## The most continuous part of the Plancherel decomposition for a reductive symmetric space.(English)Zbl 0878.43018

Let $$G/H$$ be a semisimple symmetric space; that is, $$G$$ is a connected semisimple real Lie group with involution $$\sigma$$, and $$H$$ is an open subgroup of the group of fixed points for $$\sigma$$ in $$G$$. A fundamental problem of harmonic analysis on $$G/H$$ is to obtain an explicit direct integral (Plancherel) decomposition of $$L\simeq \int^\oplus_{\widehat G} m_\pi\pi d\mu (\pi)$$ of the regular representation $$L$$ of $$G$$ in $$L^2 (G/H)$$ into irreducible unitary representations. The principal result of this paper is the determination of such a decomposition for the part of $$L^2(G/H)$$ which corresponds to the “most continuous part” of the spectrum. Let $${\mathfrak g}$$ be the Lie algebra of $$G$$ and let $${\mathfrak g}= {\mathfrak h}+ {\mathfrak q}$$ be its decomposition into $$\pm 1$$-eigenspaces for $$\sigma$$ (so that $${\mathfrak h}$$ is the Lie algebra of $$H)$$. Furthermore, let $$\theta$$ be a Cartan involution commuting with $$\sigma$$. By analogy with the group case one expects $$L^2(G/H)$$ to decompose into a finite number of parts, each attached to a particular parabolic subgroup $$P$$ satisfying $$\sigma\theta (P)=P$$. For a $$\sigma$$-minimal parabolic subgroup $$P=MAN$$ let $$\xi$$ be a finite dimensional unitary representation of $$M$$ and let $$\lambda\in {\mathfrak a}^*_{q\mathbb{C}}$$, where $${\mathfrak a}_q ={\mathfrak a} \cap {\mathfrak q}$$. Furthermore, let $$C^\infty (\xi: \lambda)$$ and $$C^{-\infty} (\xi: \lambda)$$ denote the spaces of smooth, respectively generalized, vectors for the representation $$\pi_{\xi, \lambda}$$ associated to $$\sigma$$ and $$\lambda$$, and let $$C^\infty (\xi: \lambda)^H$$ be the space of $$H$$-fixed elements in $$C^{-\infty} (\xi: \lambda)$$. In [E. P. van den Ban, Ann. Sci. Éc. Norm. Supér, IV. Sér. 21, 359-412 (1988; Zbl 0714.22009)] and [E. P. van den Ban and H. Schlichtkrull, Fourier transforms on a semisimple symmetric space (to appear in Invent. Math.)] a certain finite dimensional space $$V(\xi)$$ (independent of $$\lambda)$$ was defined together with a natural family of linear maps $$j^0(\xi: \lambda) =j^0 (P:\xi: \lambda): V(\xi)\to C^{-\infty} (\xi: \lambda)^H$$, depending meromorphically on $$\lambda\in {\mathfrak a}^*_{q\mathbb{C}}$$ and bijective for generic $$\lambda$$. The definition involved a normalization procedure which guarantees that $$\lambda \mapsto j^0 (\xi: \lambda)$$ is regular at the imaginary points $$\lambda\in i {\mathfrak a}^*_q$$. The Fourier transform $$\widehat f$$ for $$f\in C^\infty_c (G/H)$$ is then defined by $\widehat f(\xi: \lambda)= \int_{G/H} f(gH) \pi_{\xi,- \lambda} (g)j^0 (\xi: \lambda) d(gH)\in \operatorname{Hom} \bigl(V(\xi), C^\infty(\xi: -\lambda) \bigr).$ The main result of the paper asserts that if one sets $$d\mu (\xi,\lambda) =\dim(\xi)d \lambda$$, where $$d\lambda$$ is the suitably normalized Lebesgue measure on $$i{\mathfrak a}^*_q$$, then $$f\mapsto \widehat f$$ extends to a partial isometry of $$L^2(G/H)$$ onto the space $$\int^\oplus_{\xi, \lambda} m_\xi \pi_{\xi, \lambda} d\mu (\xi,\lambda)$$ which specified domains for $$\xi$$ and $$\lambda$$. The proof of these results uses uniform estimates for the coefficients of a converging expansion for the Eisenstein integral, estimates of Paley-Wiener type for the Fourier transform $${\mathcal F}f$$ of a Schwartz function $$f$$, the wave packet transform $${\mathcal I} \varphi$$ of a $$^0{\mathcal C} (\tau)$$-valued function $$\varphi$$ on $$i{\mathfrak a}^*$$ on $$G/H$$ and other methods. In the last section the authors derive a Paley-Wiener type result for spaces of split rank one. The image by $${\mathcal F}$$ of the space of compactly supported smooth functions of type $$\tau$$ is characterized by a growth condition.
Reviewer: J.Ludwig (Metz)

### MSC:

 43A85 Harmonic analysis on homogeneous spaces 22E30 Analysis on real and complex Lie groups 53C35 Differential geometry of symmetric spaces 22E46 Semisimple Lie groups and their representations

Zbl 0714.22009
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