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4 - Representation Theory of p-Adic Reductive Groups

Published online by Cambridge University Press:  06 November 2024

Roger Plymen
Affiliation:
University of Manchester
Mehmet Haluk Şengün
Affiliation:
University of Sheffield
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Summary

We present some fundamental aspects of the representation theory of reductive p-adic groups, mapping out some recent developments, including the ABPS conjecture and the description of the structure of the reduced C*-algebras of reductive groups. The exposition proceeds fairly rapidly, but is essentially self-contained.

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Publisher: Cambridge University Press
Print publication year: 2024

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References

Abramenko, P. and Brown, K. S.. Buildings, graduate texts in mathematics, vol. 248, Springer, New York, 2008.Google Scholar
Afgoustidis, A. and Aubert, A.-M.. C∗-blocks and crossed products for classical p-adic groups, IMRN 2022 (22) (2022), 17849–17908.CrossRefGoogle Scholar
Arthur, J.. On elliptic tempered characters, Acta Math. 171 (1) (1993), 73–138.Google Scholar
Aubert, A.-M., Baum, P., and Plymen, R.. Geometric structure in the principal series of the p-adic group G2, Represent. Theory 15 (2011), 126–169.CrossRefGoogle Scholar
Aubert, A.-M., Baum, P., and Plymen, R.. Geometric structure in the representation theory of reductive p-adic groups II, In: Harmonic analysis on reductive, p-adic groups, 71–90, Contemporary Mathematics, 543, American Mathematical Society, Providence, RI, 2011.Google Scholar
Aubert, A.-M., Baum, P., Plymen, R. J., and Solleveld, M.. Geometric structure in smooth dual and local Langlands correspondence, Japan. J. Math. 9 (2014), 99–136.Google Scholar
Aubert, A.-M., Baum, P., Plymen, R. J., and Solleveld, M.. Depth and the local Langlands correspondence (Arbeitstagung Bonn, 2013), In: Progress in Math., Birkhäuser, 2016, arXiv:1311.1606.Google Scholar
Aubert, A.-M., Baum, P., Plymen, R. J., and Solleveld, M.. Conjectures about p-adic groups and their noncommutative geometry, in: Around Langlands Correspondences, Contemp. Math. 691 (2017), 15–51.Google Scholar
Aubert, A.-M., Baum, P., Plymen, R. J., and Solleveld, M.. Hecke algebras for inner forms of p-adic special linear groups, J. Inst. Math. Jussieu 16 (2) (2017), 351–419.CrossRefGoogle Scholar
Aubert, A.-M., Baum, P., Plymen, R. J., and Solleveld, M.. The principal series of p-adic groups with disconnected centre, Proc. London Math. Soc. 114 (2017) 798–854.CrossRefGoogle Scholar
Aubert, A.-M., Baum, P., Plymen, R. J., and Solleveld, M.. Smooth duals of inner forms of GLn and SLn, Doc. Math. 24 (2019), 373–420.Google Scholar
Aubert, A.-M., Moussaoui, A., and Solleveld, M.. Generalizations of the Springer correspondence and cuspidal Langlands parameters, Manuscripta Math. 157 (1–2) (2018), 121–192.Google Scholar
Aubert, A.-M., Moussaoui, A., and Solleveld, M.. Affine Hecke algebras for classical p-adic groups, arXiv:2211.08196.Google Scholar
Aubert, A.-M., and Plymen, R. J.. Plancherel measure for GL(n, F) and GL(m, D): Explicit formulas and Bernstein decomposition, J. Number Theory 112 (1) (2005), 26–66.CrossRefGoogle Scholar
Aubert, A.-M. and Plymen, R. J.. Comparison of the depths on both sides of the local Langlands correspondence for Weil-restricted groups, With an appendix by Jessica Fintzen, J. Number Theory 233 (2022), 24–58.CrossRefGoogle Scholar
Aubert, A.-M.. Bruhat-Tits buildings, representations of p-adic groups and Langlands correspondence, J. Algebra, to appear.Google Scholar
Aubert, A.-M. and Xu, Y.. Hecke algebras for p-adic reductive groups and local Langlands correspondences for Bernstein blocks, special volume in the memory of Jacques Tits, Adv. Math., 436 (2024), https://doi.org/10.1016/j.aim.2023.109384.CrossRefGoogle Scholar
