zbMATH — the first resource for mathematics

Depth and the local Langlands correspondence. (English) Zbl 1388.22010
Ballmann, Werner (ed.) et al., Arbeitstagung Bonn 2013. In memory of Friedrich Hirzebruch. Proceedings of the meeting, Bonn, Germany, May, 22–28, 2013. Basel: Birkhäuser/Springer (ISBN 978-3-319-43646-3/hbk; 978-3-319-43648-7/ebook). Progress in Mathematics 319, 17-41 (2016).
Let \(F\) be a non-archimedean local field, and let \(G\) be the group of \(F\)-points of a connected reductive algebraic group over \(F\). The local Langlands correspondence (LLC) is a conjectural correspondence between the set \(\text{Irr}(G)\) of (equivalence classes of) smooth irreducible complex representations of \(G\) and the set \(\Phi(G)\) of (equivalence classes of) Langlands parameters for \(G\). This correspondence \(\text{Irr}(G)\rightarrow\Phi(G)\) is not in general one-to-one; the fibers are referred to as \(L\)-packets.
For \(\pi\in\text{Irr}(G)\), A. Moy and G. Prasad [Invent. Math. 116, No. 1–3, 393–408 (1994; Zbl 0804.22008)] associated to any irreducible smooth representation \(\pi\) of \(G\) a nonnegative real number \(d(\pi)\) known as the depth of \(\pi\). This number is defined in terms of certain filtrations of the parahoric subgroups of \(G\). One also has a notion of the depth \(d(\phi)\) of a Langlands parameter \(\phi\), namely the smallest real number \(r\) such that \(\phi\) restricts trivially to the \(r\)th ramification subgroup of \(\text{Gal}(F^{\text{sep}}/F)\). There is a growing body of evidence that in a very large collection of cases, the LLC preserves depth, i.e., \[ d(\pi) = d(\phi) \] for \(\pi\in\text{Irr}(G)\) and \(\phi\in\Phi(G)\) that correspond under the LLC. In particular, for \(G = \text{GL}_n(F)\), this equality was stated in [J.-K. Yu, Fields Institue Monographs 26, 53–77 (2009; Zbl 1179.22020)] and proved in [the authors, Res. Math. Sci. 3, Paper No. 32, 34 p. (2016; Zbl 1394.22015)]. The article under review extends this result to inner forms \(G\) of \(\text{GL}_n(F)\). The authors also prove depth preservation for a large class of Langlands parameters of inner forms \(G\) of \(\text{SL}_n(F)\), namely the essentially tame parameters; when \(\text{char}(F)\nmid n\), all parameters of \(G\) are essentially tame.
The proof for inner forms \(G\) of \(\text{GL}_n(F)\) proceeds by reducing to the case of essentially square-integrable representations using the Langlands classification. The authors then prove a uniform relation between \(d(\pi)\) and another invariant of \(\pi\), its conductor, using arguments along the lines of those in [J. Lansky and A. Raghuram, Proc. Am. Math. Soc. 131, No. 5, 1641–1648 (2003; Zbl 1027.22017)]. Since the LLC for \(G\) is the composition of the Jacquet-Langlands correspondence \(\text{Irr(G)}\rightarrow \text{Irr}(\text{GL}_n(F))\) and the LLC for \(\text{GL}_n(F)\) (for which depth preservation has already been established), it suffices to prove that the Jacquet-Langlands correspondence preserves conductors. The authors do this by proving results on the Jacquet-Langlands correspondence first stated by Deligne-Kazhdan-Vigneras.
For the entire collection see [Zbl 1364.00032].

