Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Hostname: page-component-559fc8cf4f-dxfhg Total loading time: 0.675 Render date: 2021-03-05T01:21:21.819Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }
EnglishFrançais
Compositio Mathematica Compositio Mathematica

Article contents

The pro- $p$ -Iwahori Hecke algebra of a reductive $p$ -adic group I

Part of: Forms and linear algebraic groups Linear algebraic groups and related topics Representation theory of groups

Published online by Cambridge University Press:  23 October 2015

Marie-France Vigneras
Marie-France Vigneras
Affiliation:
Institut de Mathematiques de Jussieu, 175 rue du Chevaleret, Paris 75013, France email vigneras@math.jussieu.fr
Corresponding
E-mail address:
vigneras@math.jussieu.fr
Get access
Rights & Permissions[Opens in a new window]

Abstract

Let $R$ be a commutative ring, let $F$ be a locally compact non-archimedean field of finite residual field $k$ of characteristic $p$ , and let $\mathbf{G}$ be a connected reductive $F$ -group. We show that the pro- $p$ -Iwahori Hecke $R$ -algebra of $G=\mathbf{G}(F)$ admits a presentation similar to the Iwahori–Matsumoto presentation of the Iwahori Hecke algebra of a Chevalley group, and alcove walk bases satisfying Bernstein relations. This was previously known only for a $F$ -split group $\mathbf{G}$ .

Keywords

pro-p-Iwahori Hecke algebra affine Hecke algebra Iwahori Weyl group reductive p-adic group parahoric subgroups alcove walk

MSC classification

Primary: 11E95: $p$-adic theory 20G25: Linear algebraic groups over local fields and their integers 20C08: Hecke algebras and their representations
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 4 , April 2016 , pp. 693 - 753
Copyright
© The Author 2015 

Access options

Get access to the full version of this content by using one of the access options below.
If you should have access and can't see this content please contact technical support.

References

Abe, N., Henniart, G., Herzig, F. and Vigneras, M.-F., A classification of irreducible admissible modulo $p$representations of $p$-adic reductive groups, Preprint (2014), arXiv:1412.0737.Google Scholar
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.CrossRefGoogle Scholar
Borel, A., Linear algebraic groups, second enlarged edition (Springer, New York, 1991).CrossRefGoogle Scholar
Borel, A. and Tits, J., Groupes réductifs, Publ. Math. Inst. Hautes Études Sci. 27 (1965), 55–151.CrossRefGoogle Scholar
Bourbaki, N., Groupes et algèbres de Lie, chapitres 4, 5 et 6 (Hermann, Paris, 1968).Google Scholar
Bruhat, F. and Tits, J., Groupes réductifs sur un corps local. I. Données radicielles valuées, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–252.CrossRefGoogle Scholar
Bruhat, F. and Tits, J., Groupes réductifs sur un corps local. II Schémas en groupes. Existence d’une donnée radicielle valuées, Publ. Math. Inst. Hautes Études Sci. 60 (1984), 197–376; part II.Google Scholar
Cabanes, M. and Enguehard, M., Representation theory of finite reductive groups (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Görtz, U., Alcove walks and nearby cycle on affine flag manifolds, J. Algebraic Combin. 26 (2007), 415–430.CrossRefGoogle Scholar
Haines, H. and Rapoport, M., Appendix: On parahoric subgroups, Adv. Math. 219 (2008), 188–198; appendix to: G. Pappas and M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), 118–198.CrossRefGoogle Scholar
Haines, T. and Rostami, S., The Satake isomorphism for special maximal parahoric algebras, Represent. Theory 14 (2010), 264–284.CrossRefGoogle Scholar
Henniart, G. and Vigneras, M.-F., A Satake isomorphism for representations modulo p of reductive groups over local fields, J. Reine Angew. Math. 701 (2015), 33–75.Google Scholar
Iwahori, N. and Matsumoto, H., On some Bruhat decomposition and the structure of the Hecke ring of p-adic Chevalley groups, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 5–48.CrossRefGoogle Scholar
Kottwitz, R., Isocrystals with additional structure II, Compositio Math. 109 (1997), 225–339.CrossRefGoogle Scholar
Koziol, K. and Xu, P., Hecke modules and supersingular representations of U (2, 1), Represent. Theory 19 (2015), 56–93.CrossRefGoogle Scholar
Kumar, S., Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204 (Birkhäuser, Boston, 2002).Google Scholar
Lusztig, G., Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599–635.CrossRefGoogle Scholar
Milne, J. S. and Shih, K. Y., Conjugates of Shimura varieties, in Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900 (Springer, Berlin, 1981).Google Scholar
Richarz, T., On the Iwahori–Weyl group, Preprint (2013), arXiv:1310.4635.Google Scholar
Schmidt, N. A., Generische pro-p-algebren (Dilpomarbeit, Berlin, 2009).Google Scholar
Schneider, P. and Stuhler, U., Representation theory and sheaves on the Bruhat–Tits building, Publ. Math. Inst. Hautes Études Sci. 85 (1997), 97–191.CrossRefGoogle Scholar
Steinberg, R., Lecture on Chevalley groups, Yale notes (1967).Google Scholar
Vigneras, M.-F., Représentations ℓ-modulaires d’un groupe réductif p-adique avec ℓ≠p, Progress in Mathematics, vol. 137 (Birkhäuser, Boston, 1996).Google Scholar
Vigneras, M.-F., Pro-p-Iwahori Hecke algebra and supersingular Fp -representations, Math. Ann. 331 (2005), 523–556; Erratum, Math. Ann. 333 (2005), 699–701.CrossRefGoogle Scholar
Vigneras, M.-F., Algèbres de Hecke affines génériques, Represent. Theory 10 (2006), 1–20.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 153 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 5th March 2021. This data will be updated every 24 hours.

10 Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

The pro- $p$ -Iwahori Hecke algebra of a reductive  $p$ -adic group I
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

The pro- $p$ -Iwahori Hecke algebra of a reductive  $p$ -adic group I
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

The pro- $p$ -Iwahori Hecke algebra of a reductive  $p$ -adic group I
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *