## Local coefficients as Artin factors for real groups.(English)Zbl 0674.10027

This paper deals with the local constant that appears in the functional equation of the automorphic L-function of Langlands. Let G be the real points of a quasisplit reductive algebraic group over $${\mathbb{R}}$$, $$P=MAN$$ a standard parabolic of G, $$\sigma$$ a generic representation of M; the local constant C($$\sigma)$$ (ignoring other data) is defined by the Schiffman-Knapp-Stein intertwining operator and Whittaker function. The author proves that $C(\sigma)=(i)^{2m+p}\prod^{n}_{i=1}\epsilon (a_ is,r_ i)\frac{L(1-a_ is,\tilde r_ i)}{L(a_ is,r_ i)}$ where L is the Artin L-function and $$\epsilon$$ is the root number. As an application the author gives a new proof of the function equation of the Rankin-Selberg L-function attached to pairs of cusp forms.

### MSC:

 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E30 Analysis on real and complex Lie groups 11R39 Langlands-Weil conjectures, nonabelian class field theory 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols
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### References:

 [1] J. Arthur, On some problems suggested by the trace formula , Lie group representations, II (College Park, Md., 1982/1983), Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 1-49. · Zbl 0541.22011 [2] J. Arthur, On the invariant distributions associated to weighted orbital integrals , · Zbl 0301.47014 [3] P. Delorme, Homomorphismes de Harish-Chandra liés aux $$K$$-types minimaux des séries principales généralisées des groupes de Lie réductifs connexes , Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 1, 117-156. · Zbl 0582.22009 [4] R. Goodman and N. R. Wallach, Whittaker vectors and conical vectors , J. Functional Anal. 39 (1980), no. 2, 199-279. · Zbl 0475.22010 [5] M. Hashizume, Whittaker models for real reductive groups , Japan. J. Math. (N.S.) 5 (1979), no. 2, 349-401. · Zbl 0506.22016 [6] H. Jacquet, Fonctions de Whittaker associées aux groupes de Chevalley , Bull. Soc. Math. France 95 (1967), 243-309. · Zbl 0155.05901 [7] 1 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. JSTOR: · Zbl 0473.12008 [8] 2 H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II , Amer. J. Math. 103 (1981), no. 4, 777-815. JSTOR: · Zbl 0491.10020 [9] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions , Amer. J. Math. 105 (1983), no. 2, 367-464. JSTOR: · Zbl 0525.22018 [10] D. Keys, Principal series representations of special unitary groups over local fields , Compositio Math. 51 (1984), no. 1, 115-130. · Zbl 0547.22009 [11] A. W. Knapp, Weyl group of a cuspidal parabolic , Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 2, 275-294. · Zbl 0305.22010 [12] A. W. Knapp, Commutativity of intertwining operators for semisimple groups , Compositio Math. 46 (1982), no. 1, 33-84. · Zbl 0488.22027 [13] A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups , Ann. of Math. (2) 93 (1971), 489-578. JSTOR: · Zbl 0257.22015 [14] A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups. II , Invent. Math. 60 (1980), no. 1, 9-84. · Zbl 0454.22010 [15] A. W. Knapp and N. R. Wallach, Szegö kernels associated with discrete series , Invent. Math. 34 (1976), no. 3, 163-200. · Zbl 0332.22015 [16] A. W. Knapp and Gregg J. Zuckerman, Classification of irreducible tempered representations of semisimple groups , Ann. of Math. (2) 116 (1982), no. 2, 389-455. JSTOR: · Zbl 0516.22011 [17] B. Kostant, On Whittaker vectors and representation theory , Invent. Math. 48 (1978), no. 2, 101-184. · Zbl 0405.22013 [18] J.-P. Labesse and R. P. Langlands, $$L$$-indistinguishability for $$\mathrm SL(2)$$ , Canad. J. Math. 31 (1979), no. 4, 726-785. · Zbl 0421.12014 [19] R. P. Langlands, On Artin’s $$L$$-functions , Rice University Studies 56 (1970), 23-28. · Zbl 0245.12011 [20] R. P. Langlands, On the functional equations satisfied by Eisenstein series , Lecture Notes in Mathematics, vol. 544, Springer-Verlag, Berlin, 1976. · Zbl 0332.10018 [21] R. P. Langlands, Euler products , Yale University Press, New Haven, Conn., 1971. · Zbl 0231.20016 [22] R. P. Langlands, On the classification of irreducible representations of real algebraic groups , Mimeographed notes, Institute for advanced study, 1973. [23] C. J. Moreno and F. Shahidi, The $$L$$-function $$L_ 3(s,\pi_ \Delta)$$ is entire , Invent. Math. 79 (1985), no. 2, 247-251. · Zbl 0558.10025 [24] N. S. Poulsen, On $$C^\infty$$-vectors and intertwining bilinear forms for representations of Lie groups , J. Functional Analysis 9 (1972), 87-120. · Zbl 0237.22013 [25] G. Schiffmann, Intégrales d’entrelacement et fonctions de Whittaker , Bull. Soc. Math. France 99 (1971), 3-72. · Zbl 0223.22017 [26] F. Shahidi, Functional equation satisfied by certain $$L$$-functions , Compositio Math. 37 (1978), no. 2, 171-207. · Zbl 0393.12017 [27] F. Shahidi, Whittaker models for real groups , Duke Math. J. 47 (1980), no. 1, 99-125. · Zbl 0433.22007 [28] F. Shahidi, On certain $$L$$-functions , Amer. J. Math. 103 (1981), no. 2, 297-355. JSTOR: · Zbl 0467.12013 [29] F. Shahidi, Fourier transforms of intertwining operators and Plancherel measures for $$\mathrm GL(n)$$ , Amer. J. Math. 106 (1984), no. 1, 67-111. JSTOR: · Zbl 0567.22008 [30] F. Shahidi, Some results on $$L$$-indistinguishability for $$\mathrm SL(r)$$ , Canad. J. Math. 35 (1983), no. 6, 1075-1109. · Zbl 0553.10024 [31] F. Shahidi, Artin $$L$$-functions and normalization of intertwining operators , Seminar on the Analytical Aspects of the Trace Formula II, Institute for Advanced Study, 1983-84. [32] J. A. Shalika, The multiplicity one theorem for $$\mathrm GL\sbn$$ , Ann. of Math. (2) 100 (1974), 171-193. JSTOR: · Zbl 0316.12010 [33] D. Shelstad, $$L$$-indistinguishability for real groups , Math. Ann. 259 (1982), no. 3, 385-430. · Zbl 0506.22014 [34] D. A. Vogan, Jr., Gelfand-Kirillov dimension for Harish-Chandra modules , Invent. Math. 48 (1978), no. 1, 75-98. · Zbl 0389.17002 [35] N. R. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups , Lie group representations, I (College Park, Md., 1982/1983), Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 287-369. · Zbl 0553.22005 [36] E. T. Whittaker and G. N. Watson, A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions , Fourth edition. Reprinted, Cambridge University Press, New York, 1962. · Zbl 0105.26901 [37] R. N. Wallach, Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals , · Zbl 0714.17016
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