Characteristic cycles, micro local packets and packets with cohomology

Arancibia Robert, Nicolás

doi:10.1090/tran/8492

Characteristic cycles, micro local packets and packets with cohomology
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by Nicolás Arancibia Robert PDF

Trans. Amer. Math. Soc. 375 (2022), 997-1049

Abstract:

Relying on work of Kashiwara-Schapira and Schmid-Vilonen, we describe the behaviour of characteristic cycles with respect to the operation of geometric induction, the geometric counterpart of taking parabolic or cohomological induction in representation theory. By doing this, we are able to describe to some extent the characteristic cycle associated to an induced representation, in terms of the characteristic cycle of the representation being induced. More precisely, under the hypothesis that the infinitesimal character is regular (and dominant), we show that the characteristic cycle of an induced representation splits in two terms. We describe the first term precisely, but we are not able to do the same for the second one. What we are able to say, is that this second term is supported on the boundary of the space generated by the inclusion in the flag variety of $G$, of the flag variety of the Levi subgroup. As a consequence, we prove that the cohomology packets defined by Adams and Johnson in [Compositio Math. 64 (1987), pp. 271–309] are micro-packets, that is to say that the cohomological constructions of provided by Adams and Johnson are particular cases of the sheaf-theoretic ones provided by Adams, Barbasch, and Vogan Jr. [The Langlands classification and irreducible characters for real reductive groups, Birkhäuser Boston, Inc., Boston, MA, 1992].

References

Jeffrey Adams, Dan Barbasch, and David A. Vogan Jr., The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1162533, DOI 10.1007/978-1-4612-0383-4
Jeffrey Adams and Joseph F. Johnson, Endoscopic groups and packets of nontempered representations, Compositio Math. 64 (1987), no. 3, 271–309. MR 918414
Nicolás Arancibia, Colette Mœglin, and David Renard, Paquets d’Arthur des groupes classiques et unitaires, Ann. Fac. Sci. Toulouse Math. (6) 27 (2018), no. 5, 1023–1105 (French, with English and French summaries). MR 3919547, DOI 10.5802/afst.1590
James 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. MR 748504, DOI 10.1007/BFb0073144
James Arthur, Unipotent automorphic representations: conjectures, Astérisque 171-172 (1989), 13–71. Orbites unipotentes et représentations, II. MR 1021499
James Arthur, The endoscopic classification of representations, American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013. Orthogonal and symplectic groups. MR 3135650, DOI 10.1090/coll/061
Jeffrey Adams and David A. Vogan Jr., $L$-groups, projective representations, and the Langlands classification, Amer. J. Math. 114 (1992), no. 1, 45–138. MR 1147719, DOI 10.2307/2374739
Jeffrey Adams and David A. Vogan Jr., Parameters for twisted representations, Representations of reductive groups, Progr. Math., vol. 312, Birkhäuser/Springer, Cham, 2015, pp. 51–116. MR 3495793, DOI 10.1007/978-3-319-23443-4_{3}
Alexandre Beĭlinson and Joseph Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
Walter Borho and Jean-Luc Brylinski, Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules, Invent. Math. 69 (1982), no. 3, 437–476. MR 679767, DOI 10.1007/BF01389364
W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces. III. Characteristic varieties of Harish-Chandra modules and of primitive ideals, Invent. Math. 80 (1985), no. 1, 1–68. MR 784528, DOI 10.1007/BF01388547
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
Frédéric Bien, Spherical $D$-modules and representations on symmetric spaces, Bull. Soc. Math. Belg. Sér. A 38 (1986), 37–44 (1987). MR 885522
A. Borel, Automorphic $L$-functions, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27–61. MR 546608
Leticia Barchini, Petr Somberg, and Peter E. Trapa, Reducible characteristic cycles of Harish-Chandra modules for $\textrm {U}(p,q)$ and the Kashiwara-Saito singularity, Comm. Algebra 47 (2019), no. 12, 4874–4888. MR 4019312, DOI 10.1080/00927872.2019.1596280
Alexandru Dimca, Sheaves in topology, Universitext, Springer-Verlag, Berlin, 2004. MR 2050072, DOI 10.1007/978-3-642-18868-8
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR 932724, DOI 10.1007/978-3-642-71714-7
Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. MR 2357361, DOI 10.1007/978-0-8176-4523-6
Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR 1074006, DOI 10.1007/978-3-662-02661-8
Anthony W. Knapp and David A. Vogan Jr., Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton, NJ, 1995. MR 1330919, DOI 10.1515/9781400883936
R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170. MR 1011897, DOI 10.1090/surv/031/03
R. P. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987), no. 1-4, 219–271. MR 909227, DOI 10.1007/BF01458070
I. Mirković and K. Vilonen, Characteristic varieties of character sheaves, Invent. Math. 93 (1988), no. 2, 405–418. MR 948107, DOI 10.1007/BF01394339
D. Shelstad, Embeddings of $L$-groups, Canadian J. Math. 33 (1981), no. 3, 513–558. MR 627641, DOI 10.4153/CJM-1981-044-4
Masahiro Shiota, Nash manifolds, Lecture Notes in Mathematics, vol. 1269, Springer-Verlag, Berlin, 1987. MR 904479, DOI 10.1007/BFb0078571
T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4
Wilfried Schmid and Kari Vilonen, Characteristic cycles of constructible sheaves, Invent. Math. 124 (1996), no. 1-3, 451–502. MR 1369425, DOI 10.1007/s002220050060
Olivier Taïbi, Deux applications arithmétiques des travaux d’Arthur, 2014. Thesis (Ph.D.)–Ec. Polytech. (France).
David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407
David A. Vogan Jr., Irreducible characters of semisimple Lie groups. IV. Character-multiplicity duality, Duke Math. J. 49 (1982), no. 4, 943–1073. MR 683010
David A. Vogan Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141–187. MR 750719, DOI 10.2307/2007074