zbMATH — the first resource for mathematics

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)

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
Full Text: DOI Link