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Distinguished representations and a Fourier summation formula. (English) Zbl 0778.11030

A new “Fourier” summation formula, called a relative trace formula by other authors, is proposed in the context of both \(\text{GL}(n,E)\) and the quasi-split unitary group \(U(n,E/F)\) associated with a quadratic extension \(E/F\) of global fields. The proof of this summation formula is then reduced to a local technical conjecture concerning matching of Fourier orbital integrals.
On the other hand, the author proves that the conjectured summation formula implies the following conjecture of the author: the stable (if \(n\) is odd) and the unstable (if \(n\) is even) base change is a surjection from (a) the set of irreducible automorphic discrete-series non- degenerate representations of the group of adele points on \(U(n,E/F)\), to (b) the set of automorphic irreducible nondegenerate representations of \(\text{GL}(n)\) over the adele ring of \(E\) normalizedly induced from a representation \(\rho_ 1 \times \dots \times \rho_ a\) of a parabolic subgroup of type \((n_ 1,\dots,n_ a)\), where the \(\rho_ i\) are pairwise inequivalent distinguished cuspidal nondegenerate representations of \(\text{GL}(n_ i)\) over the adele ring of \(E\).
Reviewer: Y.Ye (Iowa City)

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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References:

[1] ARTHUR (J.) . - A trace formula for reductive groups , [A1], I. Terms associated to classes in G(Q) , Duke Math. J., t. 45, 1978 , p. 911-952, and [A2], II. Applications of a truncation operator , Compositio Math., t. 40, 1980 , p. 87-121. Numdam | Zbl 0499.10032 · Zbl 0499.10032 · doi:10.1215/S0012-7094-78-04542-8
[2] BERNSTEIN (J.) and ZELEVINSKII (A.) . - Representations of the group GL(n, F) where F is a nonarchimedean local field , Uspekhi Mat. Nauk, t. 31, 1976 , p. 5-70, and [BZ1], Induced representations of reductive p-adic groups. I , Ann. Sci. École Norm. Sup., t. 10, 1977 , p. 441-472. Numdam | Zbl 0412.22015 · Zbl 0412.22015
[3] BIEN (F.) . - D-modules and spherical representations . - Princeton Univ. Press, 39, 1990 . MR 92f:22025 | Zbl 0723.22014 · Zbl 0723.22014
[4] BOPP (N.) et HARINCK (P.) . - Formule de Plancherel pour GL(n, \Bbb C)/U(p, q) , Math. Gottingensis, t. 27, 1990 , p. 1-52.
[5] DELORME (P.) . - Coefficients généralisés de séries principales sphériques et distributions sphériques sur G(\Bbb C)/G(\Bbb R) , Invent. Math., t. 105, 1991 , p. 305-346. MR 92j:22027 | Zbl 0741.43010 · Zbl 0741.43010 · doi:10.1007/BF01232269
[6] FLENSTED-JENSEN (M.) . - Discrete series for semi-simple symmetric spaces , Ann. of Math., t. 111, 1980 , p. 253-311. MR 81h:22015 | Zbl 0462.22006 · Zbl 0462.22006 · doi:10.2307/1971201
[7] FLICKER (Y.) . - [F1], On distinguished representations , J. Reine Angew. Math., t. 418, 1991 , p. 139-172 and [F2], Twisted tensors and Euler products , Bull. Soc. Math. France, t. 116, 1988 , p. 295-313, and [F3], Stable and labile base change for U(2) , Duke Math. J., t. 49, 1982 , p. 691-729, and [F4] Packets and liftings for U(3) , J. Analyse Math, t. 50, 1988 , p. 19-63, and [F5], On the local twisted tensor L-function , preprint. MR 92i:22019 · Zbl 0725.11026
[8] FLICKER (Y.) and KAZHDAN (D.) . - [FK1], Metaplectic correspondence , Publ. Math. IHES, t. 64, 1987 , p. 53-110, and [FK2], A simple trace formula , J. Analyse Math., t. 50, 1988 , p. 189-200. · Zbl 0616.10024
[9] GELFAND (I.) and KAZHDAN (D.) . - Representations of GL(n, K) where K is a local field , in Lie groups and their representations, John Wiley and Sons, 1975 , p. 95-118. MR 53 #8334 | Zbl 0348.22011 · Zbl 0348.22011
