Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T23:32:40.304Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Examples of Hecke Algebras for Two-Dimensional Local Fields

Published online by Cambridge University Press:  11 January 2016

Alexander Braverman  and
David Kazhdan
Alexander Braverman
Affiliation:
Einstein Institute of MathematicsEdmond J. Safra campus, Hebrew University of JerusalemGivat-Ram, 91904 [email protected]
David Kazhdan
Affiliation:
Mathematics DepartmentBox 1917Brown UniversityProvidence, RI [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K be a local non-archimedian field, F = K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group G = G(F) and its central extension Ĝ. For instance our spherical Hecke algebra corresponds to the subgroup G (A) ⊂ G(F) where A ⊂ F is the subring OK((t)) where OK ⊂ K is the ring of integers. It turns out that for generic level (cf. [4]) the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).

Keywords

22E5522E67
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 183 , 2006 , pp. 57 - 84
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

References

[1] Beilinson, A. and Drinfeld, V., Quantization of Hitchin’s integrable system and Hecke eigen-sheaves, preprint.Google Scholar
[2] Deligne, P., Extensions centrales de groupes algéebriques simplement connexes et co-é homologie galoisienne, Inst. Hautes Etudes Sci. Publ. Math., 84 (1996), 35–89.Google Scholar
[3] Faltings, G., Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc., 5 (2003), 41–68.Google Scholar
[4] Gaitsgory, D. and Kazhdan, D., Representations of algebraic groups over a 2-dimen-sional local field, Geom. Funct. Anal., 14 (2004), no. 3, 535–574.Google Scholar
[5] Gaitsgory, D. and Kazhdan, D., Algebraic groups over a 2-dimensional local field: some further constructions, math.RT/0406282.Google Scholar
[6] Garland, H., The arithmetic theory of loop groups. II. The Hilbert-modular case, J. Algebra, 209 (1998), no. 2, 446–532.Google Scholar
[7] Kapranov, M., The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups, math.AG/0001005.Google Scholar
[8] Kapranov, M., Double affine Hecke algebras and 2-dimensional local fields, J. Amer. Math. Soc., 14 (2001), no. 1, 239–262.Google Scholar
[9] Kumar, S., Demazure character formula in arbitrary Kac-Moody setting, Invent. Math., 89 (1987), no. 2, 395–423.Google Scholar
[10] Mathieu, O., Formules de caractères pour les algèbres de Kac-Moody géenéerales, Astéerisque No. 159–160, 1988, pp. 267.Google Scholar
[11] Quillen, D., Higher algebraic K-theory. I, Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972, Lecture Notes in Math., Vol. 341, Springer, Berlin (1973), pp. 85–147.Google Scholar