zbMATH — the first resource for mathematics

Discrete series characters for affine Hecke algebras and their formal degrees. (English) Zbl 1214.20008
The goal of this paper is to continue previous work of the authors and others to develop harmonic analysis on semisimple affine Hecke algebras (determined by a based root datum and a positive parameter function) modeled to an extent after harmonic analysis on Lie groups. The two main goals of the paper are to classify the irreducible discrete series characters and calculate their formal degrees. A key new tool used in this work is the idea of deformation of parameters. For a fixed root datum the authors consider a continuous field of affine Hecke algebras as the parameter is allowed to vary. This allows them to reduce the problem for arbitrary parameters to “generic” ones.
The results require the study of how the “central characters” behave under deformation, and also makes use of the classification of “generic residual points.” The authors achieve their second goal by obtaining a formula for the formal degree as a product of a (as yet unknown) rational number and an explicit rational function in the parameter. The main result of the paper is a classification of the irreducible discrete series characters for an affine Hecke algebra associated to an irreducible root datum (except type \(E\)) in terms of generic central characters. The classification is obtained by reducing the problem to degenerate affine Hecke algebras. The authors also discuss how to deal with the classification in the more general case of a semisimple root datum.

20C08 Hecke algebras and their representations
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
22E35 Analysis on \(p\)-adic Lie groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
43A99 Abstract harmonic analysis
Full Text: DOI arXiv
[1] Aubert, A. M., Baum, P. & Plymen, R., The Hecke algebra of a reductive p-adic group: a geometric conjecture, in Noncommutative Geometry and Number Theory, Aspects Math., E37, pp. 1–34. Vieweg, Wiesbaden, 2006. · Zbl 1120.14001
[2] Barbasch, D. & Moy, A., A unitarity criterion for p-adic groups. Invent. Math., 98 (1989), 19–37. · Zbl 0676.22012
[3] – Reduction to real infinitesimal character in affine Hecke algebras. J. Amer. Math. Soc., 6 (1993), 611–635. · Zbl 0835.22016
[4] Bernstein, I. N. & Zelevinsky, A. V., Induced representations of reductive p-adic groups. I. Ann. Sci. Ecole Norm. Sup., 10 (1977), 441–472. · Zbl 0412.22015
[5] Blackadar, B., Kumjian, A. & Rørdam, M., Approximately central matrix units and the structure of noncommutative tori. K-Theory, 6 (1992), 267–284. · Zbl 0813.46064
[6] Blondel, C., Propagation de paires couvrantes dans les groupes symplectiques. Represent. Theory, 10 (2006), 399–434. · Zbl 1133.22007
[7] Borel, A., 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] Bushnell, C. J., Henniart, G. & Kutzko, P. C., Towards an explicit Plancherel formula for reductive p-adic groups. Preprint, 2001. · Zbl 1222.22011
[9] Bushnell, C. J. & Kutzko, P. C., Types in reductive p-adic groups: the Hecke algebra of a cover. Proc. Amer. Math. Soc., 129 (2001), 601–607. · Zbl 0969.22007
[10] Carter, R. W., Finite Groups of Lie Type. Wiley, New York, 1985. · Zbl 0567.20023
[11] Cherednik, I., Double affine Hecke algebras and Macdonald’s conjectures. Ann. of Math., 141 (1995), 191–216. · Zbl 0822.33008
[12] Delorme, P. & Opdam, E. M., The Schwartz algebra of an affine Hecke algebra. J. Reine Angew. Math., 625 (2008), 59–114. · Zbl 1173.22014
[13] Emsiz, E., Opdam, E. M. & Stokman, J. V., Periodic integrable systems with delta-potentials. Comm. Math. Phys., 264 (2006), 191–225. · Zbl 1116.82020
[14] Geck, M., On the representation theory of Iwahori–Hecke algebras of extended finite Weyl groups. Represent. Theory, 4 (2000), 370–397. · Zbl 0983.20006
[15] Heckman, G. J. & Opdam, E. M., Yang’s system of particles and Hecke algebras. Ann. of Math., 145 (1997), 139–173. · Zbl 0873.43007
[16] – Harmonic analysis for affine Hecke algebras, in Current Developments in Mathematics (Cambridge, MA, 1996), pp. 37–60. International Press, Boston, MA, 1997. · Zbl 0932.22006
[17] Humphreys, J. E., Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, 9. Springer, New York, 1978. · Zbl 0447.17001
[18] Iwahori, N. & Matsumoto, H., On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups. Inst. Hautes Études Sci. Publ. Math., 25 (1965), 5–48. · Zbl 0228.20015
[19] Kato, S., An exotic Deligne–Langlands correspondence for symplectic groups. Duke Math. J., 148 (2009), 305–371. · Zbl 1183.20002
[20] Kazhdan, D. & Lusztig, G., Proof of the Deligne–Langlands conjecture for Hecke algebras. Invent. Math., 87 (1987), 153–215. · Zbl 0613.22004
[21] Lusztig, G., Cells in affine Weyl groups, in Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., 6, pp. 255–287. North-Holland, Amsterdam, 1985.
