Aubert, Anne-Marie; Baum, Paul; Plymen, Roger; Solleveld, Maarten

Smooth Duals of Inner Forms of \(\mathrm{GL}_n\) and \(\mathrm{SL}_n\)

Doc. Math. 24, 373-420 (2019)
DOI: 10.25537/dm.2019v24.373-420
Communicated by Dan Ciubotaru

Summary

Let \(F\) be a non-archimedean local field. We prove that every Bernstein component in the smooth dual of each inner form of the general linear group \(\mathrm{GL}_n(F)\) is canonically in bijection with the extended quotient for the action, given by Bernstein, of a finite group on a complex torus. For inner forms of \(\mathrm{SL}_n(F)\) we prove that each Bernstein component is canonically in bijection with the associated twisted extended quotient. In both cases, the bijections satisfy naturality properties with respect to the tempered dual, parabolic induction, central character, and the local Langlands correspondence.

Mathematics Subject Classification

20G25, 22E50

Keywords/Phrases

representation theory, Bernstein spectrum, Hecke algebras, stratified equivalence

References

  • 1. A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, On the local Langlands correspondence for non-tempered representations, Münster Journal of Mathematics 7 (2014), 27-50. zbl 06382808; MR3271238; arxiv 1303.0828.
  • 2. A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, Geometric structure in smooth dual and local Langlands conjecture, Japanese Journal of Mathematics 9 (2014), 99-136. DOI 10.1007/s11537-014-1267-x; zbl 1371.11097; MR3258616; arxiv 1211.0180.
  • 3. A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, The local Langlands correspondence for inner forms of $SL_n$, Res. Math. Sci. 3 (2016), paper 32. DOI 10.1186/s40687-016-0079-4; zbl 1394.22015; MR3579297; arxiv 1305.2638.
  • 4. A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, Hecke algebras for inner forms of $p$-adic special linear groups, J. Inst. Math. Jussieu 16 (2017), 351-419. DOI 10.1017/S1474748015000079; zbl 06704330; MR3615412; arxiv 1406.6861.
  • 5. A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, Geometric structure for the principal series of a split reductive $p$-adic group with connected centre, J. Noncommut. Geom. 10 (2016), 663-680. DOI 10.4171/JNCG/244; zbl 1347.22013; MR3519048.
  • 6. A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, The principal series of $p$-adic groups with disconnected centre, Proc. London Math. Soc. 114 (2017), 798-854. DOI 10.1112/plms.12023; zbl 1383.20033; MR3653247; arxiv 1409.8110.
  • 7. A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, On the spectra of finite type algebras, 2017. arxiv 1705.01404.
  • 8. A. I. Badulescu, Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle, Ann. Sci. Éc. Norm. Sup. 35,5 (2002), 695-747. DOI 10.1016/S0012-9593(02)01106-0; zbl 1092.11025; MR1951441.
  • 9. P.F. Baum, V. Nistor, Periodic cyclic homology of Iwahori-Hecke algebras, K-Theory 27,4 (2002), 329-357. DOI 10.1023/A:1022672218776; zbl 1056.16005; MR1962907; arxiv math/0201201.
  • 10. J. Bernstein, P. Deligne, Le ``centre'' de Bernstein, pp. 1-32, in: Représentations des groupes réductifs sur un corps local, Travaux en cours, Hermann, Paris, 1984. zbl 0599.22016; MR0771671.
  • 11. C.J. Bushnell, G. Henniart, P.C. Kutzko, Types and explicit Plancherel formulae for reductive $p$-adic groups, pp. 55-80, in: On certain L-functions, Clay Math. Proc. 13, American Mathematical Society, 2011. zbl 1222.22011; MR2767510.
