×

Langlands parameters, functoriality and Hecke algebras. (English) Zbl 1526.20073

The local Langlands correspondence is a predicted assignment \[ \mathrm{LLC}\colon\mathrm{Irr}(G) \to \Phi(G) \] of (isomorphism classes of) simple representations of \(G=\mathbf{G}(F)\) where \(F\) is a local field and \(\mathbf{G}\) is a connected reductive group defined over \(F\), to (conjugacy classes of) Langlands parameters, which are roughly speaking homomorphisms of the Galois group of \(F\) into the \(L\)-group \({}^LG\) of \(\mathbf{G}\). The local Langlands conjectures predict not only the existence of such a map, but stipulate several desiderata for it. The most basic requirement is that \(\mathrm{LLC}\) have finite fibres; for \(\phi\in\Phi(G)\) one then writes \(\Pi_\phi(G):=\mathrm{LLC}^{-1}(\phi)\) for the L-packet. Another desideratum concerning pullbacks of representations can in turn be phrased in terms of packets: If \[ \eta\colon\mathbf{G}_1\to\mathbf{G_2} \] is a homomorphism of algebraic groups over \(F\) with commutative kernel and cokernel, and \(\pi\) is an irreducible representation of \(G_2\), then it is classical that \(\eta^*\pi\) is a completely reducible representation of finite length of \(G_1\), and one then requires of \(\mathrm{LLC}\) that all direct summands of \(\eta^*\pi\) belong to the packet \(\Pi(G_2)_{\Phi(\eta)(\phi)}\), where \(\Phi(\eta)(\phi)\) is the post-composition of \(\phi\) with the induced map \({}^L\eta\) of \(L\)-groups.
In the present paper, the author proves an enhancement, in the sense of enhanced parameters, of this pullback property for the following maps \(\mathrm{LLC}\) when \(\mathbf{G_1}\) and \(\mathbf{G_2}\) belong to the following classes of groups:
1.
Split reductive groups over non-Archimedean local fields and the correspondence of [A.-M. Aubert et al., Proc. Lond. Math. Soc. (3) 114, No. 5, 798–854 (2017; Zbl 1383.20033)] for the case of principal series representations only.
2.
Unipotent representations of groups over non-Archimedean local fields which split over an unramified extension, using the correspondence of [G. Lusztig, Int. Math. Res. Not. 1995, No. 11, 517–589 (1995; Zbl 0872.20041); Represent. Theory 6, 243–289 (2002; Zbl 1016.22011)] as extended beyond adjoint groups by [M. Solleveld, Am. J. Math. 145, No. 3, 673–719 (2023; Zbl 1527.22025)].
3.
Inner twists of \(\mathrm{GL}_n\), \(\mathrm{SL}_n\), \(\mathrm{PGL}_n\) over any local field, using the correspondence of [K. Hiraga and H. Saito, On \(L\)-packets for inner forms of \(\mathrm{SL}_{n}\). Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1242.22023); A.-M. Aubert et al., Res. Math. Sci. 3, Paper No. 32, 34 p. (2016; Zbl 1394.22015)]

The enhancement is of course in terms of the component group \(\mathcal{S}_{\phi}\) of (roughly speaking) the centralizer of \(\phi\) and in the simplest case gives the decomposition of \(\eta^*\pi\) in terms of the decomposition of the irreducible constituents of \(\mathrm{ind}_{\mathcal{S}_\phi}^{\mathcal{S}_{{}^L\eta\circ\phi}}\rho\), where \((\phi,\rho)\) is the enhanced parameter of \(\pi\). Borel’s conjecture is recovered as the statement the packet \(\Pi_{{}^L\eta\circ\phi}(G_1)\) consists precisely of the direct summands of pullbacks \(\eta^*\pi\) for \(\pi\in\Pi_\phi(G_2)\). The author notes that this implies that for \(\phi\) tempered, the corresponding stable distribution supported on \(\Pi_\phi(G_2)\) is pulled back to one on \(\Pi_{{}^L\eta\circ\phi}(G_1)\).
Finally, the author shows that his enhancement of Borel’s conjecture gives a multiplicity-one criterion for \(\eta^*\pi\) in terms of representations of the centralizer groups. In particular mutliplicty-one holds whenever \(\mathcal{S}_{{}^L\eta\circ\phi}\) is abelian.
The main tools in the proofs are the affine Hecke algebras associated to Langlands parameters developed by A.-M. Aubert et al. starting in [“Affine Hecke algebras for Langlands parameters”, Preprint, arXiv:701.03593].
Finally, the author recapitulates the local Langlands correspondence for quasisplit classical groups over local fields of characteristic zero as established by J. Arthur [The endoscopic classification of representations. Orthogonal and symplectic groups. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1310.22014)] and expanded by C. P. Mok [Endoscopic classification of representations of quasi-split unitary groups. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1316.22018)] and B. Xu [Math. Ann. 370, No. 1–2, 71–189 (2018; Zbl 1384.22008)] and verifies the enhanced version of Borel’s conjecture in these cases. The recapitulation and filling in of details given in this section seems independently useful.

