The principal series of \(p\)-adic groups with disconnected center. (English) Zbl 1383.20033
Let \(\mathcal{G}\) be a split connected reductive group over a local non-Archimedean field \(F\), possibly with disconnected center. In this paper, the authors construct a local Langlands classification for all the principal series representations of \(\mathcal{G}\) under the assumption that the residual characteristic does not take on certain small values. They follow a similar approach to that of M. Reeder [Represent. Theory 6, 101–126 (2002; Zbl 0999.22021)], based on A. Roche’s realization of types, and his equivalence of categories with Iwahori-Hecke algebras [Ann. Sci. Éc. Norm. Supér. (4) 31, No. 3, 361–413 (1998; Zbl 0903.22009)]. Furthermore, they show that these representations are nicely parameterized by suitable extended quotient, in line with the ABPS conjecture (the Aubert-Baum-Plymen-Solleveld cojecture).
More specifically, let \(\mathcal{T}\) be a maximal torus in \(\mathcal{G}\), and let \(G\) and \(T\) be the Langlands dual groups of \(\mathcal{G}\) and \(\mathcal{T}\), respectively. Denote by \(\mathbf{Irr}(\mathcal{G},\mathcal{T})\) the space of all \(\mathcal{G}\)-representations in the principal series. The authors consider Langlands parameters of the form \(\Phi: F^\times \times \mathrm{SL}_2(\mathbb{C}) \to G\), and for such \(\Phi\) they define a Kazhdan-Lusztig-Reeder parameter as the ordered pair \((\Phi, \rho)\), where \(\rho\) is a “geometric” representation of \(\pi_0(Z_G(\Phi))\). Let \(\mathcal{W}^G=W(G,T)\). The authors prove that there exists a commutative, bijective triangle \[ \begin{aligned} & (\mathbf{Irr}\mathcal{T}/\!/\mathcal{W}^G)_2 \\ & \swarrow \searrow \\ \mathbf{Irr}(\mathcal{G},\mathcal{T}) & \longrightarrow \{\text{KLR parameters for }G \}/G \end{aligned} \] Here, \((\mathbf{Irr}\mathcal{T}/\!/\mathcal{W}^G)_2\) is the extendend quotient of the second kind; the right slanted map is natural, and via the bottom map any \(\pi \in \mathbf{Irr}(\mathcal{G},\mathcal{T})\) canonically determines a Langlands parameter \(\Phi\) for \(\mathcal{G}\). The triangle above is obtained as the union of corresponding triangles for all Bernstein components.
The authors explicitly verify the desiderata for the local Langlands correspondence proposed by A. Borel [Proc. Symp. Pure Math. 33, 27–61 (1979; Zbl 0412.10017)]. In particular, the constructed correspondence is functorial with respect to homomorphisms of reductive groups.
More specifically, let \(\mathcal{T}\) be a maximal torus in \(\mathcal{G}\), and let \(G\) and \(T\) be the Langlands dual groups of \(\mathcal{G}\) and \(\mathcal{T}\), respectively. Denote by \(\mathbf{Irr}(\mathcal{G},\mathcal{T})\) the space of all \(\mathcal{G}\)-representations in the principal series. The authors consider Langlands parameters of the form \(\Phi: F^\times \times \mathrm{SL}_2(\mathbb{C}) \to G\), and for such \(\Phi\) they define a Kazhdan-Lusztig-Reeder parameter as the ordered pair \((\Phi, \rho)\), where \(\rho\) is a “geometric” representation of \(\pi_0(Z_G(\Phi))\). Let \(\mathcal{W}^G=W(G,T)\). The authors prove that there exists a commutative, bijective triangle \[ \begin{aligned} & (\mathbf{Irr}\mathcal{T}/\!/\mathcal{W}^G)_2 \\ & \swarrow \searrow \\ \mathbf{Irr}(\mathcal{G},\mathcal{T}) & \longrightarrow \{\text{KLR parameters for }G \}/G \end{aligned} \] Here, \((\mathbf{Irr}\mathcal{T}/\!/\mathcal{W}^G)_2\) is the extendend quotient of the second kind; the right slanted map is natural, and via the bottom map any \(\pi \in \mathbf{Irr}(\mathcal{G},\mathcal{T})\) canonically determines a Langlands parameter \(\Phi\) for \(\mathcal{G}\). The triangle above is obtained as the union of corresponding triangles for all Bernstein components.
The authors explicitly verify the desiderata for the local Langlands correspondence proposed by A. Borel [Proc. Symp. Pure Math. 33, 27–61 (1979; Zbl 0412.10017)]. In particular, the constructed correspondence is functorial with respect to homomorphisms of reductive groups.
