## 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

### Citations:

Zbl 0582.22009; Zbl 0514.22006; Zbl 0536.43007
Full Text:

### References:

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