BF02684781
  • J. A. Shalika, The multiplicity one theorem for $\textrm {GL}_{n}$, Ann. of Math. (2) 100 (1974), 171–193. MR 348047, DOI 10.2307/1971071
  • Takuro Shintani, On an explicit formula for class-$1$ “Whittaker functions” on $GL_{n}$ over $P$-adic fields, Proc. Japan Acad. 52 (1976), no. 4, 180–182. MR 407208
  • [34] A. Weil, Dirichlet Series and Automorphic Forms, Lect. Notes in Math. 189, Springer Verlag, 1971.
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Bibliographic Information
  • E. Frenkel
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
  • MR Author ID: 257624
  • ORCID: 0000-0001-6519-8132
  • D. Gaitsgory
  • Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
  • D. Kazhdan
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
  • MR Author ID: 99580
  • K. Vilonen
  • Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254
  • MR Author ID: 178620
  • Received by editor(s): March 31, 1997
  • Received by editor(s) in revised form: November 26, 1997
  • © Copyright 1998 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 11 (1998), 451-484
  • MSC (1991): Primary 11R39, 11F70; Secondary 14H60, 22E55
  • DOI:
  • MathSciNet review: 1484882