Reviewer: Dubravka Ban (Carbondale)
MSC:
| 20G25 | Linear algebraic groups over local fields and their integers |
| 20G05 | Representation theory for linear algebraic groups |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
References:
| [1] | J. D.Adler and A.Roche, ‘An intertwining result for \(p\)‐adic groups’, Canad. J. Math.52 (2000) 449-467. · Zbl 1160.22304 |
| [2] | A.‐M.Aubert, P.Baum and R.Plymen, ‘The Hecke algebra of a reductive \(p\)‐adic group: a geometric conjecture’, Noncommutative geometry and number theory, Aspects of Mathematics E37 (eds C.Consani (ed.) and M.Marcolli (ed.); Vieweg Verlag, Berlin, 2006) 1-34. · Zbl 1120.14001 |
| [3] | A.‐M.Aubert, P.Baum, R. J.Plymen and M.Solleveld, ‘Geometric structure in smooth dual and local Langlands conjecture’, Jpn. J. Math.9 (2014) 99-136. · Zbl 1371.11097 |
| [4] | A.‐M.Aubert, P. F.Baum, R. J.Plymen and M.Solleveld, ‘Conjectures about \(p\)‐adic groups and their noncommutative geometry’, Proceedings of Around Langlands Correspondences, Orsay, 2015, to appear, Preprint, arXiv:1508.02837. |
| [5] | A.‐M.Aubert, P.Baum, R. J.Plymen and 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. · Zbl 1347.22013 |
| [6] | J. N.Bernstein, P.Deligne, ‘Le “centre” de Bernstein’, Représentations des groupes réductifs sur un corps local (Travaux en cours, Hermann, 1984) 1-32. · Zbl 0599.22016 |
| [7] | A.Borel, ‘Admissible representations of a semi‐simple group over a local field with vectors fixed under an Iwahori subgroup’, Invent. Math.35 (1976) 233-259. · Zbl 0334.22012 |
| [8] | A.Borel, ‘Automorphic L‐functions’, Automorphic forms, representations and L‐functions, Part 1, Proc. Sympos. Pure Math.33 (1979) 27-61. · Zbl 0412.10017 |
| [9] | W.Borho and R.MacPherson, ‘Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes’, C. R. Acad. Sci. Paris Sér. I292 (1981) 707-710. · Zbl 0467.20036 |
| [10] | C. J.Bushnell, G.Henniart and P. C.Kutzko, ‘Types and explicit Plancherel formulae for reductive \(p\)‐adic groups’, On certain L‐functions, Clay Mathematics Proceedings 13 (American Mathematical Society, 2011) 55-80. · Zbl 1222.22011 |
| [11] | C. J.Bushnell and P. C.Kutzko, ‘Smooth representations of reductive \(p\)‐adic groups: structure theory via types’, Proc. Lond. Math. Soc.77 (1998) 582-634. · Zbl 0911.22014 |
| [12] | R. W.Carter, Finite groups of lie type, conjugacy classes and complex characters (Wiley Classics Library, London, 1993). |
| [13] | W.Casselman, ‘The unramified principal series representations of \(p\)‐adic groups I. The spherical function’, Compos. Math.40 (1980) 387-406. · Zbl 0472.22004 |
| [14] | N.Chriss and V.Ginzburg, Representation theory and complex geometry (Birkhäuser, Basel, 2000). |
| [15] | P.Delorme and E. M.Opdam, ‘The Schwartz algebra of an affine Hecke algebra’, J. reine angew. Math.625 (2008) 59-114. · Zbl 1173.22014 |
| [16] | R.Fowler and G.Röhrle, ‘On cocharacters associated to nilpotent elements of reductive groups’, Nagoya Math. J.190 (2008) 105-128. · Zbl 1185.20050 |
| [17] | D.Goldberg and A.Roche, ‘Hecke algebras and \(S L_n\)‐types’, Proc. Lond. Math. Soc.90.1 (2005) 87-131. · Zbl 1172.22300 |
| [18] | G. J.Heckman and E. M.Opdam, ‘Harmonic analysis for affine Hecke algebras’, Current developments in mathematics (International Press, Boston, MA, 1996) 37-60. · Zbl 0932.22006 |
| [19] | R.Hotta, ‘On Springer’s representations’, J. Fac. Sci. Univ. Tokyo Sect. IA28 (1982) 863-876. · Zbl 0584.20033 |
