Annals of K-Theory

Topological K-theory of affine Hecke algebras

Maarten Solleveld

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Let ( , q ) be an affine Hecke algebra with a positive parameter function q . We are interested in the topological K-theory of its C -completion C r ( , q ) . We prove that K ( C r ( , q ) ) does not depend on the parameter q , solving a long-standing conjecture of Higson and Plymen. For this we use representation-theoretic methods, in particular elliptic representations of Weyl groups and Hecke algebras.

Thus, for the computation of these K-groups it suffices to work out the case q = 1 . These algebras are considerably simpler than for q 1 , just crossed products of commutative algebras with finite Weyl groups. We explicitly determine K ( C r ( , q ) ) for all classical root data . This will be useful for analyzing the K-theory of the reduced C -algebra of any classical p -adic group.

For the computations in the case q = 1 , we study the more general situation of a finite group Γ acting on a smooth manifold M . We develop a method to calculate the K-theory of the crossed product C ( M ) Γ . In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.

Article information

Ann. K-Theory, Volume 3, Number 3 (2018), 395-460.

Received: 6 November 2016
Revised: 18 September 2017
Accepted: 19 October 2017
First available in Project Euclid: 24 July 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20C08: Hecke algebras and their representations 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Secondary: 19L47: Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91]

topological K-theory affine Hecke algebra Weyl group crossed product algebra


Solleveld, Maarten. Topological K-theory of affine Hecke algebras. Ann. K-Theory 3 (2018), no. 3, 395--460. doi:10.2140/akt.2018.3.395.

