Kapranov, M. Double affine Hecke algebras and \(2\)-dimensional local fields. (English) Zbl 0958.20005 J. Am. Math. Soc. 14, No. 1, 239-262 (2001). Harmonic analysis on reductive groups over \(0\)- and \(1\)-dimensional local fields leads to consideration of the finite and affine Hecke algebras \(H_q\), \(\dot H_q\) associated to any finite root system \(R\) and any \(q\in\mathbb{C}^*\). These algebras can be defined in several ways, one being by generators and relations, another as the algebra of double cosets. However, harmonic analysis on groups over \(2\)-dimensional local fields has not been developed. But, recently, I. Cherednik [Ann. Math., II. Ser. 141, No. 1, 191-216 (1995; Zbl 0822.33008)] defined the double affine Hecke algebra \(\ddot H_q\) in terms of generators and relations. D. Kazhdan proposed the problem of “giving a group-theoretic construction of the Cherednik algebra” (i.e. realizing it as some algebra of double cosets). In this paper the author provides a solution to this problem by developing beginnings of harmonic analysis on reductive groups over \(2\)-dimensional local fields. Reviewer: A.Khammash (Makkah) Cited in 3 ReviewsCited in 14 Documents MSC: 20C08 Hecke algebras and their representations 43A80 Analysis on other specific Lie groups 20G25 Linear algebraic groups over local fields and their integers 22E50 Representations of Lie and linear algebraic groups over local fields Keywords:reductive groups; local fields; affine Hecke algebras; generators; relations; algebras of double cosets; harmonic analysis Citations:Zbl 0822.33008 PDFBibTeX XMLCite \textit{M. Kapranov}, J. Am. Math. Soc. 14, No. 1, 239--262 (2001; Zbl 0958.20005) Full Text: DOI arXiv References: [1] Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). 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