| [20] | N.Iwahori and H.Matsumoto, ‘On some Bruhat decomposition and the structure of the Hecke rings of the \(p\)‐adic Chevalley groups’, Publ. Math. Inst. Hautes Études Sci.25 (1965) 5-48. · Zbl 0228.20015 |
| [21] | K.Iwasawa, Local class field theory, Oxford Mathematical Monographs (Oxford University Press, Oxford, 1986). · Zbl 0604.12014 |
| [22] | J. C.Jantzen, ‘Nilpotent orbits in representation theory’, Lie theory, Lie algebras and representations, Progress in Mathematics 228 (eds J.‐P.Anker (ed.) and B.Orsted (ed.); Birkhäuser, Boston, 2004). · Zbl 1169.14319 |
| [23] | S.‐I.Kato, ‘A realization of irreducible representations of affine Weyl groups’, Indag. Math.45.2 (1983) 193-201. · Zbl 0531.20020 |
| [24] | D.Kazhdan and G.Lusztig, ‘Proof of the Deligne-Langlands conjecture for Hecke algebras’, Invent. Math.87 (1987) 153-215. · Zbl 0613.22004 |
| [25] | G.Lusztig, ‘Character sheaves V’, Adv. Math.61 (1986) 103-155. · Zbl 0602.20036 |
| [26] | G.Lusztig, ‘Cells in affine Weyl groups, II’, J. Algebra109 (1987) 536-548. · Zbl 0625.20032 |
| [27] | G.Lusztig, ‘Cells in affine Weyl groups, III’, J. Fac. Sci. Univ. Tokyo Sect. IA Math34 (1987) 223-243. · Zbl 0631.20028 |
| [28] | G.Lusztig, ‘Cells in affine Weyl groups, IV’, J. Fac. Sci. Univ. Tokyo Sect. IA Math36 (1989) 297-328. · Zbl 0688.20020 |
| [29] | G.Lusztig, ‘Affine Hecke algebras and their graded version’, J. Amer. Math. Soc.2.3 (1989) 599-635. · Zbl 0715.22020 |
| [30] | L.Morris, ‘Tamely ramified supercuspidal representations’, Ann. Sci. Éc. Norm. Supér. (4) 29 (1996) 639-667. · Zbl 0870.22012 |
| [31] | A.Moy, ‘Distribution algebras on p‐adic groups and Lie algebras’, Canad. J. Math.63.5 (2011) 1137-1160. · Zbl 1225.22017 |
| [32] | E. M.Opdam, ‘On the spectral decomposition of affine Hecke algebras’, J. Inst. Math. Jussieu3 (2004) 531-648. · Zbl 1102.22009 |
| [33] | A.Ram and J.Ramagge, ‘Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory’, Trends Math. (2003) 428-466. · Zbl 1063.20004 |
| [34] | M.Reeder, ‘Whittaker functions, prehomogeneous vector spaces and standard representations of \(p\)‐adic groups’, J. reine angew. Math.450 (1994) 83-121. · Zbl 0794.22014 |
| [35] | M.Reeder, ‘Isogenies of Hecke algebras and a Langlands correspondence for ramified principal series representations’, Represent. Theory6 (2002) 101-126. · Zbl 0999.22021 |
| [36] | A.Roche, ‘Types and Hecke algebras for principal series representations of split reductive \(p\)‐adic groups’, Ann. Sci. ��c. Norm. Supér.31 (1998) 361-413. · Zbl 0903.22009 |
| [37] | T.Shoji, ‘Green functions of reductive groups over a finite field’, Proceedings of the American Mathematical Society 47 (American Mathematical Society, Providence, RI, 1987) 289-301. · Zbl 0648.20047 |
| [38] | M.Solleveld, ‘On the classification of irreducible representations of affine Hecke algebras with unequal parameters’, Represent. Theory16 (2012) 1-87. · Zbl 1272.20003 |
| [39] | T.Springer and R.Steinberg, Conjugacy classes, Lecture Notes in Mathematics 131 (Springer‐Verlag, Berlin, 1970) 167-266. · Zbl 0249.20024 |
| [40] | R.Steinberg, ‘Regular elements of semisimple algebraic groups’, Publ. Math. Inst. Hautes Études Sci.25 (1965) 49-80. · Zbl 0136.30002 |
| [41] | R.Steinberg, ‘Torsion in reductive groups’, Adv. Math.15 (1975) 63-92. · Zbl 0312.20026 |
| [42] | J.‐L.Waldspurger, ‘La formule de Plancherel pour les groupes \(p\)‐adiques (d’après Harish‐Chandra)’, J. Inst. Math. Jussieu2 (2003) 235-333. · Zbl 1029.22016 |
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