×

zbMATH — the first resource for mathematics

Geometric structure in the principal series of the \(p\)-adic group \(\mathbf G_2\). (English) Zbl 1268.22015
Summary: In the representation theory of reductive \(p\)-adic groups \(G\), the issue of reducibility of induced representations is an issue of great intricacy.
It is our contention, expressed as a conjecture in [the authors, C. R., Math., Acad. Sci. Paris 345, No. 10, 573–578 (2007; Zbl 1128.22009)], that there exists a simple geometric structure underlying this intricate theory.
We will illustrate here the conjecture with some detailed computations in the principal series of \(G_2\).
A feature of this article is the role played by cocharacters \(h_c\) attached to two-sided cells \(c\) in certain extended affine Weyl groups.
The quotient varieties which occur in the Bernstein programme are replaced by extended quotients. We form the disjoint union \(A(G)\) of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space \(A(G)\) is a model of the smooth dual \(Irr(G)\). In this respect, our programme is a conjectural refinement of the Bernstein programme.
The algebraic deformation is controlled by the cocharacters \(h_c\). The cocharacters themselves appear to be closely related to Langlands parameters.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
20G05 Representation theory for linear algebraic groups
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] A.-M. Aubert, Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif \( p\)-adique, Trans. Amer. Math. Soc. 347 (1995) 2179-2189. Erratum: 348 (1996) 4687-4690. · Zbl 0827.22005
[2] Anne-Marie Aubert, Paul Baum, and Roger Plymen, The Hecke algebra of a reductive \?-adic group: a geometric conjecture, Noncommutative geometry and number theory, Aspects Math., E37, Friedr. Vieweg, Wiesbaden, 2006, pp. 1 – 34. · Zbl 1120.14001
[3] Anne-Marie Aubert, Paul Baum, and Roger Plymen, Geometric structure in the representation theory of \?-adic groups, C. R. Math. Acad. Sci. Paris 345 (2007), no. 10, 573 – 578 (English, with English and French summaries). · Zbl 1128.22009
[4] Paul Baum and Victor Nistor, Periodic cyclic homology of Iwahori-Hecke algebras, \?-Theory 27 (2002), no. 4, 329 – 357. · Zbl 1056.16005
[5] J. N. Bernstein, Le ”centre” de Bernstein, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 1 – 32 (French). Edited by P. Deligne. · Zbl 0599.22016
[6] I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive \?-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441 – 472. · Zbl 0412.22015
[7] A. Borel, Automorphic \?-functions, Automorphic forms, representations and \?-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27 – 61.
[8] Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 7 – 9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. · Zbl 0319.17002
[9] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5 – 251 (French). · Zbl 0254.14017
[10] Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication.
[11] David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. · Zbl 0960.14002
[12] Paul E. Gunnells, Cells in Coxeter groups, Notices Amer. Math. Soc. 53 (2006), no. 5, 528 – 535. · Zbl 1150.20024
[13] David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153 – 215. · Zbl 0613.22004
[14] Charles David Keys, On the decomposition of reducible principal series representations of \?-adic Chevalley groups, Pacific J. Math. 101 (1982), no. 2, 351 – 388. · Zbl 0438.22010
[15] George Lusztig, Singularities, character formulas, and a \?-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208 – 229. · Zbl 0561.22013
[16] George Lusztig, Cells in affine Weyl groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 255 – 287.
[17] George Lusztig, Cells in affine Weyl groups. II, J. Algebra 109 (1987), no. 2, 536 – 548. , https://doi.org/10.1016/0021-8693(87)90154-2 George Lusztig, Cells in affine Weyl groups. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 223 – 243.
[18] George Lusztig, Cells in affine Weyl groups. IV, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 2, 297 – 328. · Zbl 0688.20020
[19] G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. · Zbl 1051.20003
[20] G. Lusztig, On some partitions of a flag manifold, Preprint June 2009, arXiv:0906.1505v1. · Zbl 1220.20042
[21] Goran Muić, The unitary dual of \?-adic \?\(_{2}\), Duke Math. J. 90 (1997), no. 3, 465 – 493. · Zbl 0896.22006
[22] Arun Ram, Representations of rank two affine Hecke algebras, Advances in algebra and geometry (Hyderabad, 2001) Hindustan Book Agency, New Delhi, 2003, pp. 57 – 91. · Zbl 1062.20006
[23] Mark Reeder, Isogenies of Hecke algebras and a Langlands correspondence for ramified principal series representations, Represent. Theory 6 (2002), 101 – 126. · Zbl 0999.22021
[24] Alan Roche, Types and Hecke algebras for principal series representations of split reductive \?-adic groups, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 3, 361 – 413 (English, with English and French summaries). · Zbl 0903.22009
[25] François Rodier, Décomposition de la série principale des groupes réductifs \?-adiques, Noncommutative harmonic analysis and Lie groups (Marseille, 1980) Lecture Notes in Math., vol. 880, Springer, Berlin-New York, 1981, pp. 408 – 424 (French).
[26] Jonathan Rosenberg, Appendix to: ”Crossed products of UHF algebras by product type actions” [Duke Math. J. 46 (1979), no. 1, 1 – 23; MR 82a:46063 above] by O. Bratteli, Duke Math. J. 46 (1979), no. 1, 25 – 26.
[27] Nan Hua Xi, Representations of affine Hecke algebras, Lecture Notes in Mathematics, vol. 1587, Springer-Verlag, Berlin, 1994.
[28] Nan Hua Xi, The based ring of the lowest two-sided cell of an affine Weyl group, J. Algebra 134 (1990), no. 2, 356 – 368. · Zbl 0709.20021
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.