Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T14:55:53.068Z Has data issue: false hasContentIssue false
  • English
  • Français

Iwahori-Hecke Algebras of SL2 over 2-Dimensional Local Fields

Published online by Cambridge University Press:  20 November 2018

Kyu-Hwan Lee
Kyu-Hwan Lee*
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA
*
khlee@math.uconn.edu
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.

In this paper we construct an analogue of Iwahori–Hecke algebras of $\text{S}{{\text{L}}_{2}}$ over 2-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on $\text{S}{{\text{L}}_{2}}$, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori–Matsumoto type relations.

Keywords

22E5020G25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 6 , 14 December 2010 , pp. 1310 - 1324
Copyright
Copyright © Canadian Mathematical Society 2010

Footnotes

The author was supported in part by EPSRC grant on zeta functions and in part by KOSEF Grant #R01-2003-000-10012-0.

References

[1] Bump, D., Automorphic forms and representations. Cambridge Stud. Adv. Math. 55, Cambridge University Press, Cambridge, 1997.Google Scholar
[2] Cherednik, I., Double affine Hecke algebras and Macdonald's conjectures. Ann. of Math. 141(1995), 191–216. doi:10.2307/2118632Google Scholar
[3] Fesenko, I., Analysis on arithmetic schemes. I. Kazuya Kato's fiftieth birthday. Doc. Math. 2003, Extra Vol., 261–284 (electronic).Google Scholar
[4] Fesenko, I., Adelic approach to the zeta function of arithmetic schemes in dimension two. Mosc. Math. J. 8(2008), no. 2, 273–317, 399–400.Google Scholar
[5] Fesenko, I. and Kurihara, M., eds., Invitation to higher local fields. Papers from the conference held in Münster, August 29–September 5, 1999. Geometry & Topology Monographs, 3, Geometry & Topology Publications, Coventry, 2000.Google Scholar
[6] Gaitsgory, D. and Kazhdan, D., Representations of algebraic groups over a 2-dimensional local field. Geom. Funct. Anal. 14(2004), 535–574.Google Scholar
[7] Gaitsgory, D. and Kazhdan, D., Algebraic groups over a 2-dimensional local field: some further constructions. In: Studies in Lie theory, Progr. Math. 243, Birkhäuser Boston, Boston, MA, 2006, pp. 97–130.Google Scholar
[8] Gaitsgory, D. and Kazhdan, D., Algebraic groups over a 2-dimensional local field: irreducibility of certain induced representations. J. Differential Geom. 70(2005), no. 1, 113–127.Google Scholar
[9] Hrushovski, E. and Kazhdan, D., The value ring of geometric motivic integration and the Iwahori Hecke algebra of SL2 (with an appendix by Nir Avni). Preprint, math.AG/0609115. doi:10.1007/s00039-007-0648-1Google Scholar
[10] Iwahori, N., Generalized Tits system (Bruhat decomposition) on p-adic semisimple groups. In: 1966 Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math. IX, Boulder, Colo., 1965), Amer. Math. Soc., Providence, RI, pp. 71–83.Google Scholar
[11] Iwahori, N. and 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(1965), 5–48.Google Scholar
[12] Kapranov, M., Double affine Hecke algebras and 2-dimensional local fields. J. Amer. Math. Soc. 14(2001), 239–262. doi:10.1090/S0894-0347-00-00354-4Google Scholar
[13] Kim, H. and Lee, K.-H., Spherical Hecke algebras of SL2 over 2-dimensional local fields. Amer. J. Math. 126(2004), 1381–1399. doi:10.1353/ajm.2004.0048Google Scholar
[14] Lusztig, G., Singularities, character formulas, and a q-analog of weight multiplicities. In: Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque, 101–102, Soc. Math. France, Paris, 1983, pp. 208–229 Google Scholar
[15] Parshin, A. N., Vector bundles and arithmetic groups I. Trudy Mat. Inst. Steklov. 208(1995), Teor. Chisel, Algebra i Algebr. Geom., 240–265.Google Scholar
[16] Saito, K. and Takebayashi, T., Extended affine root systems III (Elliptic Weyl groups). Publ. Res. Inst. Math. Sci. 33(1997), 301–329. doi:10.2977/prims/1195145453Google Scholar
[17] Shimura, G., Introduction to the arithmetic theory of automorphic functions. Kanô Memorial Lectures, 1, Publications of the Mathematical Society of Japan, 11, Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, NJ, 1971.Google Scholar
[18] Zhukov, I., Higher dimensional local fields. Invitation to higher local fields ( Münster, 1999), 5–18 (electronic), Geom. Topol. Monogr. 3, Geom. Topol. Publ., Coventry, 2000.Google Scholar