# zbMATH — the first resource for mathematics

Geometric structure for the principal series of a split reductive $$p$$-adic group with connected centre. (English) Zbl 1347.22013
Let $$\mathcal{G}$$ be a split reductive $$p$$-adic group with connected centre. The authors show that each Bernstein block in the principal series of $$\mathcal{G}$$ admits a geometric structure of an extended quotient. For the Iwahori-spherical block, this extended quotient has the form $$T//W$$ where $$T$$ is a maximal torus in the Langlands dual group of $$\mathcal{G}$$ and $$W$$ is the Weyl group of $$\mathcal{G}$$.
Reviewer: Hu Jun (Beijing)

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 20G05 Representation theory for linear algebraic groups
Full Text:
##### References:
 [1] A.-M. Aubert, P. Baum and R. J. Plymen, The Hecke algebra of a reductive p-adic group: a geometric conjecture, in Noncommutative geometry and number theory, eds. C. Consani and M. Marcolli, 1–34, Aspects of Mathematics, E37, Vieweg Verlag, 2006.Zbl 1120.14001 MR 2327297 · Zbl 1120.14001 [2] A.-M. Aubert, P. Baum and R. J. Plymen, Geometric structure in the principal series of the p-adic group G2, Represent. Theory, 15 (2011), 126–169. Zbl 1268.22015 MR 2772586 · Zbl 1268.22015 [3] A.-M. Aubert, P. Baum, R. J. Plymen and M. Solleveld, Geometric structure in smooth dual and local Langlands conjecture, Japanese J. Math., 9 (2014), no. 2, 99–136.Zbl 06363092 MR 3258616 · Zbl 1371.11097 [4] R. W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Pure and Applied Mathematics, John Wiley & Sons, New York NJ, 1985.Zbl 0567.20023 MR 0794307 · Zbl 0567.20023 [5] N. Chriss and V. Ginzburg, Representation theory and complex geometry, Birkhäuser, 2000.Zbl 0879.22001 MR 1433132 · Zbl 0879.22001 [6] S.-I. Kato, A realization of irreducible representations of affine Weyl groups, Indag. Math., 45 (1983), no. 2, 193–201.Zbl 0531.20020 MR 0705426 · Zbl 0531.20020 [7] D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. math., 87 (1987), 153–215.Zbl 0613.22004 MR 0862716 · Zbl 0613.22004 [8] M. Khalkhali, Basic noncommutative geometry, EMS Lecture Series 2009. Zbl 1210.58006 MR 2567651 · Zbl 1210.58006 [9] E. M. Opdam, On the spectral decompostion of affine Hecke algebras, J. Inst. Math. Jussieu, 3 (2004), 531–648.Zbl 1102.22009 MR 2094450 ramified principal series representations, Represent. Theory, 6 (2002), 101–126. Zbl 0999.22021 MR 1915088 · Zbl 1102.22009 [10] A. Roche, Types and Hecke algebras for principal series representations of split reductive p-adic groups, Ann. scient. Éc. Norm. Sup., 31 (1998), 361–413. Zbl 0903.22009 MR 1621409 · Zbl 0903.22009 [11] J.-P. Serre, Linear representations of finite groups, Springer Verlag, 1977. Zbl 0355.20006 MR 0450380 [12] T. Shoji, Green functions of reductive groups over a finite field, in The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 289– 301, Proc. Sympos. Pure Math., 47, Part 1, Amer. Math. Soc., Providence, RI, 1987.Zbl 0648.20047 MR 0933366 [13] M. Solleveld, On the classification of irreducible representations of affine Hecke algebras with unequal parameters, Represent. Theory, 16 (2012), 1–87. Zbl 1272.20003 MR 2869018 · Zbl 1272.20003 [14] M. Solleveld, Hochschild homology of affine Hecke algebras, Journal of Algebra, 384 (2013), 1–35.Zbl 1305.20002 MR 3045149 · Zbl 1305.20002
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.