On CAP representations for even orthogonal groups I: A correspondence of unramified representations

Abstract

The authors prove the local unramified correspondence for a new type of construction of CAP representations of even orthogonal groups by a generalized automorphic descent method. This method is expected to work for all classical groups.

Keywords

Automorphic representations / Fourier coefficients / Satake parameter transfer

References

[1] Arthur, J., The Endoscopic Classification of Representations. Orthogonal and Symplectic Groups, American Mathematical Society Colloquium Publications, 61, American Mathematical Society, Providence, RI, 2013.

[2] Cogdell J. W., Kim H. H., Piatetski-Shapiro I. I., Shahidi F.. Functoriality for the classical groups. Publ. Math. Inst. Hautes Etudes Sci., 2004, 99: 163-233

[3] Ginzburg D.. Endoscopic lifting in classical groups and poles of tensor L functions. Duke Math. Journal, 2008, 141(3): 447-503

[4] Ginzburg D.. Constructing automorphic representations in split classical groups. Electron. Res. Announc. Math. Sci., 2012, 19: 18-32

[5] Ginzburg D., Jiang D., Soudry D.. Periods of automorphic forms, poles of L-functions, and functorial lifting. Science in China: Mathematics, 2010, 53(10): 2215-2238

[6] Ginzburg, D., Jiang, D. and Soudry, D., On CAP representations for even orthogonal groups II: Global constructions, in preparation.

[7] Ginzburg D., Rallis S., Soudry D.. On explicit lifts of cusp forms from GLm to classical groups. Annals of Math., 1999, 150: 807-866

[8] Ginzburg D., Rallis S., Soudry D.. On Fourier coefficients of automorphic forms of symplectic groups. Manuscripta Math., 2003, 111: 1-16

[9] Ginzburg, D., Rallis, S. and Soudry, D., The Descent Map from Automorphic Representation of GL(n) to Classical Groups, World Scientific, Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.

[10] Jacquet H., Shalika J.. On Euler products and classification of automorphic forms I. American J. Math., 1981, 103: 499-588

[11] Jiang D.. Automorphic Integral Transforms for Classical Groups I: Endoscopy Correspondences, Automorphic Forms: L-functions and Related Geometry: Assessing the Legacy of I.I. Piatetski-Shapiro. Contemporary Mathematics, American Mathematical Society, 2014, 614: 179-242

[12] Jiang D., Liu B., Zhang L.. Poles of certain residual Eisenstein series of classical groups. Pacific Journal of Mathematics, 2013, 264(1): 83-123

[13] Lapid E., Muic G., Tadic M.. On the generic unitary dual of quasi-split classical groups. IMRN, 2004, 26: 1335-1354

[14] Moeglin C., Waldspurger J. L.. Modéles de Whittaker dégénérés pour des groupes p-adiques. Math. Z., 1987, 196: 427-452

[15] Piatetski-Shapiro I., Rallis S.. L-Functions for the Classical Groups in Explicit Constructions of Automorphic L-Functions, Leture Notes in Mathematrics, 1254. Springer-Verlag, 1987 1-52

[16] Zelevinsky, A., Representations of Finite Classical Groups, Leture Notes in Mathematics, 869, Springer-Verlag, 1981.