Skip to main content Accessibility help
×
Home

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

  • Kyu-Hwan Lee (a1)

Abstract

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.

    • 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.

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

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

      Iwahori-Hecke Algebras of SL2 over 2-Dimensional Local Fields
      Available formats
      ×

Copyright

Corresponding author

Footnotes

Hide All

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

Footnotes

References

Hide All
[1] Bump, D., Automorphic forms and representations. Cambridge Stud. Adv. Math. 55, Cambridge University Press, Cambridge, 1997.
[2] Cherednik, I., Double affine Hecke algebras and Macdonald's conjectures. Ann. of Math. 141(1995), 191–216. doi:10.2307/2118632
[3] Fesenko, I., Analysis on arithmetic schemes. I. Kazuya Kato's fiftieth birthday. Doc. Math. 2003, Extra Vol., 261–284 (electronic).
[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.
[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.
[6] Gaitsgory, D. and Kazhdan, D., Representations of algebraic groups over a 2-dimensional local field. Geom. Funct. Anal. 14(2004), 535–574.
[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.
[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.
[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/s00039-007-0648-1
[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.
[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.
[12] Kapranov, M., Double affine Hecke algebras and 2-dimensional local fields. J. Amer. Math. Soc. 14(2001), 239–262. doi:10.1090/S0894-0347-00-00354-4
[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.2004.0048
[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
[15] Parshin, A. N., Vector bundles and arithmetic groups I. Trudy Mat. Inst. Steklov. 208(1995), Teor. Chisel, Algebra i Algebr. Geom., 240–265.
[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/1195145453
[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.
[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.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

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

  • Kyu-Hwan Lee (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed