ar X iv : m at h / 03 09 16 8 v 1 [ m at h . R T ] 9 S ep 2 00 3 IWAHORI - HECKE ALGEBRAS
نویسنده
چکیده
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, including Bernstein’s presentation and description of the center, Macdonald’s formula, the CasselmanShalika formula, and the Kato-Lusztig 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. The following notation will be used throughout this paper. We work over a padic field F with valuation ring O and prime ideal P = (π). We denote by k the residue field O/P and by q the cardinality of k. Consider a split connected reductive group G over F , with split maximal torus A and Borel subgroup B = AN containing A. We write B̄ = AN̄ for the Borel subgroup containing A that is opposite to B. We assume that G,A,N are defined over O. We write K for G(O) and I for the Iwahori subgroup of K defined as the inverse image under G(O) → G(k) of B(k). For μ ∈ X∗(A) we write π for the element μ(π) ∈ A(F ). Note that μ 7→ π gives an isomorphism from X∗(A) to A/AO. (We will often abbreviate A(F ) to A and A(O) to AO, etc.)
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