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Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs. (The invariant Paley-Wiener theorem for reductive Lie groups). (French) Zbl 0584.22005

Let G be a real reductive Lie group satisfying a few extra conditions. Let K be a maximal compact subgroup. Let \(M_ iA_ iN_ i\), \(i=1,...,R\), be representatives of the different classes of cuspidal parabolic subgroups. Let \(\pi_{\delta,\lambda}\), \((\delta,\lambda)\in (M_ i){\hat{\;}}_ d\times {\mathfrak a}^*_{i{\mathbb{C}}}\) be the generalized principal series related to \(M_ iA_ iN_ i\). Let \(C_ c^{\infty}(G,K)\) denote the set of (two-sided) K-finite, compactly supported \(C^{\infty}\)-functions on G. The invariant Fourier transform of \(f\in C_ c^{\infty}(G,K)\) is defined as the function \[ (i,\delta,\lambda)\to F_ i(\delta,\lambda)=<trace\quad \pi_{\delta,\lambda},\quad f>. \] Let PW(\({\mathfrak a}^*_ i)\), the Paley-Wiener space on \({\mathfrak a}^*_ i\), be the space of entire, rapidly decreasing functions of exponential type on \({\mathfrak a}^*_{i{\mathbb{C}}}\), and let \(W_ i\) be the Weyl group for \((G,A_ i)\) which acts on \((M_ i){\hat{\;}}_ d\) and on \({\mathfrak a}^*_{i{\mathbb{C}}}.\)
The main result of the paper, the invariant Paley-Wiener theorem, states that a function \((i,\delta,\lambda)\to F_ i(\delta,\lambda)\) is the invariant Fourier transform of a function f in \(C_ c^{\infty}(G,K)\) if and only if the following three conditions hold: (i) \(F_ i\) has finite support in \(\delta\), (ii) \(\lambda \to F_ i(\delta,\lambda)\) belongs to PW(\({\mathfrak a}^*_ i)\) and (iii) \(F_ i(w\delta,w\lambda)=F_ i(\delta,\lambda)\) for all \(w\in W_ i.\)
The paper also gives a result relating the exponential type to the size of the support. This result about the support has been sharpened by M. Cowling [ibid. 83, 403-404 (1986)].
In an appendix by the first author an application is given of the theorem to the theory of orbital integrals and base change, improving on a result by Shelstad from Schwartz-functions to functions in \(C_ c^{\infty}.\)
There is a mistake in the proof of the main theorem since Lemma 4, which is wrongly attributed to Vogan, is simply false. This lemma is used in the proof of Proposition 2 (ii) and Lemma 11 (ii). In an Erratum, to appear, these two results are proved in another way. In particular, for G connected this is rather easy.
Apart from this the proof is based on a study of the action of U(\({\mathfrak g})^ K\) on the minimal K-types of \(\pi_{\delta,\lambda}\) [cf. P. Delorme, Ann. Sci. Éc. Norm. Supér., IV. Sér. 17, 117-156 (1984; Zbl 0582.22009)] and on Theorem 3, which is derived from Arthur’s Paley- Wiener theorem [cf. J. Arthur, Acta Math. 150, 1-89 (1983; Zbl 0514.22006)], which yield the existence of many functions \(f\in C_ c^{\infty}(G,K)\) of specified K-type and specified invariant Fourier transform. Another and more direct proof of Theorem 3 is given in the paper. This proof relies on a result from Arthur (op. cit.) about multipliers. This multiplier theorem has an elementary proof [cf. P. Delorme, Invent. Math. 75, 9-23 (1984; Zbl 0536.43007)]. But the use of the multiplier theorem involves the fact that Harish-Chandra’s Plancherel measure is tempered. In a paper to appear by the second author and the reviewer a direct elementary proof of Theorem 3 is given.
Reviewer: M.Flensted-Jensen

MSC:

22E30 Analysis on real and complex Lie groups
43A80 Analysis on other specific Lie groups
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References:

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