Noncommutative Geometry and Number Theory pp 301-321 | Cite as

# A non-commutative geometry approach to the representation theory of reductive *p*-adic groups: Homology of Hecke algebras, a survey and some new results

## Abstract

We survey some of the known results on the relation between the homology of the *full* Hecke algebra of a reductive *p*-adic group *G*, and the representation theory of *G*. Let us denote by *C* _{c} ^{∞} (*G*) the full Hecke algebra of *G* and by HP*(*C* _{c} ^{∞} (*G*)) its periodic cyclic homology groups. Let *Ĝ* denote the admissible dual of *G*. One of the main points of this paper is that the groups HP*(*C* _{c} ^{∞} (*G*)) are, on the one hand, directly related to the topology of *Ĝ* and, on the other hand, the groups HP*(*C* _{c} ^{∞} (*G*)) are explicitly computable in terms of *G* (essentially, in terms of the conjugacy classes of *G* and the cohomology of their stabilizers). The relation between HP*(*C* _{c} ^{∞} (*G*)) and the topology of *Ĝ* is established as part of a more general principle relating HP*(*A*) to the topology of Prim(*A*), the primitive ideal spectrum of *A*, for any finite typee algebra *A*. We provide several new examples illustrating in detail this principle. We also prove in this paper a few new results, mostly in order to better explain and tie together the results that are presented here. For example, we compute the Hochschild homology of O(*X*) ⋊ Г, the crossed product of the ring of regular functions on a smooth, complex algebraic variety *X* by a finite group Г. We also outline a very tentative program to use these results to construct and classify the cuspidal representations of *G*. At the end of the paper, we also recall the definitions of Hochschild and cyclic homology.

## Keywords

Conjugacy Class Spectral Sequence Noncommutative Geometry Chern Character Cyclic Homology## Preview