Barbasch, D. and Moy, A.. Reduction to real infinitesimal character in affine Hecke algebras, J. Amer. Math. Soc. 6 (3) (1993), 611–635.Google Scholar
Barbasch, D. and Moy, A.. Unitary spherical spectrum for p-adic classical groups, Acta Appl. Math. 44 (1–2) (1996), 3–37, Representations of Lie groups, Lie algebras and their quantum analogues.CrossRefGoogle Scholar
Barbasch, D., Ciubotaru, D., and Trapa, P.. Dirac cohomology for graded affine Hecke algebras, Acta Math. 209 (2) (2012), 197–227.Google Scholar
Bernstein, J.. “Centre”, Le de Bernstein (rédigé par P. Deligne), Representations of reductive groups over a local field, 1984, pp. 1–32.Google Scholar
Bernstein, J.. Draft of: Representations of p-adic groups, Lectures by Joseph Bernstein Harvard University, Fall 1992, Notes by Karl E. Rumelhart.Google Scholar
Bernstein, J. and Zelevinsky, A.. Representations of the group GL(n, F), where F is a local non-Archimedean field, Uspehi Mat. Nauk. 31 (3) (1976), 5–70.CrossRefGoogle Scholar
Bernstein, J. and Zelevinsky, A.. Induced representations of reductive p-adic groups. I, Ann. Sci. École Norm. Sup. 10 (4) (1977), 441–472.Google Scholar
Blondel, C., Critère d’injectivité pour l’application de Jacquet, C. R. Acad. Sci. Paris Sér. I Math. 325 (11) (1997), 1149–1152.Google Scholar
Borel, A.. Linear algebraic groups, 2nd edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991.Google Scholar
Borel, A.. Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233–259.CrossRefGoogle Scholar
Bourbaki, N.. Éléments de mathématique. Groupes et algèbres de Lie, Chapitres 4-5-6, Springer, Berlin, 2007.Google Scholar
Bruhat, F. and Tits, J.. Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251.Google Scholar
Bushnell, C. J. and Henniart, G.. The local Langlands conjecture for GL(2). Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335. Springer-Verlag, Berlin, 2006.Google Scholar
Bushnell, C. and Kutzko, P.. The admissible dual of GL(N ) via compact open subgroups, vol. 129, Annals of Math Studies, Princeton University Press, Princeton, NJ, 1993.Google Scholar
Bushnell, C. and Kutzko, P.. The admissible dual of SL(N ). I, Ann. Sci. École Norm. Sup. 26 (2) (1993), 261–280.CrossRefGoogle Scholar
Bushnell, C. and Kutzko, P.. Smooth representations of reductive p-adic groups: structure theory via types, Proc. London Math. Soc. 77 (3) (1998), 582–634.CrossRefGoogle Scholar
Cartier, P.. Representations of p-adic groups: a survey, in: Automorphic forms, representations and L-functions (Proceedings of Symposia in Pure Mathematics, Oregon State University, Corvallis, OR, 1977), Part 1, 111–155, Proceedings of Symposia in Pure Mathematics, XXXIII, American Mathematical Society, Providence, RI, 1979.Google Scholar
Casselman, W.. Introduction to the theory of admissible representations of p-adic reductive groups, 1995.Google Scholar
Chern, S.S. and Hirzebruch, F.. eds, Note concerning Jacques Tits, Wolf Prize in mathematics, vol. 2, 703–754, World Scientific Publishing Co. Inc., River Edge, NJ, 2001.CrossRefGoogle Scholar
Ciubotaru, D., Opdam, E., and Trapa, P.. Algebraic and analytic Dirac induction for graded affine Hecke algebras, J. Inst. Math. Jussieu 13 (3) (2014), 447–486.CrossRefGoogle Scholar
Ciubotaru, D.. Spin representations of Weyl groups and the Springer correspondence, J. ReineAngew. Math. 671 (2012) 199–222.Google Scholar
Ciubotaru, D. and Trapa, P.. Characters of Springer representations on elliptic conjugacy classes, Duke Math. J. 162 (2) (2013), no. 2 201–223.Google Scholar
DeBacker, S.. Some applications of Bruhat-Tits theory to harmonic analysis on a reductive p-adic group, Michigan Math. J. 50 (2) (2002), 241–261.Google Scholar
Deligne, P., and Lusztig, G.. Representations of reductive groups over finite fields, Ann. Math. 103 (1) (1976), 103–161.CrossRefGoogle Scholar