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  • A.-M. Aubert, A. Moussaoui, and M. Solleveld, “Affine Hecke algebras for Langlands parameters”, preprint, 2016. To appear in Proceedings of the Simons Symposium.
  • A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld, “Conjectures about $p$-adic groups and their noncommutative geometry”, pp. 15–51 in Around Langlands correspondences, edited by F. Brumley et al., Contemporary Math. 691, American Mathematical Society, Providence, RI, 2017.
  • A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld, “Hecke algebras for inner forms of $p$-adic special linear groups”, J. Inst. Math. Jussieu 16:2 (2017), 351–419.
  • A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld, “The principal series of $p$-adic groups with disconnected center”, Proc. Lond. Math. Soc. $(3)$ 114:5 (2017), 798–854.
  • D. Barbasch and A. Moy, “Reduction to real infinitesimal character in affine Hecke algebras”, J. Amer. Math. Soc. 6:3 (1993), 611–635.
  • P. Baum and A. Connes, “Chern character for discrete groups”, pp. 163–232 in A fête of topology, edited by Y. Matsumoto et al., Academic Press, Boston, 1988.
  • P. Baum, A. Connes, and N. Higson, “Classifying space for proper actions and K-theory of group $C^\ast$-algebras”, pp. 240–291 in $C^\ast$-algebras: 1943–1993 (San Antonio, TX, 1993), edited by R. S. Doran, Contemporary Math. 167, American Mathematical Society, Providence, RI, 1994.
  • 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.
  • W. Borho and R. MacPherson, “Représentations des groupes de Weyl et homologie d'intersection pour les variétés nilpotentes”, C. R. Acad. Sci. Paris Sér. I Math. 292:15 (1981), 707–710.
  • G. E. Bredon, Equivariant cohomology theories, Lecture Notes in Math. 34, Springer, 1967.
  • C. J. Bushnell, G. Henniart, and P. C. Kutzko, “Types and explicit Plancherel formulæ for reductive $p$-adic groups”, pp. 55–80 in On certain $L$-functions, edited by J. Arthur et al., Clay Math. Proceedings 13, American Mathematical Society, Providence, RI, 2011.
  • H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, 1956.
  • R. W. Carter, “Conjugacy classes in the Weyl group”, Compositio Math. 25 (1972), 1–59.
  • R. W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, John Wiley & Sons, New York, 1985.
  • D. Ciubotaru, “On unitary unipotent representations of $p$-adic groups and affine Hecke algebras with unequal parameters”, Represent. Theory 12 (2008), 453–498.
  • D. Ciubotaru and E. Opdam, “Formal degrees of unipotent discrete series representations and the exotic Fourier transform”, Proc. Lond. Math. Soc. $(3)$ 110:3 (2015), 615–646.
  • D. Ciubotaru and E. Opdam, “A uniform classification of discrete series representations of affine Hecke algebras”, Algebra Number Theory 11:5 (2017), 1089–1134.
  • D. Ciubotaru, E. M. Opdam, and P. E. Trapa, “Algebraic and analytic Dirac induction for graded affine Hecke algebras”, J. Inst. Math. Jussieu 13:3 (2014), 447–486.
  • C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics XI, Interscience Publishers, New York, 1962.
  • P. Delorme and E. M. Opdam, “The Schwartz algebra of an affine Hecke algebra”, J. Reine Angew. Math. 625 (2008), 59–114.
  • P. Delorme and E. Opdam, “Analytic $R$-groups of affine Hecke algebras”, J. Reine Angew. Math. 658 (2011), 133–172.
  • D. Goldberg, “Reducibility of induced representations for ${\rm Sp}(2n)$ and ${\rm SO}(n)$”, Amer. J. Math. 116:5 (1994), 1101–1151.
  • G. J. Heckman and E. M. Opdam, “Yang's system of particles and Hecke algebras”, Ann. of Math. $(2)$ 145:1 (1997), 139–173.
  • V. Heiermann, “Opérateurs d'entrelacement et algèbres de Hecke avec paramètres d'un groupe réductif $p$-adique: le cas des groupes classiques”, Selecta Math. $($N.S.$)$ 17:3 (2011), 713–756.
  • S. Illman, “Smooth equivariant triangulations of $G$-manifolds for $G$ a finite group”, Math. Ann. 233:3 (1978), 199–220.
  • N. Iwahori and H. Matsumoto, “On some Bruhat decomposition and the structure of the Hecke rings of ${\mathfrak p}$-adic Chevalley groups”, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48.
  • P. Julg, “K-théorie équivariante et produits croisés”, C. R. Acad. Sci. Paris Sér. I Math. 292:13 (1981), 629–632.
  • G. G. Kasparov, “Equivariant KK-theory and the Novikov conjecture”, Invent. Math. 91:1 (1988), 147–201.
  • S.-I. Kato, “A realization of irreducible representations of affine Weyl groups”, Nederl. Akad. Wetensch. Indag. Math. 45:2 (1983), 193–201.
  • D. Kazhdan and G. Lusztig, “Proof of the Deligne–Langlands conjecture for Hecke algebras”, Invent. Math. 87:1 (1987), 153–215.
  • M. Langer and W. Lück, “Topological K-theory of the group $C^*$-algebra of a semi-direct product $\mathbb Z^n\rtimes\Bbb Z/m$ for a free conjugation action”, J. Topol. Anal. 4:2 (2012), 121–172.
  • W. Lück and B. Oliver, “Chern characters for the equivariant K-theory of proper $G$-CW-complexes”, pp. 217–247 in Cohomological methods in homotopy theory (Bellaterra, Spain, 1998), edited by J. Aguadé et al., Progress in Math. 196, Birkhäuser, Basel, 2001.
  • G. Lusztig, “Affine Hecke algebras and their graded version”, J. Amer. Math. Soc. 2:3 (1989), 599–635.
  • P. A. Mischenko, “Invariant tempered distributions on the reductive $p$-adic group ${\rm GL}\sb{n}(F\sb{p})$”, C. R. Math. Rep. Acad. Sci. Canada 4:2 (1982), 123–127.
  • E. M. Opdam, “On the spectral decomposition of affine Hecke algebras”, J. Inst. Math. Jussieu 3:4 (2004), 531–648.
  • E. Opdam and M. Solleveld, “Discrete series characters for affine Hecke algebras and their formal degrees”, Acta Math. 205:1 (2010), 105–187.
  • E. Opdam and M. Solleveld, “Extensions of tempered representations”, Geom. Funct. Anal. 23:2 (2013), 664–714.
  • N. C. Phillips, Equivariant K-theory and freeness of group actions on $C^*$-algebras, Lecture Notes in Math. 1274, Springer, 1987.
  • R. J. Plymen, “The reduced $C^\ast$-algebra of the $p$-adic group ${\rm GL}(n)$”, J. Funct. Anal. 72:1 (1987), 1–12.
  • R. J. Plymen, “Noncommutative geometry: Illustrations from the representation theory of ${\rm GL}(n)$”, lecture notes, 1993,
  • A. Ram and J. Ramagge, “Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory”, pp. 428–466 in A tribute to C. S. Seshadri (Chennai, 2002), edited by V. Lakshmibai et al., Birkhäuser, Basel, 2003.
  • M. Reeder, “Euler–Poincaré pairings and elliptic representations of Weyl groups and $p$-adic groups”, Compositio Math. 129:2 (2001), 149–181.
  • A. Roche, “Parabolic induction and the Bernstein decomposition”, Compositio Math. 134:2 (2002), 113–133.
  • C. Schochet, “Topological methods for $C\sp{\ast} $-algebras, II: Geometric resolutions and the Künneth formula”, Pacific J. Math. 98:2 (1982), 443–458.
  • J. Słomińska, “On the equivariant Chern homomorphism”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24:10 (1976), 909–913.
  • K. Slooten, A combinatorial generalization of the Springer correspondence for classical type, Ph.D. thesis, Universiteit van Amsterdam, 2003,
  • K. Slooten, “Generalized Springer correspondence and Green functions for type $B/C$ graded Hecke algebras”, Adv. Math. 203:1 (2006), 34–108.
  • M. Solleveld, Periodic cyclic homology of affine Hecke algebras, Ph.D. thesis, Universiteit van Amsterdam, 2007.
  • M. Solleveld, “Periodic cyclic homology of reductive $p$-adic groups”, J. Noncommut. Geom. 3:4 (2009), 501–558.
  • M. Solleveld, “Homology of graded Hecke algebras”, J. Algebra 323:6 (2010), 1622–1648.
  • M. Solleveld, “On the classification of irreducible representations of affine Hecke algebras with unequal parameters”, Represent. Theory 16 (2012), 1–87.
  • M. Solleveld, “Parabolically induced representations of graded Hecke algebras”, Algebr. Represent. Theory 15:2 (2012), 233–271.
  • T. A. Springer, “A construction of representations of Weyl groups”, Invent. Math. 44:3 (1978), 279–293.
  • R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society 80, American Mathematical Society, Providence, RI, 1968.