Abstract
The cocenter of an affine Hecke algebra plays an important role in the study of representations of the affine Hecke algebra and the geometry of affine Deligne–Lusztig varieties (see for example, He and Nie in Compos Math 150(11):1903–1927, 2014; He in Ann Math 179:367–404, 2014; Ciubotaru and He in Cocenter and representations of affine Hecke algebras, 2014). In this paper, we give a Bernstein–Lusztig type presentation of the cocenter. We also obtain a comparison theorem between the class polynomials of the affine Hecke algebra and those of its parabolic subalgebras, which is an algebraic analog of the Hodge–Newton decomposition theorem for affine Deligne–Lusztig varieties. As a consequence, we present a new proof of the emptiness pattern of affine Deligne–Lusztig varieties (Görtz et al. in Compos Math 146(5):1339–1382, 2010; Görtz et al. in Ann Sci Ècole Norm Sup, 2012).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
There is a more complicated formula for \(\alpha ^\vee \in 2 Y\) as well. However, we do not need it in this paper.
References
Bernstein, J., Deligne, P., Kazhdan, D.: Trace Paley–Wiener theorem for reductive \(p\)-adic groups. J. d’Anal. Math. 47, 180–192 (1986)
Ciubotaru, D., He, X.: Cocenter and representations of affine Hecke algebras, arXiv:1409.0902 (2014)
Ciubotaru, D., He, X.: The cocenter of graded affine Hecke algebra and the density theorem, arXiv:1208.0914 (2012)
Görtz, U., Hem, X., Nie, S.: \(P\)-alcoves and nonemptiness of affine Deligne–Lusztig varieties. Ann. Sci. Ècole Norm. Sup. (To appear); arXiv:1211.3784 (2012)
Görtz, U., Haines, T., Kottwitz, R., Reuman, D.: Affine Deligne–Lusztig varieties in affine flag varieties. Compos. Math. 146(5), 1339–1382 (2010)
He, X., Nie, S.: Minimal length elements of extended affine Weyl groups. Compos. Math. 150(11), 1903–1927 (2014)
He, X.: Minimal length elements in some double cosets of Coxeter groups. Adv. Math. 215(2), 469–503 (2007)
He, X., Nie, S.: Minimal length elements of finite Coxeter groups. Duke Math. J. 161, 2945–2967 (2012)
He, X.: Geometric and homological properties of affine Deligne–Lusztig varieties. Ann. Math. 179, 367–404 (2014)
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, 5–48 (1965)
Kazhdan, D.: Representations groups over close local fields. J. d’Anal. Math. 47, 175–179 (1986)
Kottwitz, R.: Isocrystals with additional structure. II. Compos. Math. 109, 255–339 (1997)
Lusztig, G.: Parabolic character sheaves. I. Mosc. Math. J. 4(1), 153–179 (2004)
Lusztig, G.: Affine Hecke algebras and their graded version. J. Am. Math. Soc. 2, 599–635 (1989)
Lusztig, G.: Hecke Algebras with Unequal Parameters, CRM Monograph Series, 18. American Mathematical Society, Providence (2003)
Opdam, E., Solleveld, M.: Homological algebra for affine Hecke algebras. Adv. Math. 220(5), 1549–1601 (2009)
Opdam, E., Solleveld, M.: Discrete series characters for affine Hecke algebras and their formal degrees. Acta Math. 205, 105–187 (2010)
Acknowledgments
We thank the referee for his/her thorough review and many useful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Xuhua He was partially supported by HKRGC grant 602011.
Rights and permissions
About this article
Cite this article
He, X., Nie, S. \(P\)-alcoves, parabolic subalgebras and cocenters of affine Hecke algebras. Sel. Math. New Ser. 21, 995–1019 (2015). https://doi.org/10.1007/s00029-014-0170-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-014-0170-x