Fourier inversion on a reductive symmetric space
Let \(X=G/H\) be a reductive symmetric space, that is, \(G\) is a connected reductive Lie group of Harish-Chandra's class with an involution \(\sigma\), and \(H\) an open subgroup of the group of \(\sigma\)-fixed elements in \(G\). As an important step towards a complete Plancherel theorem (i.e. an explicit direct integral decomposition) for the regular representation of \(G\) in \(L^2(X)\), the authors define in the article at hand a ''pseudo-wave packet'' transform which they show to be an inverse of a suitably normalized Fourier transform. It is part of a series of papers by the authors on topics circling around this issue; in particular, it should be viewed as a direct continuation of \textit{E.P. van den Ban} and \textit{H. Schlichtkrull} [Ann. Math. (2) 145, No. 2, 267--364 (1997; Zbl 0878.43018)] and \textit{E.P. van den Ban} and \textit{H. Schlichtkrull} [Invent. Math. 130, No. 3, 517--574 (1997; Zbl 0936.22008)], which treat the ''most continuous part'' of the spectrum and the semisimple case, respectively. The article is structured as follows. Let \(K\) be a maximal compact subgroup of \(G\) invariant under \(\sigma\), let \((\tau,V_{\tau})\) be a f.d. unitary representation of \(K\), \(\lambda\in{\mathfrak a}^*_{q\mathbb C}\) and \(x\in X\). In Sections 1 and 2, the \(\tau\)-spherical Fourier transform \(\mathcal{F}\) is defined using normalized Eisenstein integrals \(E^0(\lambda:x)\) from one of the papers mentioned previously. In Section 3 (and part of the appendix), it is shown that \(E^0(\lambda:x)\) is singular only along real root hyperplanes in \({\mathfrak a}^*_{q\mathbb C}\). Pseudo-wave packets are defined in Section 4 on a suitable dense subset \(X_+\) of \(X\) by \[ \mathcal{T}_{\eta}\varphi(x)= |W|\int_{\eta+i{\mathfrak a}^*_q} E_+(\lambda:x)\varphi(x) d\lambda, \] where \(E_+\) is a linear operator into \(V_{\tau}\) appearing in the expansion of \(E^0\) and defined by its asymptotic behaviour and a vanishing property relative to some invariant differential operators. By Cauchy's theorem, it does not depend on \(\eta\) and will then be denoted by \(\mathcal{T}\). Some estimates of \(E_+\) as a function of \(\lambda\), convergence of \(\mathcal{T}\) and a relation with the ``classical'' Fourier inversion \(\mathcal{I}\) on Riemannian symmetric spaces follow. Sections 5 to 9 are devoted to the proof of the main result (Theorem 4.7): For \(f\in C^{\infty}_c(X:\tau)\), one has \(\mathcal{T}\mathcal{F}f(x)=f(x)\) for \(x\in X_+\). The proof heavily relies on the authors' papers [Represent. Theory 5, 615--712 (2001; Zbl 1049.22004) and Compos. Math. 123, No. 1, 27--72 (2002; Zbl 0965.22011)]. Roughly speaking, the difference \(\mathcal{T}\mathcal{F}f - \mathcal{I}\mathcal{F}f \) is expressed as a sum of residues, where summation is over a non-empty subset \(F\) of the set of simple roots \(\Delta\) (in fact, what remains for the Plancherel formula is essentially to identify these residues in terms of generalized principal series representations). As applications of their main result, the authors treat the Plancherel formula for a reductive symmetric space having only one conjugacy class of Cartan subspaces (Section 11) and the Fourier transform of rapidly decreasing functions (Section 12).
scientific article; zbMATH DE number 1541201
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