The action of intertwining operators on spherical vectors in the minimal principal series of a reductive symmetric space
Let \(G\) be a real semisimple Lie group with finite center, \(\sigma\) an involution of \(G\) and \(H\) an open subgroup of the group \(G^\sigma\) of fixed points for \(\sigma\). Moreover, let \(\Theta\) be a Cartan involution commuting with \(\sigma\). The minimal principal series which appear in the most continuous part of the Plancherel decomposition of \(L^2(G/H)\) are parabolically induced representations \(\pi_{\xi,\lambda}=\text{ind}^G_P(\xi\oplus\lambda\oplus 1)\), where \(P=MAN\) is a minimal \(\sigma\Theta\)-stable parabolic subgroup of \(G,\xi\) a finite dimensional unitary representation of \(M\) and \(\lambda\in{\mathfrak a}^*_{q\mathbb{C}}\) the space of complex characters of \(A\) with \(\sigma\nu=-\nu\). Let \(C^{-\infty}(P:\xi:\lambda)\) denote the space of generalized sections of the homogeneous vector bundle in which \(\pi_{\xi,\lambda}\) is naturally realized and let \(C^{-\infty}(P:\xi:\lambda)^H\) be its \(H\)-fixed elements. In the theory an important role is played by the standard intertwining operator \(A(Q:P:\xi:\lambda):C^{-\infty}(P:\xi:\lambda)\to C^{-\infty}(Q:\xi:\lambda)\), with \(Q\) a parabolic subgroup associated to \(P\). By equivariance the standard intertwining operator maps the spherical vectors for \(\pi_{P,\xi,\lambda}\) to those for \(\pi_{Q,\xi,\lambda}\). In [Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, 359-412 (1988; Zbl 0714.22009)] the author had established the existence of a finite-dimensional Hilbert space \(V(\xi)\) and a linear map \(j(P:\xi:\lambda)\to C^{-\infty}(P:\xi:\lambda)^H\) depending meromorphically on \(\lambda\in{\mathfrak a}^*_{q\mathbb{C}}\) and bijective for generic \(\lambda\). He also established the existence of a unique endomorphism \(B(Q:P:\xi:\lambda)\) of \(V(\xi)\) such that \[ A(Q:P:\xi:\lambda)\circ j(P:\xi:\lambda)=j(P:\xi:\lambda)\circ B(Q:P:\xi:\lambda). \] In the paper mentioned above the author proved that \(B(Q:P:\xi:-\overline\lambda)^*=B(Q:P:\xi:\lambda)\). For part of the argument leading to this formula the author needed the assumptions that \(H=G^\sigma\) and that all Cartan subgroups of \(G\) are abelian. The main result of the present paper is that the formula holds without restrictions on \(G,H\). A major part of the paper is thus devoted to the description of connected components of parabolic subgroups and the action of \(\sigma\) on them. The results of the paper can also be obtained by an application of a paper of \textit{J. Carmona} and \textit{P. Delorme} [Transformation de Fourier pour les espaces symétriques réductifs, Preprint, Luminy (1996)].
scientific article; zbMATH DE number 1136775