Harmonic analysis on the Iwahori-Hecke algebra
type: Item
instance of: scholarly article
title: Harmonic analysis on the Iwahori-Hecke algebra
zbMATH Open document ID: 1323.11028
author: Yuval Z. Flicker
published in: Japanese Journal of Mathematics. 3rd Series
publication date: 2014-10-31T00:00:00Z
review text: 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, su…
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11R39
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 11S37
zbMATH DE Number: 6363094
zbMATH Keywords: Iwahori-Hecke algebra
zbMATH Keywords: Bernstein presentation
zbMATH Keywords: intertwining operators
zbMATH Keywords: trace
zbMATH Keywords: generating function
zbMATH Keywords: Bernstein forms
zbMATH Keywords: trace forms
zbMATH Keywords: macdonald's spherical forms
MaRDI profile type: Publication
full work available at URL: s11537-014-1365-9
OpenAlex ID: W1981010095
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DOI: 10.1007/S11537-014-1365-9
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description: scientific article; zbMATH DE number 6363094