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The archimedean theory of the exterior square $ L$-functions over $ \mathbb{Q}$

Authors: Stephen D. Miller and Wilfried Schmid
Journal: J. Amer. Math. Soc. 25 (2012), 465-506
MSC (2010): Primary 11F55, 11F66
Published electronically: December 6, 2011
MathSciNet review: 2869024
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Abstract: The analytic properties of automorphic $ L$-functions have historically been obtained either through integral representations (the ``Rankin-Selberg method'') or properties of the Fourier expansions of Eisenstein series (the ``Langlands-Shahidi method''). We introduce a method based on pairings of automorphic distributions that appears to be applicable to a wide variety of $ L$-functions, including all which have integral representations. In some sense our method could be considered a completion of the Rankin-Selberg method because of its common features. We consider a particular but representative example, the exterior square $ L$-functions on $ GL(n)$, by constructing a pairing which we compute as a product of this $ L$-function times an explicit ratio of Gamma functions. We use this to deduce that exterior square $ L$-functions, when multiplied by the Gamma factors predicted by Langlands, are holomorphic on $ \mathbb{C}-\{0,1\}$ with at most simple poles at 0 and 1, proving a conjecture of Langlands which has not been obtained by the existing two methods.

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  • [1] Dustin Belt, On Local Exterior-Square L-functions, Ph.D. Thesis, 2011.
  • [2] Daniel Bump and Solomon Friedberg, The exterior square automorphic $ L$-functions on $ {\rm GL}(n)$, Festschrift in honor of I. I. Piatetski-Shapiro, Part II, 1990, pp. 47-65. MR 1159108 (93d:11050)
  • [3] W. Casselman, Jacquet modules for real reductive groups, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), 1980, pp. 557-563. MR 562655 (83h:22025)
  • [4] W. Casselman, Canonical extensions of Harish-Chandra modules to representations of $ G$, Canad. J. Math. 41 (1989), no. 3, 385-438. MR 1013462 (90j:22013)
  • [5] William Casselman, Henryk Hecht, and Dragan Miličić, Bruhat filtrations and Whittaker vectors for real groups, The mathematical legacy of Harish-Chandra, 2000, pp. 151-190. MR 1767896 (2002b:22023)
  • [6] Jacques Dixmier and Paul Malliavin, Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bull. Sci. Math. (2) 102 (1978), no. 4, 307-330. MR 517765 (80f:22005)
  • [7] Sergey Fomin and Andrei Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), no. 2, 335-380. MR 1652878 (2001f:20097)
  • [8] Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Springer-Verlag, Berlin, 1972. MR 0342495 (49:7241)
  • [9] Roe Goodman and Nolan R. Wallach, Whittaker vectors and conical vectors, J. Funct. Anal. 39 (1980), no. 2, 199-279. MR 597811 (82i:22018)
  • [10] Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, 2001. MR 1876802 (2002m:11050)
  • [11] Guy Henniart, Une preuve simple des conjectures de Langlands pour $ {\rm GL}(n)$ sur un corps $ p$-adique, Invent. Math. 139 (2000), no. 2, 439-455 (French, with English summary). MR 1738446 (2001e:11052)
  • [12] H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499-558. MR 618323 (82m:10050a)
  • [13] Hervé Jacquet and Joseph Shalika, Exterior square $ L$-functions, Automorphic forms, Shimura varieties, and $ L$-functions, Vol.II (Ann Arbor, MI, 1988), 1990, pp. 143-226. MR 1044830 (91g:11050)
  • [14] Hervé Jacquet, Ilja Iosifovitch Piatetski-Shapiro, and Joseph Shalika, Automorphic forms on $ {\rm GL}(3)$, Ann. of Math. (2) 109 (1979), no. 1 and  2, 169-258. MR 519356 (80i:10034a); MR 0528964 (80i:10034b)
  • [15] Dihua Jiang, On the fundamental automorphic $ L$-functions of $ {\rm SO}(2n+1)$, Int. Math. Res. Not. (2006). MR 2211151 (2007c:11063)