  • 12. C.J. Bushnell, P.C. Kutzko, Smooth representations of reductive $p$-adic groups: structure theory via types, Proc. London Math. Soc. 77,3 (1998), 582-634. DOI 10.1112/S0024611598000574; zbl 0911.22014; MR1643417.
  • 13. P. Deligne, D. Kazhdan, M.-F.~Vignéras, Représentations des algèbres centrales simples $p$-adiques, pp. 33-117, in: Représentations des groupes réductifs sur un corps local, Travaux en cours, Hermann, 1984. zbl 0583.22009; MR0771672.
  • 14. P. Delorme, E.M. Opdam, The Schwartz algebra of an affine Hecke algebra, J. reine angew. Math. 625 (2008), 59-114. DOI 10.1515/CRELLE.2008.090; zbl 1173.22014; MR2482216; arxiv math/0312517.
  • 15. G. Henniart, Une preuve simple des conjectures de Langlands pour $GL(n)$ sur un corps $p$-adique, Inv. Math. 139 (2000), 439-455. DOI 10.1007/s002220050012; zbl 1048.11092; MR1738446.
  • 16. K. Hiraga, H. Saito, On L-packets for inner forms of $SL_n$, Mem. Amer. Math. Soc. 1013, Vol. 215 (2012). DOI 10.1090/S0065-9266-2011-00642-8; zbl 1242.22023; MR2918491.
  • 17. S.-I. Kato, A realization of irreducible representations of affine Weyl groups, Indag. Math. 45,2 (1983), 193-201. zbl 0531.20020; MR0705426.
  • 18. D. Kazhdan, G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153-215. DOI 10.1007/BF01389157; zbl 0613.22004; MR0862716.
  • 19. G. Lusztig, Cells in affine Weyl groups, pp. 255-287, in: Algebraic groups and related topics, Adv. Stud. Pure Math. 6, North Holland, Amsterdam, 1985. zbl 0569.20032; MR0803338.
  • 20. G. Lusztig, Cells in affine Weyl groups III, J. Fac. Sci. Univ. Tokyo 34,2 (1987), 223-243. zbl 0631.20028; MR0914020.
  • 21. G. Lusztig, Cells in affine Weyl groups IV, J. Fac. Sci. Univ. Tokyo 36,2 (1989), 297-328. zbl 0688.20020; MR1015001.
  • 22. E.M. Opdam, On the spectral decomposition of affine Hecke algebras, J. Inst. Math. Jussieu 3,4 (2004), 531-648. DOI 10.1017/S1474748004000155; zbl 1102.22009; MR2094450; arxiv math/0101007.
  • 23. A. Ram, J. Rammage, Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory, pp. 428-466, in: A tribute to C.S. Seshadri (Chennai 2002), Trends in Mathematics, Birkhäuser, 2003. zbl 1063.20004; MR2017596; arxiv math/0401322.
  • 24. V. Sécherre, Représentations lisses de $GL_m (D)$ III: types simples, Ann. Scient. Éc. Norm. Sup. 38 (2005), 951-977. DOI 10.1016/j.ansens.2005.10.003; zbl 1106.22014; MR2216835.
  • 25. V. Sécherre, S. Stevens, Représentations lisses de $GL_m (D)$ IV: représentations supercuspidales, J. Inst. Math. Jussieu 7,3 (2008), 527-574. DOI 10.1017/S1474748008000078; zbl 1140.22014; MR2427423.
  • 26. V. Sécherre, S. Stevens, Smooth representations of $GL(m,D)$ VI: semisimple types, Int. Math. Res. Notices 13 (2012), 2994-3039. DOI 10.1093/imrn/rnr122; zbl 1246.22023; MR2946230.
  • 27. A.J. Silberger, Special representations of reductive $p$-adic groups are not integrable, Ann. Math 111 (1980), 571-587. DOI 10.2307/1971110; zbl 0437.22015; MR0577138.
  • 28. J.-L. Waldspurger, La formule de Plancherel pour les groupes $p$-adiques (d'après Harish-Chandra), J. Inst. Math. Jussieu 2,2 (2003), 235-333. DOI 10.1017/S1474748003000082; zbl 1029.22016; MR1989693.

Affiliation

Aubert, Anne-Marie
CNRS, Sorbonne Université, Université Paris Diderot, Institut de Mathématiques de Jussieu, Paris Rive Gauche, IMJ-PRG F-75005 Paris, France
Baum, Paul
Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA
Plymen, Roger
School of Mathematics, Manchester University, Manchester M13 9PL, England
Solleveld, Maarten
IMAPP, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands

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