ar X iv : m at h / 03 09 16 8 v 3 [ m at h . R T ] 1 6 A pr 2 00 9 IWAHORI - HECKE ALGEBRAS

Our aim here is to give a fairly self-contained exposition of some basic facts about the Iwahori-Hecke algebra H of a split p-adic group G, including Bernstein’s presentation and description of the center, Macdonald’s formula, the CasselmanShalika formula, and the Lusztig-Kato formula. There are no new results here, and the same is essentially true of the proofs. We have been strongly influenced by the notes [1] of a course given by Bernstein. In the spirit of Bernstein’s work, we approach the material with an emphasis on the “universal unramified principal series” module M = Cc(AON\G/I), which is a right module over the Iwahori-Hecke algebra H = Cc(I\G/I). We use M to develop the theory of intertwining operators in a purely algebraic framework. Once this framework is established, we adapt it to produce rather efficient proofs of the above results, following closely at times earlier proofs. In particular, in our treatment of Macdonald’s formula and the Casselman-Shalika formula, we follow the method introduced by Casselman [6] and Casselman-Shalika [7]. We follow Kato’s strategy from [14] in proving a fundamental formula of Lusztig [16], which nowadays has come to be known as the Lusztig-Kato formula. The reader may find in [21] another survey article which proves some of the results of the present paper by different methods.