22E50 Representations of Lie and linear algebraic groups over local fields
20G25 Linear algebraic groups over local fields and their integers
Full Text: DOI
[1] A.-M. Aubert, P. Baum, R.J. Plymen, M. Solleveld, Geometric structure and the local Langlands conjecture (2013). arXiv:1211.0180 · Zbl 1371.11097
[2] A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, The local Langlands correspondence for inner forms of SL n . Res. Math. Sci. (2013, To appear). Preprint · Zbl 1394.22015
[3] A.I. Badulescu, Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle. Ann. Sci. Éc. Norm. Sup. (4) 35, 695–747 (2002) · Zbl 1092.11025
[4] A.I. Badulescu, Un théorème de finitude (with an appendix by P. Broussous). Compos. Math. 132, 177–190 (2002) · Zbl 1013.20036
[5] A. Borel, Automorphic L-functions. Proc. Symp. Pure Math 33 (2), 27–61 (1979) · Zbl 0412.10017
[6] P. Broussous, Minimal strata for GL m (D). J. Reine Angew. Math. 514, 199–236 (1999) · Zbl 0936.22011
[7] P. Broussous, B. Lemaire, Building of GL(m, D) and centralizers. Transform. Groups 7 (1), 15–50 (2002) · Zbl 1001.22016
[8] C.J. Bushnell, A. Frölich, Gauss Sums and p-adic Division Algebras. Lecture Notes in Mathematics, vol. 987 (Springer, Berlin, 1983) · Zbl 0507.12008
[9] C.J. Bushnell, A. Frölich, Non-abelian congruence Gauss sums and p-adic simple algebras. Proc. Lond. Math. Soc. (3) 50, 207–264 (1985) · Zbl 0558.12007
[10] C.J. Bushnell, G. Henniart, The essentially tame local Langlands correspondence, I. J. Am. Math. Soc. 18.3, 685–710 (2005) · Zbl 1073.11070
[11] C.J. Bushnell, G. Henniart, The Local Langlands Conjecture for GL(2). Grundlehren der mathematischen Wissenschaften, vol. 335 (Springer, Berlin, 2006) · Zbl 1100.11041
[12] C.J. Bushnell, P.C. Kutzko, The admissible dual of SL(N) I. Ann. Scient. Éc. Norm. Sup. (4) 26, 261–280 (1993) · Zbl 0787.22017
[13] T.H. Chen, M. Kamgarpour, Preservation of depth in local geometric Langlands correspondence (2014). arXiv: 1404.0598 · Zbl 1401.17020
[14] S. DeBacker, M. Reeder, Depth zero supercuspidal L-packets and their stability. Ann. Math. 169 (3), 795–901 (2009) · Zbl 1193.11111
[15] P. Deligne, D. Kazhdan, M.-F. Vigneras, Représentations des algèbres centrales simples p-adiques, in Représentations des Groupes réductifs sur un Corps Local (Travaux en cours, Hermann, 1984), pp. 33–117
[16] R. Ganapathy, The local Langlands correspondence for GSp 4 over local function fields. Am. J. Math. 137 (6), 1441–1534 (2015) · Zbl 1332.22018
[17] R. Godement, H. Jacquet, Zeta Functions of Simple Algebras. Lecture Notes in Mathematics, vol. 260 (Springer, New York, 1972) · Zbl 0244.12011
[18] M. Grabitz, A.J. Silberger, W. Zink, Level zero types and Hecke algebras for local central simple algebras. J. Number Theory 91, 92–125 (2001) · Zbl 1009.22016
[19] B.H. Gross, M. Reeder, ”Arithmetic invariants of discrete Langlands parameters”, Duke Math. J. 154 (2010), 431–508. · Zbl 1207.11111
[20] M. Harris, R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, vol. 151 (Princeton University Press, Princeton NJ, 2001) · Zbl 1036.11027
[21] G. Henniart, On the local Langlands conjecture for GL(n): the cyclic case. Ann. Math. 123, 143–203 (1986) · Zbl 0588.12010
[22] G. Henniart, Une preuve simple de conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math. 139 (2000), 439–455. · Zbl 1048.11092
