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.


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
Full Text: DOI EuDML


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