Harmonic analysis for certain representations of graded Hecke algebras (Q1899822)

For a fixed positive root \(R_+ \subseteq R\) of a root system \(R \subset {\mathfrak a}^*\) of a Euclidean space \(\mathfrak a\) \textit{I. Cherednik} [Invent. Math. 106, 411-432 (1991; Zbl 0742.20019)] has defined differential operators \(D_\zeta\), \(\zeta \in {\mathfrak h}\) (= complexification of \(\mathfrak a\)), acting in the space of functions on \({\mathfrak h}\) which are invariant for the translations in the lattice \(2 \pi i ZR^\vee\) (\(R^\vee\) = the coroot system). These operators and the Weyl group generate an operator algebra, which is isomorphic to the graded Hecke algebra \(\mathcal H\) in the sense of \textit{G. Lusztig} [J. Am. Math. Soc. 2, 599-635 (1989; Zbl 0715.22020)]. In this paper the author analyses the properties of the spaces \(C^\infty_c ({\mathfrak a})\) and \(C^\infty (T)\) \((T = i{\mathfrak a}/2\pi iZ R^\vee\)), which are \(\mathcal H\)-modules with natural + and \(*\)-invariant inner products. Decompositions of the spaces into irreducible parts are given, the Paley- Wiener theorem is proved and the inversion and the Plancherel formula are established.