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Fourier and Poisson transformation associated to a semisimple symmetric space. (English) Zbl 0665.43004

Let G/H be a semisimple symmetric space: G is a semisimple Lie group and H is the fixed point group of an involution \(\tau\) of G. Let \(\theta\) be a Cartan involution commuting with \(\tau\) and K be the fixed point group of \(\theta\). One considers a parabolic subgroup P with a decomposition \(P=MAN\) such that MA is \(\tau\) and \(\theta\) stable, and \(M/M\cap H\) is compact. By a result of Matsuki there is a finite number of open orbits \({\mathcal O}_ 1,...,{\mathcal O}_ r\) of P in G/H. Using the finite- dimensional H-K-spherical representations of G, the author gives a system of equations for the set \(G/H-{\mathcal O}\), with \({\mathcal O}=\cup {\mathcal O}_ j.\)
For a character \(\nu\) of A, and an irreducible finite-dimensional \(M\cap H\)-spherical representation \(\sigma\) of M one defines the representation \(\sigma_{\nu}\) of P, \(\sigma_{\nu}(man)=a^{\nu}\sigma (m)\), and the induced representation \(\pi_{\sigma,\nu}\) of G, acting in a space \(I(\sigma,\nu)\) of functions on G with values in the space of \(\sigma\). The Poisson kernel \(p^ j_{\sigma,\nu}(x)\) is an analytic function on \({\mathcal O}_ j\), extended by 0 outside of \({\mathcal O}_ j\). For \(\nu\) in a certain tube, \(p^ j_{\sigma,\nu}(x)\) is locally integrable and defines a distribution \(u^ j_{\sigma,\nu}\) belonging to the space \(I(\sigma,-\nu)^ H_{-\infty}\). With respect to suitable coordinates \(p^ j_{\sigma,\nu}\) is a product of powers of polynomials. By the Bernstein theorem it follows that \(\nu \mapsto u^ j_{\sigma,\nu}\) extends to a meromorphic function with values in \(I(\sigma,\nu)_{- \infty}.\)
The Poisson transformation \({\mathcal P}^ j_{\sigma,\nu}\) is the map from \(I(\sigma,\nu)_{\infty}\) into \({\mathcal C}^{\infty}(G)\) defined by \({\mathcal P}^ j_{\sigma,\nu}(f)(x)=<\pi^{-\infty}_{\sigma,- \nu}(x)u^ j_{\sigma,\nu},f>\), and the Fourier transformation, essentially its dual, is the map from \({\mathcal C}_ c^{\infty}(G/H)\) into \(I({\bar\sigma},\nu)_{\infty}\) given by \({\mathcal F}^ j_{\sigma,\nu}(\phi)(x)= \int_{G/H}\phi(y)p^ j_{\sigma,\nu}(y^{-1}x)dy\). A Fatou type theorem is proved for the Poisson transformation which involves intertwining operators.

MSC:

43A85 Harmonic analysis on homogeneous spaces
22E30 Analysis on real and complex Lie groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
32D15 Continuation of analytic objects in several complex variables
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References:

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