MSC:

20G25 Linear algebraic groups over local fields and their integers
11S37 Langlands-Weil conjectures, nonabelian class field theory
20C08 Hecke algebras and their representations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 10.1353/ajm.2016.0024 · Zbl 1354.11039 · doi:10.1353/ajm.2016.0024
[2] 10.1007/978-1-4612-0383-4 · doi:10.1007/978-1-4612-0383-4
[3] 10.1007/BF02771784 · Zbl 1137.22009 · doi:10.1007/BF02771784
[4] 10.4310/PAMQ.2006.v2.n1.a9 · Zbl 1158.22017 · doi:10.4310/PAMQ.2006.v2.n1.a9
[5] 10.1090/coll/061 · doi:10.1090/coll/061
[6] ; Aubert, Münster J. Math., 7, 27 (2014)
[7] 10.1186/s40687-016-0079-4 · Zbl 1394.22015 · doi:10.1186/s40687-016-0079-4
[8] ; Aubert, Around Langlands correspondences. Contemp. Math., 691, 15 (2017)
[9] 10.1112/plms.12023 · Zbl 1383.20033 · doi:10.1112/plms.12023
[10] 10.1007/s00229-018-1001-8 · Zbl 1414.11161 · doi:10.1007/s00229-018-1001-8
[11] ; Aubert, Geometric aspects of the trace formula, 23 (2018)
[12] 10.1016/S0012-9593(02)01106-0 · Zbl 1092.11025 · doi:10.1016/S0012-9593(02)01106-0
[13] 10.2140/pjm.2012.255.281 · Zbl 1284.22008 · doi:10.2140/pjm.2012.255.281
[14] ; Ban, Geometry, algebra, number theory, and their information technology applications. Springer Proc. Math. Stat., 251, 59 (2018)
[15] ; Bernstein, Representations of reductive groups over a local field, 1 (1984) · Zbl 0544.00007
[16] ; Borel, Automorphic forms, representations and L-functions. Proc. Sympos. Pure Math., 33, 27 (1979) · Zbl 0414.22020
[17] 10.1007/978-1-4612-0941-6 · doi:10.1007/978-1-4612-0941-6
[18] ; Borel, Inst. Hautes Études Sci. Publ. Math., 55 (1965)
[19] ; Borel, Continuous cohomology, discrete subgroups, and representations of reductive groups. Annals of Mathematics Studies, 94 (1980) · Zbl 0443.22010
[20] ; Bruhat, Inst. Hautes Études Sci. Publ. Math., 5 (1972)
[21] 10.1112/S0024611598000574 · Zbl 0911.22014 · doi:10.1112/S0024611598000574
[22] 10.1007/s00209-018-2091-4 · Zbl 1446.11096 · doi:10.1007/s00209-018-2091-4
[23] ; Deligne, Representations of reductive groups over a local field, 33 (1984) · Zbl 0544.00007
[24] 10.1016/0001-8708(82)90030-5 · Zbl 0493.22005 · doi:10.1016/0001-8708(82)90030-5
[25] ; Gelfand, Lie groups and their representations, 95 (1975)
[26] ; Haines, Automorphic forms and Galois representations. London Math. Soc. Lecture Note Ser., 415, 118 (2014)
[27] ; Harris, The geometry and cohomology of some simple Shimura varieties. Annals of Mathematics Studies, 151 (2001) · Zbl 1036.11027
[28] 10.1007/s002220050012 · Zbl 1048.11092 · doi:10.1007/s002220050012
[29] 10.24033/bsmf.2431 · Zbl 1029.22023 · doi:10.24033/bsmf.2431
[30] 10.1090/S0065-9266-2011-00642-8 · Zbl 1242.22023 · doi:10.1090/S0065-9266-2011-00642-8