Digne, F. and Michel, J.. Representations of finite groups of Lie type, London Mathematical Society student texts, vol. 95, Cambridge University Press, Cambridge, 2020.Google Scholar
Fintzen, J.. Types for tame p-adic groups, Ann. Math. 193 (1) (2021), 303–346.CrossRefGoogle Scholar
Fintzen, J.. On the construction of tame supercuspidal representations, Compos. Math. 157 (12) (2021), 2733–2746.CrossRefGoogle Scholar
Fintzen, J., Kaletha, T., and Spice, L.. A twisted Yu construction, Harish-Chandra characters, and endoscopy, Duke Math. J. 172 (12) (2023), 2241–2301.Google Scholar
Feng, Y., Opdam, E., and Solleveld, M.. Supercuspidal unipotent representations: L-packets and formal degrees, J. Ec. Polytec. Math. 7 (2020), 1133–1193.Google Scholar
Gelfand, I. M. and Graev, M. I.. The group of matrices of second order with coefficients in a locally compact field and special functions on locally compact fields, Uspekhi Mat. Nauk 18 (1963), 29–99.Google Scholar
Harish-Chandra, Harmonic analysis on reductive p-adic groups, Proceedings of Symposia in Pure Mathematics., vol. 26, American Mathematical Society, Providence, RI, 1973, pp. 167–192.Google Scholar
Harish-Chandra, Admissible invariant distributions on reductive p-adic groups, Lie Theories and Their Applications, Queen's Papers in Pure Appl. Math., vol. 48, Queen's University Kingston, Ontario, 1978, pp. 281–347.Google Scholar
Humphreys, J. E., Reflection groups and coxeter groups, Cambridge studies in advanced mathematics, vol. 29, Cambridge University Press, Cambridge, 1990.CrossRefGoogle Scholar
Kaletha, T. and Prasad, G.. Bruhat–Tits theory: A new approach, New Math. Monographs, vol. 44, Cambridge University Press, Cambridge, 2023.Google Scholar
Kaletha, T. and Taibi, O.. The local Langlands conjecture, Proceedings of Symposia in Pure Mathematics (to appear).Google Scholar
Kazhdan, D. and Lusztig, G.. Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987) 153–215.CrossRefGoogle Scholar
Kim, J.-L.. Supercuspidal representations: An exhaustion theorem, J. Amer. Math. Soc. 20 (2007), 273–320.Google Scholar
Kim, J.-L. and Yu, J.-K. Construction of tame types. Representation theory, number theory, and invariant theory, 337–357, Progress in Mathematics, 323, Birkhäuser/Springer, Cham, 2017.Google Scholar
Kutzko, P. and Morris, L.. Explicit Plancherel theorems for H (q1, q2) and SL2(F), Pure and Applied Mathematics Quarterly 5(1) (2009), 435–467.CrossRefGoogle Scholar
Lusztig, G.. Characters of reductive groups over a finite field, Ann. Math. Stud., Princeton, NJ, 1984.Google Scholar
Lusztig, G.. On the representations of reductive groups with disconnected center, Astérisque 168 (1988), 157–166.Google Scholar
Lusztig, G.. Classification of unipotent representations of simple p-adic groups Internat. Math. Res. Notices 1995 (11) (1995), 517–589.CrossRefGoogle Scholar
Lusztig, G.. Classification of unipotent representations of simple p-adic groups. II, Represent. Theory 6 (2002), 243–289.Google Scholar
Lusztig, G.. Affine Hecke algebras and their graded version, 2 (3) (1989), 599–635.CrossRefGoogle Scholar
A. Mayeux Bruhat-Tits theory from Berkovich's point of view. Analytic filtrations, Ann. H. Lebesgue 5 (2022), 813–839.CrossRefGoogle Scholar
Mayeux, A. and Yamamoto, Y.. Comparing Bushnell-Kutzko and Sécherre's constructions of types for GLN and its inner forms with Yu's construction, arXiv:2112.12367.Google Scholar
Miyauchi, M. and Stevens, S.. Semisimle types for p-adic classical groups, Mathematische Annalen 358 (2014), 257–288.CrossRefGoogle Scholar
Morris, L.. Tamely ramified supercuspidal representations, Ann. Scien. Éc. Norm. Sup. 4e série 29 (1996), 639–667.Google Scholar
Morris, L.. Level zero G-types, Compositio Mathematica 118 (1999), 135–157.CrossRefGoogle Scholar
Moy, A. and Prasad, G.. Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994), 393–408.CrossRefGoogle Scholar
