These are purely expository notes of Opdam’s analysis [O1] of the trace form τ(f) = f(e) on the Hecke algebra H = C _c (I\G/I) of compactly supported functions f on a connected reductive split p-adic group G which are biinvariant under an Iwahori subgroup I, extending Macdonald’s work. We attempt to give details of the proofs, and choose notations which seem to us more standard. Many objects of harmonic analysis are met: principal series, Macdonald’s spherical forms, trace forms, Bernstein forms. The latter were introduced by Opdam under the name Eisenstein series for H. The idea of the proof is that the last two linear forms are proportional, and the proportionality constant is computed by projection to Macdonald’s spherical forms. Crucial use is made of Bernstein’s presentation of the Iwahori–Hecke algebra by means of generators and relations, as an extension of a finite dimensional algebra by a large commutative subalgebra. We give a complete proof of this using the universal unramified principal series right H-module M = C _c (A(O)N\G/I) to develop a theory of intertwining operators algebraically.

中文翻译：

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Iwahori-Hecke代数的调和分析

这些纯粹是Opdam对Hecke代数上的迹线形式τ（f）=  f（e）的分析[O1]的纯粹说明性注释H  =  C _c（I \ G / I）关于连接的归约分解p上的紧支撑函数f -adic组G在Iwahori亚组I下是双不变的 ，扩展了麦克唐纳的工作。我们尝试提供证明的细节，并选择在我们看来更标准的符号。满足了谐波分析的许多对象：主序列，麦克唐纳的球形，迹线形式，伯恩斯坦形式。后者由Opdam以H的Eisenstein系列的名称引入。证明的思想是最后两个线性形式是成比例的，并且比例常数是通过投影到麦克唐纳德的球形形式来计算的。伯恩斯坦通过生成器和关系对Iwahori-Hecke代数的表示是至关重要的用途，它是大型可交换子代数对有限维代数的扩展。我们使用通用无分支主体序列权H-模块M对此进行了完整的证明。 =  C _c（A（O）N \ G / I）来发展一个代数交织算子的理论。