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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
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##### 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
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