×

Automorphic forms on \(\text{SO}(4)\). (English) Zbl 1061.11022

This paper is an announcement of the results on automorphic forms on \(\text{SO}(4)\) from the preprint [Y. Flicker, Lifting automorphic forms of \(\text{PGSp}(2)\) and \(\text{SO}(4)\) to \(\text{PGL}(4)\) (2001)].
The study of the stabilization of the twisted trace formula in [loc. cit.] leads to a theory of lifting of automorphic representations of \(\text{PGSp}(2)\) and \(\text{SO}(4)\) to \(\text{ PGL}(4)\). The announcement of the results for \(\text{PGSp}(2)\) is published in [Y. Flicker, Automorphic forms on \(\text{PGSp}(2)\), Electron. Res. Announc. Am. Math. Soc. 10, 39–50 (2004; Zbl 1062.11027)].
The paper under review states the results on endoscopic lifting from \(\text{SO}(4)\) to \(\text{PGL}(4)\). The first result is the statement that the functorial product of two automorphic representations \(\pi_1\) and \(\pi_2\) of \(\text{GL}(2, \mathbb{A})\) whose central characters \(\omega_1\), \(\omega_2\) satisfy \(\omega_1 \omega_2=1\) exists as an automorphic representation of \(\text{PGL}(4, \mathbb{A})\). The second statement is a rigidity theorem for the automorphic representations of \(\text{SO}(4)\).

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
22E46 Semisimple Lie groups and their representations
11F55 Other groups and their modular and automorphic forms (several variables)

Citations:

Zbl 1062.11027
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] Bernstein, J., and Zelevinsky, A.: Induced representations of reductive \(p\)-adic groups. I. Ann. Sci. École Norm. Sup., 10 , 441-472 (1977). · Zbl 0412.22015
[2] Blasius, D.: On multiplicities for \(\SL(n)\). Israel J. Math., 88 , 237-251 (1994). · Zbl 0826.11023 · doi:10.1007/BF02937513
[3] Borel, A.: Automorphic \(L\)-functions. Proc. Sympos. Pure Math., 33 II. Amer. Math. Soc., Providence, R.I., pp. 27-63 (1979). · Zbl 0412.10017
[4] Flicker, Y.: Lifting automorphic forms of PGSp(2) and SO(4) to PGL(4). (2001). (Preprint). · Zbl 1062.11027 · doi:10.1090/S1079-6762-04-00128-3
[5] Flicker, Y.: Automorphic forms on PGSp(2). Elect. Res. Announc. AMS, 10, 39-50 (2004); http://www.ams.org/era. · Zbl 1062.11027 · doi:10.1090/S1079-6762-04-00128-3
[6] Flicker, Y.: On the symmetric square. IV. Applications of a trace formula. Trans. Amer. Math. Soc., 330 , 125-152 (1992). · Zbl 0761.11027 · doi:10.2307/2154157
[7] Flicker, Y.: On the symmetric square. V. Unit elements. Pacific J. Math., 175 , 507-526 (1996). · Zbl 0865.11045
[8] Flicker, Y.: On the symmetric square. VI. Total global comparison. J. Funct. Analyse, 122 , 255-278 (1994). · Zbl 0815.11030 · doi:10.1006/jfan.1994.1068
[9] Flicker, Y., and Zinoviev, D.: Twisted character of a small representation of PGL(4). Moscow Math. J. (2004). (To appear). · Zbl 1066.11022
[10] Harish-Chandra: Admissible invariant distributions on reductive \(p\)-adic groups. Preface and notes by Stephen DeBacker and Paul J. Sally, Jr. University Lecture Series, no. 16, Americal Mathematicsl Society, Providence, RI, xiv,+,97,pp. (1999). · Zbl 0928.22017
[11] Harish-Chandra: Admissible invariant distributions on reductive \(p\)-adic groups. Lie theories and their applications (Proc. Ann. Sem. Canad. Math. Congr., Queen’s Univ., Kingston, Ont., 1977), Queen’s Papers in Pure Appl. Math. no.,48, Queen’s Univ., Kingston, Ont., pp. 281-347 (1978). · Zbl 0433.22012
[12] Jacquet, H., and Shalika, J.: On Euler products and the classification of automorphic forms II. Amer. J. Math., 103 , 777-815 (1981). · Zbl 0491.10020 · doi:10.2307/2374050
[13] Kottwitz, R.: Stable trace formula: cuspidal tempered terms. Duke Math. J., 51 , 611-650 (1984). · Zbl 0576.22020 · doi:10.1215/S0012-7094-84-05129-9
[14] Kottwitz, R., and Shelstad, D.: Foundations of twisted endoscopy. Asterisque, 255 , vi,+,190,pp. (1999). · Zbl 0958.22013
[15] Moeglin, C., and Waldspurger, J.-L.: Le spectre résiduel de \(\GL(n)\). Ann. Sci. Ecole Norm. Sup., 22 , 605-674 (1989). · Zbl 0696.10023
[16] Ramakrishnan, D.: Modularity of the Rankin-Selberg \(L\)-series, and multiplicity one for \(\SL(2)\). Ann. of Math., 152 , 45-111 (2000). · Zbl 0989.11023 · doi:10.2307/2661379
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.