Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T06:11:51.919Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraic and analytic Dirac induction for graded affine Hecke algebras

Published online by Cambridge University Press:  13 March 2013

Dan Ciubotaru ,
Eric M. Opdam  and
Peter E. Trapa
Dan Ciubotaru
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA ([email protected]; [email protected])
Eric M. Opdam
Affiliation:
Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands ([email protected])
Peter E. Trapa
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We define the algebraic Dirac induction map ${\mathrm{Ind} }_{D} $ for graded affine Hecke algebras. The map ${\mathrm{Ind} }_{D} $ is a Hecke algebra analog of the explicit realization of the Baum–Connes assembly map in the $K$-theory of the reduced ${C}^{\ast } $-algebra of a real reductive group using Dirac operators. The definition of ${\mathrm{Ind} }_{D} $ is uniform over the parameter space of the graded affine Hecke algebra. We show that the map ${\mathrm{Ind} }_{D} $ defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded affine Hecke algebra analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

Keywords

graded affine Hecke algebra; Dirac index; elliptic representations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 3 , July 2014 , pp. 447 - 486
Copyright
©Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Atiyah, M. and Schmid, W., A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62.Google Scholar
Borel, A., Admissible representations of a semisimple p-adic group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233–259.Google Scholar
Barbasch, D., Ciubotaru, D. and Trapa, P., The Dirac operator for graded affine Hecke algebras, Acta Math. 209 (2) (2012), 197–227.Google Scholar
Barbasch, D. and Moy, A., A unitarity criterion for $p$ -adic groups, Invent. Math. 98 (1989), 19–38.Google Scholar
Baum, P. and Connes, A., Geometric K-theory for Lie groups and foliations, Enseign. Math. (2) 46 (1–2) (2000), 3–42.Google Scholar
Baum, P., Connes, A. and Higson, N., Classifying space for proper actions and K-theory of group ${C}^{\ast } $ -algebras, in C ∗-algebras: 1943–1993 (San Antonio, TX, 1993), Contemp. Math., Volume 167, pp. 240–291 (Amer. Math. Soc., 1994).Google Scholar
Beynon, W. and Spaltenstein, N., Green functions of finite Chevalley groups of type ${E}_{n} $ ( $n= 6, 7, 8$ ), J. Algebra 88 (2) (1984), 584–614.Google Scholar
Carter, R. W., Finite Groups of Lie Type, Pure and Applied Math. (New York), p. xii+544pp (Wiley-Interscience, NY, 1985).Google Scholar
Ciubotaru, D., Spin representations of Weyl groups and Springer’s correspondence, J. Reine Angew. Math. 671 (2012), 199–222.Google Scholar
Ciubotaru, D. and Kato, S., Tempered modules in the exotic Deligne–Langlands classification, Adv. Math. 226 (2011), 1538–1590.Google Scholar
Ciubotaru, D., Kato, M. and Kato, S., On characters and formal degrees of discrete series of classical affine Hecke algebras, Invent. Math. 187 (2012), 589–635.Google Scholar
Ciubotaru, D. and Trapa, P., Characters of Springer representations on elliptic conjugacy classes, Duke Math. J. 162 (2) (2013), 201–223.Google Scholar
Delorme, P. and Opdam, E., Schwartz algebra of an affine Hecke algebra, J. Reine Angew. Math. 625 (2008), 59–114.Google Scholar
Drinfeld, V. G., Degenerate affine Hecke algebras and yangians, Funktsional. Anal. i Prilozhen. 20 (1) (1986), 69–70 (in Russian); Engl. transl.: Functional Anal. Appl. 20(1) (1986), 62–64.Google Scholar
Drinfeld, V. G., Quasi-Hopf algebras, Algebra i Analiz 1 (6) (1989), 114–148 (in Russian); Engl. transl. in Leningrad Math, J. 1 (1990), 1419–1457.Google Scholar
Emsiz, E., Opdam, E. M. and Stokman, J. V., Trigonometric Cherednik algebra at critical level and quantum many-body problems, Selecta Math. (N.S.) 14 (3–4) (2009), 571–605.Google Scholar
Heckman, G. J. and Opdam, E. M., Yang’s system of particles and Hecke algebras, Ann. of Math. 45 (1997), 139–173.Google Scholar
Huang, J. S. and Pandžić, P., Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), 185–202.Google Scholar
Kato, S., An exotic Deligne–Langlands correspondence for symplectic groups, Duke Math. J. 148 (2) (2009), 305–371.Google Scholar
Kazhdan, D. and Lusztig, G., Proof of Deligne–Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153–215.Google Scholar
Lafforgue, V., Banach $KK$ -theory and the Baum–Connes conjecture, ICM III (1–3) (2002), 795–811.Google Scholar
Lusztig, G., Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (3) (1989), 599–635.Google Scholar
Lusztig, G., Cuspidal local systems and graded Hecke algebras III, Represent. Theory 6 (2002), 202–242.Google Scholar
Morris, A., Projective characters of exceptional Weyl groups, J. Algebra 29 (1974), 567–586.Google Scholar
Opdam, E. M., On the spectral decomposition of affine Hecke algebras, J. Math. Jussieu 3 (4) (2004), 531–648.CrossRefGoogle Scholar
Opdam, E. M., The central support of the Plancherel measure of an affine Hecke algebra, Mosc. Math. J. 7 (2007), 723–741, 767–768.Google Scholar
Opdam, E. and Solleveld, M., Homological algebra for affine Hecke algebras, Adv. Math. 220 (5) (2009), 1549–1601.Google Scholar
Opdam, E. and Solleveld, M., Discrete series characters for affine Hecke algebras and their formal degrees, Acta Math. 205 (2010), 105–187.Google Scholar
Parthasarathy, R., Dirac operator and the discrete series, Ann. Math. (2) 96 (1972), 1–30.Google Scholar
Read, E. W., On projective representations of the finite reflection groups of type ${B}_{l} $ and ${D}_{l} $ , J. Lond. Math. Soc. (2) 10 (1975), 129–142.Google Scholar
Reeder, M., Euler–Poincaré pairings and elliptic representations of Weyl groups and p-adic groups, Compositio Math. 129 (2) (2001), 149–181.Google Scholar
Slooten, K., Induced discrete series representations for Hecke algebras of types ${ B}_{n}^{ \mathsf{aff} } $ and ${ C}_{n}^{ \mathsf{aff} } $ , Int. Math. Res. Not. (10) (2008), Art. ID rnn023, 41 pp.Google Scholar
Solleveld, M., On the classification of irreducible representations of affine Hecke algebras with unequal parameters, Represent. Theory 16 (2012), 1–87.Google Scholar