Harmonic analysis on the Iwahori-Hecke algebra
Let \(G\) be a split connected reductive reductive group over a \(p\)-adic field \(F\). Any \(f \in C^\infty_c (G(F ))\) also defines a map on the set \(\hat{G}\) of irreducible admissible representations of \(G(F )\), by the prescription \( \pi \mapsto \operatorname{tr} \pi (f )\). According to Harish-Chandra's Plancherel decomposition, one can construct an explicit (modulo knowledge of the essentially square integrable representations of \(G(F ))\) measure on \(\hat{G}\) such that for all \(f \in C^\infty_c (G(F )), f (1)\) can be recovered as the integral of the function \( \pi \mapsto \operatorname{tr}\pi (f )\) over \(\hat{G}\). In [J. Inst. Math. Jussieu 3, No. 4, 531--648 (2004; Zbl 1102.22009)] \textit{E. M. Opdam} formulated and proved an analogous Plancherel decomposition for affine Hecke algebras. Before that, in [Prog. Math. 210, 301--323 (2003; Zbl 1162.22301)], he had initiated a study of this problem in the case of the Iwahori Hecke algebra \(H(I\;G/I)\), making crucial use of its Bernstein presentation, and as a consequence recovered Macdonald's decomposition of the `spherical part' of the Plancherel formula, namely, a spectral decomposition for functions biinvariant under \(K = G(\mathcal O)\) (for a suitable realization of \(G\) as a split reductive group over the ring \(\mathcal O\) of integers of \(F\)). While [Zbl 1102.22009] is much more general and harder than [Zbl 1162.22301], the latter presents an important idea occurring at the starting point for the investigations in the former. In the present paper, the author gives an exposition of the results of [Zbl 1162.22301], after recalling the Bernstein presentation for Iwahori Hecke algebras following [\textit{T. J. Haines} et al., J. Ramanujan Math. Soc. 25, No. 2, 113--145 (2010; Zbl 1202.22013)] so as to make everything more self-contained. The author adds that working in the explicit context of Iwahori Hecke algebras would hopefully make the presentation in well with more geometric ideas, such as those given recently by \textit{R. Bezrukavnikov} and \textit{D. Kazhdan} [``Geometry of second adjointness for p-adic groups'', Preprint, \url{arXiv:1112.6340}], regarding formulas from [Zbl 1162.22301], in their recent work concerning Bernstein's second adjointness theorem.
scientific article; zbMATH DE number 6363094
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