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| In mathematics, the '''Langlands–Shahidi method''' provides the means to define [[automorphic L-function]]s in many cases that arise with connected [[reductive group]]s over a [[Algebraic number field|number field]]. This includes [[Rankin–Selberg method|Rankin–Selberg]] products for cuspidal [[automorphic representation]]s of [[general linear group]]s. The method develops the theory of the '''local coefficient''', which links to the global theory via [[Eisenstein series]]. The resulting ''L''-functions satisfy a number of analytic properties, including an important functional equation.
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| == The local coefficient ==
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| The setting is in the generality of a connected quasi-split reductive group ''G'', together with a '''Levi''' subgroup ''M'', defined over a [[local field]] ''F''. For example, if ''G'' = ''G<sub>l</sub>'' is a [[classical group]] of rank ''l'', its maximal Levi subgroups are of the form GL(''m'') × ''G<sub>n</sub>'', where ''G<sub>n</sub>'' is a classical group of rank ''n'' and of the same type as ''G<sub>l</sub>'', ''l'' = ''m'' + ''n''. [[Freydoon Shahidi|F. Shahidi]] develops the theory of the '''local coefficient''' for irreducible generic representations of ''M(F)''.<ref name=''sh1>F. Shahidi, ''On certain ''L''-functions'', American Journal of Mathematics '''103''' (1981), 297–355.</ref> The local coefficient is defined by means of the uniqueness property of [[Whittaker model]]s paired with the theory of intertwining operators for representations obtained by parabolic induction from generic representations.
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| The global intertwining operator appearing in the functional equation of [[Robert Langlands|Langlands]]' theory of Eisenstein series<ref name="L1">R. P. Langlands, ''On the Functional Equations Satisfied by Eisenstein Series'', Lecture Notes in Math., Vol. 544, Springer-Verlag, Berlin-Heidelberg-New York, 1976.</ref> can be decomposed as a product of local intertwining operators. When ''M'' is a maximal Levi subgroup, local coefficients arise from Fourier coefficients of appropriately chosen Eisenstein series and satisfy a crude functional equation involving a product of partial ''L''-functions.
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| == Local factors and functional equation ==
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| An induction step refines the crude functional equation of a globally generic cuspidal automorphic representation <math>\pi = \otimes' \pi_v</math> to individual functional equations of partial ''L''-functions and '''γ-factors''':<ref name="sh2">F. Shahidi, ''A proof of Langlands conjecture on Plancherel measures; Complementary series for ''p''-adic groups'', Annals of Mathematics '''132''' (1990), 273–330.</ref>
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| :<math>L^S(s,\pi,r_i) = \prod_{v \in S} \gamma_i(s,\pi_v,\psi_v) L^S(1-s,\tilde{\pi},r_i).</math>
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| The details are technical: ''s'' a complex variable, ''S'' a finite set of places (of the underlying global field) with <math>\pi_v</math> unramified for ''v'' outside of ''S'', and <math>r = \oplus r_i</math> is the adjoint action of ''M'' on the complex Lie algebra of a specific subgroup of the Langlands [[dual group]] of ''G''. When ''G'' is the [[special linear group]] SL(2), and ''M'' = ''T'' is the maximal torus of diagonal matrices, then π is a [[Hecke character|Größencharakter]] and the corresponding γ-factors are the local factors of [[John Tate|Tate's thesis]].
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| The γ-factors are uniquely characterized by their role in the functional equation and a list of local properties, including multiplicativity with respect to parabolic induction. They satisfy a relationship involving [[Artin L-function]]s and [[Artin root number]]s when ''v'' gives an archimedean local field or when ''v'' is non-archimedean and <math>\pi_v</math> is a constituent of an unramified principal series representation of ''M(F)''. Local ''L''-functions and root numbers ε<math>(s,\pi_v,r_{i,v},\psi_v)</math> are then defined at every place, including <math>v \in S</math>, by means of Langlands classification for ''p''-adic groups. The functional equation takes the form
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| :<math>L(s,\pi,r_i) = \epsilon(s,\pi,r_i) L(1-s,\tilde{\pi},r_i),</math> | |
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| where <math>L(s,\pi,r_i)</math> and <math>\epsilon(s,\pi,r_i)</math> are the completed global ''L''-function and root number.
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| == Examples of automorphic ''L''-functions ==
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| * <math>L(s,\pi_1 \times \pi_2)</math>, the Rankin–Selberg ''L''-function of cuspidal automorphic representations <math>\pi_1</math> of GL(''m'') and <math>\pi_2</math> of GL(''n'').
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| * <math>L(s,\tau \times \pi)</math>, where τ is a cuspidal automorphic representation of GL(''m'') and π is a globally generic cuspidal automorphic representation of a classical group ''G''.
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| * <math>L(s,\tau,r)</math>, with τ as before and ''r'' a symmetric square, an exterior square, or an Asai representation of the dual group of GL(''n'').