  • [16] Henry H. Kim, Langlands-Shahidi method and poles of automorphic $ L$-functions: application to exterior square $ L$-functions, Canad. J. Math. 51 (1999), no. 4, 835-849. MR 1701344 (2000f:11058)
  • [17] Henry H. Kim, Functoriality for the exterior square of $ {\rm GL}\sb 4$ and the symmetric fourth of $ {\rm GL}\sb 2$, J. Amer. Math. Soc. 16 (2003), no. 1, 139-183 (electronic). MR 1937203 (2003k:11083)
  • [18] Robert P. Langlands, Problems in the theory of automorphic forms, Lectures in modern analysis and applications, III, 1970, pp. 18-61. Lecture Notes in Math., Vol. 170. MR 0302614 (46:1758)
  • [19] Robert P. Langlands, Euler products, Yale University Press, New Haven, Conn., 1971. MR 0419366 (54:7387)
  • [20] Robert P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, 1989, pp. 101-170. MR 1011897 (91e:22017)
  • [21] Stephen D. Miller and Wilfried Schmid, Distributions and analytic continuation of Dirichlet series, J. Funct. Anal. 214 (2004), no. 1, 155-220. MR 2079889 (2005i:11132)
  • [22] Stephen D. Miller and Wilfried Schmid, Automorphic distributions, $ L$-functions, and Voronoi summation for $ {\rm GL}(3)$, Ann. of Math. (2) 164 (2006), no. 2, 423-488. MR 2247965 (2007j:11065)
  • [23] Stephen D. Miller and Wilfried Schmid, The Rankin-Selberg method for automorphic distributions, Representation theory and automorphic forms, Progr. Math., vol. 255, Birkhäuser Boston, Boston, MA, 2008, pp. 111-150. MR 2369497 (2009a:11119)
  • [24] Stephen D. Miller and Wilfried Schmid, Pairings of automorphic distributions. To appear in Mathematische Annalen,
  • [25] Stephen D. Miller and Wilfried Schmid, Adelization of Automorphic Distributions and Mirabolic Eisenstein Series. To appear in Contemporary Mathematics, volume in honor of Gregg Zuckerman's 60th birthday, Jeff Adams, Bong Lian, and Siddhartha Sahi, editors,
  • [26] Stephen D. Miller and Wilfried Schmid, On the rapid decay of cuspidal automorphic forms.
  • [27] Wilfried Schmid, Automorphic distributions for $ {\rm SL}(2,\mathbb{R})$, Conférence Moshé Flato 1999, Vol. I (Dijon), 2000, pp. 345-387. MR 1805897 (2002g:11061)
  • [28] Freydoon Shahidi, A proof of Langlands' conjecture on Plancherel measures; complementary series for $ p$-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273-330. MR 1070599 (91m:11095)
  • [29] Eric Stade, Mellin transforms of $ {\rm GL}(n,\mathbb{R})$ Whittaker functions, Amer. J. Math. 123 (2001), no. 1, 121-161. MR 1827280 (2003a:22010)
  • [30] Marko Tadić, $ \widehat {\rm GL}(n,\mathbb{C})$ and $ \widehat {\rm GL}(n,\mathbb{R})$, Automorphic forms and $ L$-functions II. Local aspects, Contemp. Math., vol. 489, Amer. Math. Soc., Providence, RI, 2009, pp. 285-313. MR 2537046 (2010j:22020)
  • [31] David A. Vogan Jr., Gelfand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), no. 1, 75-98. MR 0506503 (58:22205)
  • [32] Nolan R. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, Lie group representations, I, 1983, pp. 287-369. MR 727854 (85g:22029)
  • [33] D. P. Želobenko, Compact Lie groups and their representations, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 40. MR 0473098 (57:12776b)

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Additional Information

Stephen D. Miller
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854-8019

Wilfried Schmid
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Received by editor(s): November 24, 2010
Received by editor(s) in revised form: September 12, 2011
Published electronically: December 6, 2011
Additional Notes: The first author was partially supported by NSF grant DMS-0901594 and an Alfred P. Sloan Foundation Fellowship
The second author was partially supported by DARPA grant HR0011-04-1-0031 and NSF grant DMS-0500922
Dedicated: In memory of Joseph Shalika
Article copyright: © Copyright 2011 by Stephen D. Miller and Wilfried Schmid

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