[22] – Affine Hecke algebras and their graded version. J. Amer. Math. Soc., 2 (1989), 599–635. · Zbl 0715.22020
[23] – Classification of unipotent representations of simple p-adic groups. Int. Math. Res. Not., 1995 (1995), 517–589. · Zbl 0872.20041
[24] – Cuspidal local systems and graded Hecke algebras. II, in Representations of Groups (Banff, AB, 1994), CMS Conf. Proc., 16, pp. 217–275. Amer. Math. Soc., Providence, RI, 1995.
[25] – Cuspidal local systems and graded Hecke algebras. III. Represent. Theory, 6 (2002), 202–242. · Zbl 1031.22007
[26] – Hecke Algebras with Unequal Parameters. CRM Monograph Series, 18. Amer. Math. Soc., Providence, RI, 2003.
[27] Macdonald, I. G., Spherical Functions on a Group of p-adic Type. Publications of the Ramanujan Institute, 2. Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras, Madras, 1971. · Zbl 0302.43018
[28] Matsumoto, H., Analyse harmonique dans les systèmes de Tits bornologiques de type affine. Lecture Notes in Mathematics, 590. Springer, Berlin–Heidelberg, 1977. · Zbl 0366.22001
[29] Morris, L., Tamely ramified intertwining algebras. Invent. Math., 114 (1993), 1–54. · Zbl 0854.22022
[30] – Level zero G-types. Compositio Math., 118 (1999), 135–157.
[31] Opdam, E. & Solleveld, M., Homological algebra for affine Hecke algebras. Adv. Math., 220 (2009), 1549–1601. · Zbl 1195.20004
[32] Opdam, E. M., On the spectral decomposition of affine Hecke algebras. J. Inst. Math. Jussieu, 3 (2004), 531–648. · Zbl 1102.22009
[33] – Hecke algebras and harmonic analysis, in International Congress of Mathematicians. Vol. II, pp. 1227–1259. Eur. Math. Soc., Zürich, 2006.
[34] – The central support of the Plancherel measure of an affine Hecke algebra. Mosc. Math. J., 7 (2007), 723–741, 767–768. · Zbl 1158.22003
[35] Phillips, N. C., K-theory for Fréchet algebras. Internat. J. Math., 2 (1991), 77–129. · Zbl 0744.46065
[36] Ram, A. & Ramagge, J., Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory, in A Tribute to C. S. Seshadri (Chennai, 2002), Trends Math., pp. 428–466. Birkhäuser, Basel, 2003. · Zbl 1063.20004
[37] Reeder, M., Formal degrees and L-packets of unipotent discrete series representations of exceptional p-adic groups. J. Reine Angew. Math., 520 (2000), 37–93. · Zbl 0947.20026
[38] – Euler–Poincaré pairings and elliptic representations of Weyl groups and p-adic groups. Compositio Math., 129 (2001), 149–181. · Zbl 1037.22039
[39] Schneider, P. & Stuhler, U., Representation theory and sheaves on the Bruhat–Tits building. Inst. Hautes Études Sci. Publ. Math., 85 (1997), 97–191. · Zbl 0892.22012
[40] Slooten, K., A Combinatorial Generalization of the Springer Correspondence for Classical Type. Ph.D. Thesis, University of Amsterdam, Amsterdam, 2003. http://dare.uva.nl/en/record/119136 . · Zbl 1110.20007
[41] – Generalized Springer correspondence and Green functions for type B/C graded Hecke algebras. Adv. Math., 203 (2006), 34–108. · Zbl 1151.20004
[42] Solleveld, M. S., Periodic Cyclic Homology of Affine Hecke Algebras. Ph.D. Thesis, University of Amsterdam, Amsterdam, 2007. http://dare.uva.nl/en/record/217308 . · Zbl 1305.20002
[43] Takesaki, M., Theory of Operator Algebras. I. Springer, New York, 1979. · Zbl 0436.46043
[44] Vignéras, M. F., On formal dimensions for reductive p-adic groups, in Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday (Ramat Aviv, 1989), Part I, Israel Math. Conf. Proc., 2, pp. 225–266. Weizmann, Jerusalem, 1990.
[45] Waldspurger, J. L., La formule de Plancherel pour les groupes p-adiques (d’après Harish-Chandra). J. Inst. Math. Jussieu, 2 (2003), 235–333. · Zbl 1029.22016
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.