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| A full list of '''Langlands–Shahidi L-functions'''<ref name="sh3">F. Shahidi, ''Eisenstein Series and Automorphic ''L''-functions'', Colloquium Publications, Vol. 58, American Mathematical Society, Providence, Rhode Island, 2010. ISBN 978-0-8218-4989-7</ref> depends on the quasi-split group ''G'' and maximal Levi subgroup ''M''. More specifically, the decomposition of the adjoint action <math>r = \oplus r_i</math> can be classified using [[Dynkin diagram]]s. A first study of automorphic ''L''-functions via the theory of Eisenstein Series can be found in Langlands' '''Euler Products''',<ref name="la">R. P. Langlands, ''Euler Products'', Yale Univ. Press, New Haven, 1971</ref> under the assumption that the automorphic representations are everywhere unramified. What the Langlands–Shahidi method provides is the definition of ''L''-functions and root numbers with no other condition on the representation of ''M'' other than requiring the existence of a Whittaker model.
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| == Analytic properties of ''L''-functions ==
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| Global ''L''-functions are said to be '''nice'''<ref name="CPS">J. W. Cogdell and I. I. Piatetski–Shapiro, ''Converse theorems for GL(''n'')'', Publications Mathématiques de l'IHÉS '''79''' (1994), 157–214.</ref> if they satisfy:
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| # <math>L(s,\pi,r), \ L(s,\tilde{\pi}, r) \ </math> extend to entire functions of the complex variable ''s''.
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| # <math>L(s,\pi,r), \ L(s,\tilde{\pi},r) \ </math> are bounded in vertical strips.
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| # (Functional Equation) <math>L(s,\pi,r) = \epsilon(s,\pi,r) L(1-s,\tilde{\pi},r)</math>.
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| Langlands–Shahidi ''L''-functions satisfy the functional equation. Progress towards boundedness in vertical strips was made by S. S. Gelbart and F. Shahidi.<ref name="GeSh">S. Gelbart and F. Shahidi, ''Boundedness of automorphic ''L''-functions in vertical strips'', Journal of the American Mathematical Society, '''14''' (2001), 79–107.</ref> And, after incorporating twists by highly ramified characters, Langlands–Shahidi ''L''-functions do become entire.<ref name="KSh">H. H. Kim and F. Shahidi, ''Functorial products for GL(2) × GL(3) and the symmetric cube for GL(2)'', Annals of Mathematics '''155''' (2002), 837–893.</ref>
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| Another result is the non-vanishing of ''L''-functions. For Rankin–Selberg products of general linear groups it states that <math>L(1+it,\pi_1 \times \pi_2)</math> is non-zero for every real number ''t''.<ref name="Shahidi">F. Shahidi, On nonvanishing of L-functions.
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| Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 3, 462–464.</ref>
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| == Applications to functoriality and to representation theory of ''p''-adic groups ==
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| * '''Functoriality for the classical groups''': A cuspidal globally generic automorphic representation of a classical group admits a [[Langlands program|Langlands functorial]] lift to an automorphic representation of GL(''N''),<ref name="CKPSS">J. W. Cogdell, H. H. Kim, I. I. Piatetski–Shapiro, and F. Shahidi, ''Functoriality for the classical groups'', Publications Mathématiques de l'IHÉS '''99''' (2004), 163–233</ref> where ''N'' depends on the classical group. Then, the Ramanujan bounds of W. Luo, Z. Rudnick and P. Sarnak<ref name="LRS">W. Luo, Z. Rudnick, and P. Sarnak, ''On the generalized Ramanujan conjecture for GL(''n'')'', Proceedings of Symposia in Pure Mathematics '''66''', part 2 (1999), 301–310.</ref> for GL(''N'') over number fields yield non-trivial bounds for the [[Ramanujan–Peterson conjecture|generalized Ramanujan conjecture]] of the classical groups.
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| * '''Symmetric powers for GL(2)''': Proofs of functoriality for the symmetric cube and for the symmetric fourth<ref>H. H. Kim, ''Functoriality for the exterior square of GL(4) and the symmetric fourth of GL(2)'', Journal of the American Mathematical Society '''16''' (2002), 131–183.</ref> powers of cuspidal automorphic representations of GL(2) were made possible by the Langlands–Shahidi method. Progress towards higher Symmetric powers leads to the best possible bounds towards the [[Ramanujan–Peterson conjecture]] of automorphic cusp forms of GL(2).
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| * '''Representations of ''p''-adic groups''': Applications involving [[Harish-Chandra]] μ functions (from the Plancherel formula) and to complementary series of ''p''-adic reductive groups are possible. For example, GL(''n'') appears as the Siegel Levi subgroup of a classical group G. If π is a smooth irreducible ramified supercuspidal representation of GL(''n'', ''F'') over a field ''F'' of ''p''-adic numbers, and <math>I(\pi) = I(0,\pi)</math> is irreducible, then:
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| # <math>I(s,\pi)</math> is irreducible and in the complementary series for 0 < ''s'' < 1;
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| # <math>I(1,\pi)</math> is reducible and has a unique generic non-supercuspidal discrete series subrepresentation;
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| # <math>I(s,\pi)</math> is irreducible and never in the complementary series for ''s'' > 1.
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| Here, <math>I(s,\pi)</math> is obtained by unitary parabolic induction from
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| :*<math>\pi \otimes |\det|^s</math> if ''G'' = SO(2''n''), Sp(2''n''), or U(''n''+1, ''n'');
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| :*<math>\pi \otimes |\det|^{s/2}</math> if ''G'' = SO(2''n''+1) or U(''n'', ''n''). | |
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| == References ==
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| <!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
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| {{Reflist}}
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| <!--- Categories --->
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| {{DEFAULTSORT:Langlands-Shahidi method}}
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| [[Category:Automorphic forms]]
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| [[Category:Representation theory]]
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