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Arthur packets for pure real forms of symplectic and special orthogonal groups

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  12 March 2025

Nicolas Arancibia Robert  and
Paul Mezo
Nicolas Arancibia Robert
Affiliation:
Laboratoire AGM - Analyse, Géométrie, Modélisation, CY Cergy Paris Université - site Saint Martin 2 av. Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France e-mail: [email protected]
Paul Mezo*
Affiliation:
The School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada
*
e-mail: [email protected]
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Abstract

Arthur packets have been defined for pure real forms of symplectic and special orthogonal groups following two different approaches. The first approach, due to Arthur, Moeglin, and Renard uses harmonic analysis. The second approach, due to Adams, Barbasch, and Vogan uses microlocal geometry. We prove that the two approaches produce essentially equivalent Arthur packets. This extends previous work of the authors and J. Adams for the quasisplit real forms.

Keywords

EndoscopyArthur packetsGroup representations

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 22
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The second author was supported in part by the NSERC grant RGPIN-06361.

References

Achar, P. N., Perverse sheaves and applications to representation theory, Mathematical Surveys and Monographs, 258, American Mathematical Society, Providence, RI, 2021. MR4337423Google Scholar
Adams, J., Arancibia Robert, N., and Mezo, P., Equivalent definitions of Arthur packets for real classical groups . Mem. Amer. Math. Soc. 300(2024), no. 1503, v+110.Google Scholar
Arthur, J., Unipotent automorphic representations: Conjectures . Astérisque 171–172(1989), 13–71. MR1021499Google Scholar
Arthur, J., The endoscopic classification of representations, American Mathematical Society Colloquium Publications, 61, American Mathematical Society, Providence, RI, 2013. MR3135650Google Scholar
Adams, J., Barbasch, D., and Vogan, D. A. Jr., The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, 104, Birkhäuser Boston, Inc., Boston, MA, 1992. MR1162533Google Scholar
Arancibia, N., Mœglin, C., and Renard, D., Paquets d’Arthur des groupes classiques et unitaires . Ann. Fac. Sci. Toulouse Math. (6) 27(2018), no. 5, 1023–1105. MR3919547Google Scholar
Arancibia Robert, N. and Mezo, P., L-packets over strong real forms , Represent. Theory 28(2024), 20–48. MR4685815Google Scholar
Arancibia Robert, N. and Mezo, P., Equivalent definitions of Arthur packets for real unitary groups. Preprint, 2022. arXiv:2204.10715v2.Google Scholar
Adams, J. and Taïbi, O., Galois and Cartan cohomology of real groups . Duke Math. J. 167(2018), no. 6, 1057–1097.Google Scholar
Borel, A., Grivel, P.-P., Kaup, R., Haefliger, A., Malgrange, B., and Ehlers, F., Algebraic  $D$ -modules, Perspectives in Mathematics, 2, Academic Press, Inc., Boston, MA, 1987.Google Scholar
Bernstein, J. and Lunts, V., Equivariant sheaves and functors, Lecture Notes in Mathematics, 1578, Springer-Verlag, Berlin, 1994.Google Scholar
Clifton, L. R., Cunningham, A. F., Moussaoui, A., Mracek, J., and Xu, B., Arthur packets for $p$ -adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples . Mem. Amer. Math. Soc. 276(2022), no. 1353, ix+216. MR4391878Google Scholar
Christie, A. and Mezo, P., Twisted endoscopy from a sheaf-theoretic perspective . In: Geometric aspects of the trace formula, Springer, Cham, Switzerland, 2018, pp. 121–161.Google Scholar
Cunningham, C. and Ray, M., Proof of Vogan’s conjecture on Arthur packets: irreducible parameters of $p$ -adic general linear groups . Manuscripta Math. 173(2024), nos. 3–4, 1073–1097. MR4704767Google Scholar
Goresky, M. and MacPherson, R., Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 14, Springer-Verlag, Berlin, 1988.Google Scholar
Jacobson, N., Basic algebra. I. 2nd ed., W. H. Freeman and Company, New York, NY, 1985. MR780184Google Scholar
Kaletha, T., Rigid inner forms of real and $p$ -adic groups . Ann. Math. (2) 184(2016), no. 2, 559–632.Google Scholar
Knapp, A. W., Lie groups beyond an introduction. 2nd ed., Progress in Mathematics, 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR1920389Google Scholar
Knus, M.-A., Merkurjev, A., Rost, M., and Tignol, J.-P., The book of involutions, American Mathematical Society Colloquium Publications, 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR1632779Google Scholar
Kottwitz, R. E. and Shelstad, D., Foundations of twisted endoscopy . Astérisque 255(1999), vi+190. MR1687096Google Scholar
Langlands, R. P., On the classification of irreducible representations of real algebraic groups . In: Representation theory and harmonic analysis on semisimple Lie groups, American Mthematical Society, Providence, RI, 1989, pp. 101–170. MR1011897Google Scholar
Langlands, R. P. and Shelstad, D., On the definition of transfer factors . Math. Ann. 278(1987), nos. 1–4, 219–271. MR909227Google Scholar
Mezo, P., Character identities in the twisted endoscopy of real reductive groups . Mem. Amer. Math. Soc. 222(2013), no. 1042, vi+94.Google Scholar
Moeglin, C. and Renard, D., Sur les paquets d’Arthur des groupes classiques et unitaires non quasi-d’eployés . In: Relative aspects in representation theory. Langlands functoriality and automorphic forms, Springer, Cham, Switzerland, 2018, pp. 341–361. MR3839702Google Scholar
Moeglin, C. and Renard, D., Sur les paquets d’Arthur des groupes classiques réels . J. Eur. Math. Soc. (JEMS) 22(2020), no. 6, 1827–1892. MR4092900Google Scholar
Shelstad, D., $L$ -indistinguishability for real groups . Math. Ann. 259(1982), no. 3, 385–430. MR661206Google Scholar
Shelstad, D., Tempered endoscopy for real groups. III. Inversion of transfer and $L$ -packet structure . Represent. Theory 12(2008), 369–402. MR2448289Google Scholar
Shelstad, D., On geometric transfer in real twisted endoscopy . Ann. Math. (2) 176(2012), no. 3, 1919–1985. MR2979862Google Scholar
Springer, T. A., Reductive groups . In: Automorphic forms, representations and $L$ -functions. Proceedings of symposia in pure mathematics, Oregon State University, Corvallis, Oregon, 1977 Part 1, American Mathematical Society, Providence, RI, 1979, pp. 3–27. MR546587Google Scholar
Springer, T. A., Linear algebraic groups. 2nd ed., Progress in Mathematics, 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR1642713 Google Scholar
Vogan, D. A. Jr., Representations of real reductive Lie groups, Progress in Mathematics, 15, Birkhäuser, Boston, MA, 1981.Google Scholar