[23] K. Hiraga, H. Saito, On L-packets for inner forms of SL n . Mem. Am. Math. Soc. 215 (1013) (2012) · Zbl 1242.22023
[24] H. Jacquet, Principal L-functions of the linear group. Proc. Symp. Pure Math. 33 (2), 63–86 (1979) · Zbl 0413.12007
[25] H. Jacquet, R. Langlands, Automorphic Forms on GL(2). Lecture Notes in Mathematics, vol. 114 (Springer, New York, 1970) · Zbl 0236.12010
[26] R.E. Kottwitz, Stable trace formula: cuspidal tempered terms. Duke Math. J. 51 (3), 611–650 (1984) · Zbl 0576.22020
[27] J. Lansky, A. Raghuram, On the correspondence of representations between GL(n) and division algebras. Proc. Am. Math. Soc. 131 (5), 1641–1648 (2002) · Zbl 1027.22017
[28] G. Laumon, M. Rapoport, U. Stuhler, \[ \mathcal{D} \] -elliptic sheaves and the Langlands correspondence. Invent. Math. 113, 217–238 (1993) · Zbl 0809.11032
[29] G. Lusztig, Classification of unipotent representations of simple p-adic groups. Int. Math. Res. Not. 11, 517–589 (1995) · Zbl 0872.20041
[30] G. Lusztig, Classification of unipotent representations of simple p-adic groups II. Represent. Theory 6, 243–289 (2002) · Zbl 1016.22011
[31] C. Moeglin, Stabilité en niveau 0, pour les groupes orthogonaux impairs p-adiques. Doc. Math. 9, 527–564 (2004) · Zbl 1074.22006
[32] L. Morris, Tamely ramified intertwining algebras. Invent. Math. 114 (1), 1–54 (1993) · Zbl 0854.22022
[33] A. Moy, G. Prasad, Unrefined minimal K-types for p-adic groups. Invent. Math. 116, 393–408 (1994) · Zbl 0804.22008
[34] A. Moy, G. Prasad, Jacquet functors and unrefined minimal K-types. Comment. Math. Helv. 71, 98–121 (1996) · Zbl 0860.22006
[35] M. Reeder, Supercuspidal L-packets of positive depth and twisted Coxeter elements. J. Reine Angew. Math. 620, 1–33 (2008) · Zbl 1153.22021
[36] M. Reeder, J.-K. Yu, Epipelagic representations and invariant theory. J. Am. Math. Soc. 27, 437–477 (2014) · Zbl 1284.22011
[37] V. Sécherre, S. Stevens, Représentations lisses de GL m (D) IV: représentations supercuspidales. J. Inst. Math. Jussieu 7 (3), 527–574 (2008) · Zbl 1140.22014
[38] J.-P. Serre, Corps Locaux (Hermann, Paris, 1962) · Zbl 0137.02601
[39] M. Tadić, Induced representations of GL(n, A) for p-adic division algebras A. J. Reine Angew. Math. 405, 48–77 (1990) · Zbl 0684.22008
[40] J. Tate, Number theoretic background. Proc. Symp. Pure Math 33 (2), 3–26 (1979) · Zbl 0422.12007
[41] D. Vogan, The local Langlands conjecture, in Representation Theory of Groups and Algebras. Contemporary Mathematics, vol. 145 (American Mathematical Society, Providence, RI, 1993), pp. 305–379 · Zbl 0802.22005
[42] A. Weil, Exercices dyadiques. Invent. Math. 27, 1–22 (1974) · Zbl 0307.12017
[43] J.-K. Yu, Bruhat-Tits theory and buildings, in Ottawa Lectures on Admissible Representations of Reductive p-adic Groups. Fields Institute Monographs (American Mathematical Society, Providence, RI, 2009), pp. 53–77
[44] J.-K. Yu, On the local Langlands correspondence for tori, in Ottawa Lectures on Admissible Representations of Reductive p-adic Groups. Fields Institute Monographs (American Mathematical Society, Providence, RI, 2009), pp. 177–183
[45] A.V. Zelevinsky, Induced representations of reductive p-adic groups II. On irreducible representations of GL(n). Ann. Sci. École Norm. Sup. (4) 13 (2), 165–210 (1980) · Zbl 0441.22014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.