[10] JACQUET (H.) , PIATETSKI-SHAPIRO (I.) and SHALIKA (J.) . - Rankin-Selberg convolutions , Amer. J. Math., t. 105, 1983 , p. 367-464. MR 85g:11044 | Zbl 0525.22018 · Zbl 0525.22018 · doi:10.2307/2374264
[11] JACQUET (H.) and SHALIKA (J.) . - On Euler products and the classification of automorphic representations , Amer. J. Math., t. 103, 1981 , p. I : 499-558, II : 777-815, and [JS1] Rankin-Selberg convolutions : Archimedean theory , in Piatetski-Shapiro Festchrift I, IMCP, t. 2, 1990 , p. 125-207. MR 82m:10050a | Zbl 0473.12008 · Zbl 0473.12008 · doi:10.2307/2374103
[12] KEYS (D.) . - [K1], Principal series representations of special unitary groups over local field , Compositio Math., t. 51, 1984 , p. 115-130, and [K2] L-indistinguishability and R-groups for quasi-split groups ; unitary groups in even dimension , Ann. Sci. École Norm. Sup., t. 20, 1987 , p. 31-64. Numdam | MR 85d:22031 | Zbl 0547.22009 · Zbl 0547.22009
[13] LANGLANDS (R.) . - [L1], Problems in the theory of automorphic forms , in Modern Analysis and applications, SLN, t. 170, 1970 , p. 18-86, and [L2], On the functional equations satisfied by Eisenstein series , SLN, t. 544, 1976 , and [L3], Book review of The Theory of Eisenstein Systems , by S. Osborne and G. Warner, Academic Press, New-York 1981 , Bull. AMS, t. 9, 1983 , p. 351-361. Article | MR 46 #1758
[14] MOEGLIN (C.) et WALDSPURGER (J.-L) . - [MW1], Le spectre résiduel de GL(n) , Ann. Sci. École Norm. Sup., t. 22, 1989 , p. 605-674, and [MW2], Décomposition spectrale et séries d’Eisenstein . Numdam | MR 91b:22028 | Zbl 0696.10023 · Zbl 0696.10023
[15] MORRIS (L.) . - Eisenstein series for reductive groups over global function fields , Canad. J. Math., t. 34, 1982 , I : p. 91-168, II : p. 1112-1182. MR 84j:22024a | Zbl 0499.22021 · Zbl 0499.22021 · doi:10.4153/CJM-1982-009-2
[16] OSHIMA (T.) and MATSUKI (T.) . - A description of discrete series for semisimple symmetric spaces , Adv. Stud. Pure Math., t. 4, 1984 , p. 331-390. MR 87m:22042 | Zbl 0577.22012 · Zbl 0577.22012
[17] SANO (S.) . - Invariant spherical distributions and the Fourier inversion formula on GL(n, \Bbb C)/GL(n, \Bbb R) , J. Math. Soc. Japan, t. 36, 1984 , p. 191-219. Article | MR 85h:43011 | Zbl 0545.43010 · Zbl 0545.43010 · doi:10.2969/jmsj/03620191
[18] SHAHIDI (F.) . - [Sh1], On certain L-functions , Amer. J. Math., t. 103, 1981 , p. 297-355, and [Sh2], Fourier transforms of intertwining operators and Plancherel measures for GL(n) , Amer. J. Math., t. 104, 1982 , p. 67-111, and [Sh3], Local coefficients and normalization of intertwining operators for GL(n) , Compositio Math., t. 48, 1983 , p. 271-295. Numdam | MR 82i:10030 · Zbl 0467.12013
[19] TADIC (M.) . - Classification of unitary representations in irreducible representations of general linear group , Ann. Sci. École Norm. Sup., t. 19, 1986 , p. 335-382. Numdam | MR 88b:22021 | Zbl 0614.22005 · Zbl 0614.22005
[20] VOGAN (D.) . - Gelfand-Kirillov dimension for Harish-Chandra modules , Invent. Math., t. 48, 1978 , p. 75-98. MR 58 #22205 | Zbl 0389.17002 · Zbl 0389.17002 · doi:10.1007/BF01390063
[21] WALDSPURGER (J.-L.) . - Correspondence de Shimura , J. Math. Pures Appl., t. 59, 1980 , p. 1-113, and Sur les coefficients de Fourier des formes modulaires de poids demi-entier , J. Math. Pures Appl., t. 60, 1981 , p. 375-484. Zbl 0431.10015 · Zbl 0431.10015
[22] ZELEVINSKY (A.) . - Induced representations of reductive p-adic groups, II. On irreducible representations of GL(n) , Ann. Sci. École Norm. Sup., t. 13, 1980 , p. 165-210. Numdam | MR 83g:22012 | Zbl 0441.22014 · Zbl 0441.22014
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