Moy, A. and Prasad, G.. Jacquet functors and unrefined minimal K-types, Comment. Math. Helv. 71 (1) (1996), 98–121.CrossRefGoogle Scholar
Moy, A. and Tadic’, M.. Some algebras of essentially compact distributions of a reductive p-adic group. In: Harmonic analysis, group representations, automorphic forms and invariant theory, 247–275, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, vol. 12, World Scientific Publishing, Hackensack, NJ, 2007.Google Scholar
Plymen, R. J.. Reduced C∗-algebra for reductive p-adic groups, J. Funct. Anal. 88 (2) (1990), 251–266.CrossRefGoogle Scholar
Rémy, B., Thuillier, A., and Werner, A.. An intrinsic characterization of Bruhat-Tits buildings inside analytic groups, Michigan Math. J. 72 (2022), 543–557.Google Scholar
Renard, D.. Représentations des groupes réductifs p-adiques, Cours Spécialisés, 17, Société Mathématique de France, Paris, 2010. pp.vi+332. ISBN: 978-285629278-5.Google Scholar
Roche, A.. Parabolic induction and the Bernstein decomposition, Compositio Math. 134 (2) (2002), 113–133.Google Scholar
Roche, A.. The Bernstein decomposition and the Bernstein centre. In: Ottawa lectures on admissible representations of reductive p-adic groups, Fields Institute Monographs, vol. 26, American Mathematical Society, Providence, RI, 2009, 3–52.Google Scholar
Ronan, M.. Lectures on Buildings: Updated and Revised, University of Chicago Press, Chicago, IL, 2009.Google Scholar
Sécherre, V. and Stevens, S.. Représentations lisses de GLm (D) IV: représentations supercuspidales, J. Inst. Math. Jussieu 7 (2008), 527–574.CrossRefGoogle Scholar
Shahidi, F.. Fourier transforms of intertwining operators and Plancherel measures for GL(n), Amer. J. Math. 106(1) (1984), 67–111.CrossRefGoogle Scholar
Shahidi, F.. A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups, Ann. Math. 132(2) (1990), 273–330.CrossRefGoogle Scholar
Shahidi, F.. Langlands’ conjecture on Plancherel measures for p-adic groups. In: “Harmonic analysis on reductive groups”, Birkhäuser Boston, Boston, MA, 1991, pp. 277–295.Google Scholar
Shalika, J. A.. Representations of the two by two unimodular group over local fields. In: Contributions to automorphic forms, geometry, and number theory, Johns Hopkins University Press, Baltimore, MD, 2004, pp. 1–38.Google Scholar
Silberger, A. J.. Introduction to harmonic analysis on reductive p-adic Groups, Princeton University Press, Princeton, NJ, 1979.Google Scholar
Solleveld, M.. Topological K-theory of affine Hecke algebras, Ann. K-Theory 3 (2018), 395–460.Google Scholar
Solleveld, M.. On unipotent representations of ramified p-adic groups, Represent. Theory 27 (2023), 669–716.CrossRefGoogle Scholar
Solleveld, M.. Endomorphism algebras and Hecke algebras for reductive p-adic groups, J. Algebra 606 (2022), 371–470.CrossRefGoogle Scholar
Solleveld, M.. Hochschild homology of reductive p-adic groups, J. Noncommutative Geometry, to appear.Google Scholar
Stevens, S.. The supercuspidal representations of p-adic classical groups, Invent. Math. 172 (2008), 289–352.CrossRefGoogle Scholar
Tadic, M.. Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case), Ann. Sci. École Norm. Sup. 19 (3) (1986), 335–382.CrossRefGoogle Scholar
Tits, J.. Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, vol. 386, Springer-Verlag, Berlin-New York, 1974.Google Scholar
Waldspurger, J.-L.. La formule de Plancherel pour les groupes p-adiques (d’après Harish-Chandra), J. Inst. Math. Jussieu 22 (2003), 235–333.CrossRefGoogle Scholar
Yu, J.-K.. Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (3) (2001), 579–622.Google Scholar
Yu, J.-K.. Smooth models associated to concave functions in Bruhat-Tits theory. In: “Autour des schémas en groupes, III”, 227–258, Panor. Synthèses, vol. 47, Soc. Math. France, Paris, 2015.Google Scholar

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