Iwahori--Hecke Algebras of $SL_2$ over $2$-Dimensional Local Fields

http://dx.doi.org/10.4153/CJM-2010-072-x
Canad. J. Math. 62(2010), 1310-1324

  Published:2010-09-15
 Printed: Dec 2010

In this paper we construct an analogue of Iwahori--Hecke algebras of $\operatorname{SL}_2$ 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 $\operatorname{SL}_2$, 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.