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## References

- [1]P. Baum, N. Higson, and R. Plymen, Representation theory of
*p*-adic groups: a view from operator algebras,*The mathematical legacy of Harish-Chandra*(Baltimore, MD, 1998), 111–149, Proc. Symp. Pure Math. 69, A. M. S. Providence RI, 2000.MathSciNetGoogle Scholar - [2]P. Baum and V. Nistor,
*Periodic cyclic homology of Iwahori-Hecke algebras*, C. R. Acad. Sci. Paris**332**(2001), 783–788.zbMATHMathSciNetGoogle Scholar - [3]P. Baum and V. Nistor,
*Periodic cyclic homology of Iwahori-Hecke algebras*,*K*-theory**27**(2003), 329–358.CrossRefMathSciNetGoogle Scholar - [4]M-T. Benameur and V. Nistor, Homology of complete symbols and noncommutative geometry, In
*Quantization of Singular Symplectic Quotients*, N.P. Landsman, M. Pflaum, and M. Schlichenmaier, ed.,*Progress in Mathematics***198**, pages 21–46, Birkhäuser, Basel-Boston-Berlin, 2001.Google Scholar - [5]M. Benameur, J. Brodzki, and V. Nistor,
*Cyclic homology and pseudodifferential operators: a survey*.Google Scholar - [6]J. Bernstein, P. Deligne, and D. Kazhdan,
*Trace Paley-Wiener theorem for reductive p-adic groups*, J. Analyse Math.**47**(1986), 180–192.zbMATHMathSciNetGoogle Scholar - [7]J. N. Bernstein,
*Le “centre” de Bernstein*, Representations of reductive groups over a local field (Paris) (P. Deligne, ed.), Hermann, Paris, 1984, pp. 1–32.Google Scholar - [8]P. Blanc and J. L. Brylinski,
*Cyclic Homology and the Selberg Principle*, J. Funct. Anal.**109**(1992), 289–330.zbMATHCrossRefMathSciNetGoogle Scholar - [9]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.zbMATHCrossRefMathSciNetGoogle Scholar - [10]J. Brodzki, An introduction to
*K*-theory and cyclic cohomology, Advanced Topics in Mathematics. PWN—Polish Scientific Publishers, Warsaw, 1998.Google Scholar - [11]J. Brodzki and V. Nistor,
*Cyclic homology of crossed products*, work in progress.Google Scholar - [12]J. Brodzki and Z. Lykova,
*Excision in cyclic type homology of Fréchet algebras*, Bull. London Math. Soc. 33 (2001), no. 3, 283–291.zbMATHMathSciNetGoogle Scholar - [13]J. Brodzki, R. Plymen,
*Chern character for the Schwartz algebra of p-adic GL(n)*, Bulletin of the LMS, 34, (2002) 219–228.zbMATHMathSciNetGoogle Scholar - [14]J.-L. Brylinski,
*Central localization in Hochschild homology*, J. Pure Appl. Algebra**57**(1989), 1–4.zbMATHCrossRefMathSciNetGoogle Scholar - [15]J.-L. Brylinski and V. Nistor,
*Cyclic cohomology of etale groupoids*, K-Theory**8**(1994), 341–365.zbMATHCrossRefMathSciNetGoogle Scholar - [16]A. Connes,
*Noncommutative differential geometry*, Publ. Math. IHES**62**(1985), 41–144.zbMATHGoogle Scholar - [17]A. Connes, Noncommutative Geometry, Academic Press, New York-London, 1994.zbMATHGoogle Scholar
- [18]J. Cuntz,
*Excision in periodic cyclic theory for topological algebras*, In:*Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995)*, 43–53, Amer. Math. Soc., Providence, RI, 1997.Google Scholar - [19]J. Cuntz and D. Quillen,
*Excision in bivariant periodic cyclic cohomology*, Invent. Math.**127**(1997), 67–98.zbMATHCrossRefMathSciNetGoogle Scholar - [20]B. Feigin and B. L. Tsygan,
*Additive K-Theory and cristaline cohomology*, Funct. Anal. Appl.**19**(1985), 52–62.CrossRefMathSciNetGoogle Scholar - [21]B. Feigin and B. L. Tsygan, Cyclic homology of algebras with quadratic relations, universal enveloping algebras and group algebras, in
*K-theory, arithmetic and geometry (Moscow, 1984–1986)*, 210–239 Lect. Notes Math.**1289**, Springer, Berlin, 1987.CrossRefGoogle Scholar - [22]T. G. Goodwillie,
*Cyclic homology, derivations, and the free loopspace*, Topology**24**(1985), 187–215.zbMATHCrossRefMathSciNetGoogle Scholar - [23]G. Hochschild, B. Kostant, and A. Rosenberg,
*Differential forms on regular affine algebras*, Trans. AMS**102**(1962), 383–408.zbMATHCrossRefMathSciNetGoogle Scholar - [24]John D. Jones and C. Kassel,
*Bivariant cyclic theory*,*K*-Theory**3**(1989), 339–365.zbMATHCrossRefMathSciNetGoogle Scholar - [25]
- [26]C. Kassel,
*A Knneth formula for the cyclic cohomology of*ℤ/2-*graded algebra*, Math. Ann.**275**(1986), 683–699.zbMATHCrossRefMathSciNetGoogle Scholar - [27]C. Kassel,
*Cyclic homology, comodules, and mixed complexes*, J. Algebra**107**(1987), 195–216.zbMATHCrossRefMathSciNetGoogle Scholar - [28]D. Kazhdan,
*Cuspidal geometry of p-adic groups*, J. Analyse Math.**47**(1986), 1–36.zbMATHMathSciNetCrossRefGoogle Scholar - [29]D. Kazhdan, V. Nistor, and P. Schneider,
*Hochschild and cyclic homology of finite type algebras*, Selecta Math. (N.S.)**4**(1998), 321–359.zbMATHCrossRefMathSciNetGoogle Scholar - [30]M. Kontsevich and A. Rosenberg,
*Noncommutative smooth spaces*, The Gelfand Mathematical Seminar, 1996–1999, 85–108, Birkhäuser Boston, Boston, Ma, 2000.Google Scholar - [31]S. Lang,
*Algebra*, Revised third edition, Springer, New York, 2002zbMATHGoogle Scholar - [32]J.-L. Loday, Cyclic Homology, Springer-Verlag, Berlin-Heidelberg-New York, 1992.zbMATHGoogle Scholar
- [33]J.-L. Loday and D. Quillen,
*Cyclic homology and the Lie homology of matrices*, Comment. Math. Helv.**59**(1984), 565–591.zbMATHCrossRefMathSciNetGoogle Scholar - [34]G. Lusztig,
*Cells in affine Weyl groups*, In “Algebraic Groups and Related Topics,” pp. 255–287, Advanced studies in Pure Math., vol.**6**, Kinokunia and North Holland, 1985.MathSciNetGoogle Scholar - [35]G. Lusztig,
*Cells in affine Weyl groups II*, J. Algebra**109**(1987), 536–548.zbMATHCrossRefMathSciNetGoogle Scholar - [36]G. Lusztig,
*Cells in affine Weyl groups III*, J. Fac. Sci. Univ. Tokyo Sect. IAMath.**34**(1987), 223–243.zbMATHMathSciNetGoogle Scholar - [37]G. Lusztig,
*Cells in affine Weyl groups IV*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**36**(1989), 297–328.zbMATHMathSciNetGoogle Scholar - [38]S. Mac Lane,
*Homology*, Springer-Verlag, Berlin-Heidelberg-New York, 1995.zbMATHGoogle Scholar - [39]S. Mac Lane and I. Moerdijk,
*Sheaves in geometry and logic. A first introduction to topos theory*, Universitext, Springer-Verlag, Berlin-Heidelberg-New York, 1994.Google Scholar - [40]Yu. Manin, Topics in noncommutative geometry, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1991, viii+164pp.zbMATHGoogle Scholar
- [41]Yu. Manin,
*Real multiplication and non-commutative geometry*, Lectures at M.P.I, May 2001.Google Scholar - [42]Yu. Manin and M. Marcolli,
*Continued fractions, modular symbols, and noncommutative geometry*, Selecta Math. New Ser.,**8**(2002), 475–521.Google Scholar - [43]V. Nistor,
*Group cohomology and the cyclic cohomology of crossed products*, Invent. Math.**99**(1990), 411–424.zbMATHCrossRefMathSciNetGoogle Scholar - [44]V. Nistor,
*Cyclic cohomology of crossed products by algebraic groups*, Invent. Math.**112**(1993), 615–638.zbMATHCrossRefMathSciNetGoogle Scholar - [45]N. Higson and V. Nistor,
*Cyclic homology of totally disconnected groups acting on buildings*, J. Funct. Anal.**141**(1996), 466–495.zbMATHCrossRefMathSciNetGoogle Scholar - [46]V. Nistor,
*Higher index theorems and the boundary map in cyclic homology*, Documenta**2**(1997), 263–295.zbMATHMathSciNetGoogle Scholar - [47]V. Nistor,
*Higher orbital integrals, Shalika germs, and the Hochschild homology of Hecke algebras of p-adic groups*, Int. J. Math. Math. Sci. 26 (2001), 129–160.zbMATHCrossRefMathSciNetGoogle Scholar - [48]P. Schneider,
*The cyclic homology of p-adic reductive groups*, J. Reine Angew. Math.**475**(1996), 39–54.zbMATHMathSciNetGoogle Scholar - [49]S. Dave,
*Equivariant yclic homology of pseudodifferential operators*, Thesis, Pennsylvania State University, in preparation.Google Scholar - [50]J.-P. Serre,
*Linear representations of finite groups*, Translated from the second French edition by Leonard L. Scott, Springer, New York, 1977.Google Scholar - [51]B. L. Tsygan,
*Homology of matrix Lie algebras over rings and Hochschild homology*, Uspekhi Math. Nauk.,**38**(1983), 217–218.zbMATHMathSciNetGoogle Scholar - [52]C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics
**38**,*Cambridge University Press, Cambridge*, 1994.Google Scholar - [53]M. Wodzicki,
*Excision in cyclic homology and in rational algebraic K-theory*, Annals of Mathematics**129**(1989), 591–640.CrossRefMathSciNetGoogle Scholar