[31] ; Jacquet, Automorphic forms on GL(2). Lecture Notes in Mathematics, 114 (1970) · Zbl 0236.12010
[32] 10.2140/ant.2013.7.2447 · Zbl 1371.11148 · doi:10.2140/ant.2013.7.2447
[33] 10.4007/annals.2016.184.2.6 · Zbl 1393.22009 · doi:10.4007/annals.2016.184.2.6
[34] 10.24033/asens.1523 · Zbl 0634.22014 · doi:10.24033/asens.1523
[35] 10.1090/pspum/061/1476501 · doi:10.1090/pspum/061/1476501
[36] 10.1090/surv/031/03 · doi:10.1090/surv/031/03
[37] 10.1007/BF01244308 · Zbl 0809.11032 · doi:10.1007/BF01244308
[38] ; Lusztig, Representations of finite Chevalley groups. CBMS Regional Conference Series in Mathematics, 39 (1978) · Zbl 0418.20037
[39] ; Lusztig, Inst. Hautes Études Sci. Publ. Math., 145 (1988)
[40] 10.2307/1990945 · Zbl 0715.22020 · doi:10.2307/1990945
[41] 10.1155/S1073792895000353 · Zbl 0872.20041 · doi:10.1155/S1073792895000353
[42] 10.1090/S1088-4165-02-00173-5 · Zbl 1016.22011 · doi:10.1090/S1088-4165-02-00173-5
[43] 10.2969/aspm/00610289 · doi:10.2969/aspm/00610289
[44] ; Matsumoto, Analyse harmonique dans les systèmes de Tits bornologiques de type affine. Lecture Notes in Mathematics, 590 (1977) · Zbl 0366.22001
[45] 10.1090/memo/1108 · Zbl 1316.22018 · doi:10.1090/memo/1108
[46] 10.1023/A:1001019027614 · Zbl 0937.22011 · doi:10.1023/A:1001019027614
[47] 10.4153/CJM-2011-025-3 · Zbl 1225.22017 · doi:10.4153/CJM-2011-025-3
[48] 10.1017/S1474748004000155 · Zbl 1102.22009 · doi:10.1017/S1474748004000155
[49] 10.2140/pjm.2002.206.451 · Zbl 1052.22014 · doi:10.2140/pjm.2002.206.451
[50] 10.1090/S1088-4165-02-00167-X · Zbl 0999.22021 · doi:10.1090/S1088-4165-02-00167-X
[51] ; Renard, Représentations des groupes réductifs p-adiques. Cours Spécialisés, 17 (2010) · Zbl 1186.22020
[52] 10.1016/S0012-9593(98)80139-0 · Zbl 0903.22009 · doi:10.1016/S0012-9593(98)80139-0
[53] 10.1023/A:1000216412619 · Zbl 0882.22019 · doi:10.1023/A:1000216412619
[54] 10.2307/2042302 · Zbl 0368.22009 · doi:10.2307/2042302
[55] 10.1016/j.jalgebra.2018.06.012 · Zbl 1404.22045 · doi:10.1016/j.jalgebra.2018.06.012
[56] 10.1090/S1088-4165-2012-00406-X · Zbl 1272.20003 · doi:10.1090/S1088-4165-2012-00406-X
[57] 10.1007/978-0-8176-4840-4 · doi:10.1007/978-0-8176-4840-4
[58] 10.2140/pjm.1992.152.375 · Zbl 0724.22017 · doi:10.2140/pjm.1992.152.375
[59] ; Tits, Automorphic forms, representations and L-functions. Proc. Sympos. Pure Math., 33, 29 (1979)
[60] 10.1090/conm/145/1216197 · doi:10.1090/conm/145/1216197
[61] 10.1112/S0010437X16007545 · Zbl 1375.22011 · doi:10.1112/S0010437X16007545
[62] 10.1007/s00208-016-1515-x · Zbl 1384.22008 · doi:10.1007/s00208-016-1515-x
[63] 10.24033/asens.1379 · Zbl 0441.22014 · doi:10.24033/asens.1379
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.