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The principal series for a reductive symmetric space. I: H-fixed distribution vectors. (English) Zbl 0714.22009

Let G/H be a reductive symmetric space. We discuss the main topics treated in this paper under separate headings. (Unexplained notation is mostly standard and explained in the paper.)
1. The principal series. Let \(P=MAN\) be a \(\sigma\theta\)-stable parabolic subgroup with \(a=L(A)\) containing the maximal abelian \(a_{oq}\) in \(p\cap q\) where \(g=k+p=h+q\). (These \(P\in:\) \({\mathcal P}_{\sigma}(A_ q)\) are classified in terms of root and Weyl group data.) Let \(\hat M_{ps}\) be the set of (unitary) \(\xi\in \hat M\) for which there is \(w\in W=W(g,a_ 0)\), \(a_ 0\supset a_{0q}\) maximal abelian in p, so that \(w\xi\) has nontrivial (M\(\cap H)\)-distribution vectors in \({\mathcal K}^{- \infty}_{w\xi}\). By definition, the (non-unitary) principal series for G/H consists of representations \(Ind^ G_ P(\xi \otimes e^{\lambda}\otimes 1)\), \(\xi\in \hat M_{ps}\), \(\lambda \in a^*_{qc}\). It follows from results of Bruhat that \(Ind^ G_ P(\xi \otimes e^{\lambda}\otimes 1)\) is irreducible when \(\lambda \in ia^*_ q\) satisfies \(<\lambda,\alpha >\neq 0\) for all \(\alpha \in \Sigma =\Sigma (g,a_ p).\)
2. Intertwining operators. Let \(P_ 1=MAN_ 1\) and \(P_ 2=MAN_ 2\) be two P’s as above. Let \(A(P_ 2:P_ 1:\xi:\lambda): C^{\infty}(P_ 1:\xi:\lambda)\to C^{\infty}(P_ 2:\xi:\lambda)\) be the intertwining operator of the \(C^{\infty}\) induced representations defined as those of Knapp and Stein (K.-S.). Theorem. As for \(f\in C^{\infty}(K,\xi)\), the \(C^{\infty}(K,\xi)\)-valued function \(\lambda \to A(P_ 1:P_ 2:\xi:\lambda)f\), initially defined for \(\lambda\in {\mathcal A}(P_ 2| P_ 1)\) extends meromorphically to \(a^*_{qc}\). For each \(\lambda_ 0\in a^*_{qc}\) there is a neighbourhood \(N(\lambda_ 0)\) and a non- zero holomorphic \(\phi\) : N(\(\lambda\) \({}_ 0)\to {\mathbb{C}}\) such that the map \((\lambda,g)\to \phi (\lambda)A(P_ 1:P_ 2:\xi:\lambda)f\) of \(N(\lambda_ 0)\times C^{\infty}(K:\xi)\) into \(C^{\infty}(K:\xi)\) is continuous. The proof of this follows K.-S. with modifications and new arguments needed for the case at hand. The transformation properties of the K.-S. operators carry over as well.
3. H-fixed distributions. Fix \(P\in {\mathcal P}_{\sigma}(A_ q)\) and \(\xi\in \hat M_{fu}\) (finite-dimensional unitary). Let \({\mathcal O}(P)\) denote the union of the open H-orbits in \(P\setminus G\), \[ C^{\infty}({\mathcal O}(P):P:\xi:\lambda)=\{f: {\mathcal O}(P)\to {\mathcal K}_{\xi}| C^{\infty},\quad f(man\cdot x)=a^{\lambda +\rho_ P}f(x)\}, \] and r: \({\mathcal D}'(G:P:\xi:\lambda)^ H\to C^{\infty}({\mathcal O}(P):P:\xi:\lambda)\) the restriction map. Theorem. If \(k\in {\mathbb{N}}\), then for \(\lambda\) in the complement \({\mathcal B}_ k\) of a finite union of hyperplanes, r maps \({\mathcal D}'_ k(G:P:\xi:\lambda)^ H\) (distributions of order \(\leq k)\) injectively into \(C^{\infty}({\mathcal O}(P):P:\xi:\lambda)\). As a consequence, there is an injective evaluation map ev: \({\mathcal D}'(G:P:\xi:\lambda)^ H\to V(\xi)\) into a finite- dimensional space V(\(\xi\)), which leads further to a map j(P:\(\xi\) :\(\lambda\) :\(\eta\)): V(\(\xi\))\(\to {\mathcal D}'(G:P:\xi:\lambda)^ H\) that is bijective for \(\lambda\in {\mathcal B}=\cap {\mathcal B}_ k\) and with inverse ev.
4. The matrix B. It is defined by the relation \(A(P_ 2:P_ 1:\xi:\lambda)\circ j(P_ 1:\xi:\lambda)=j(P_ 2:\xi:\lambda)\circ B(P_ 2:P_ 1:\xi:\lambda)\) and is a meromorphic function of \(\lambda \in a^*_{qc}\) with values in End V(\(\xi\)). Theorem. Assume all Cartan subgroups of G are abelian. Then \(B(P_ 2:P_ 1:\xi:\lambda)^*=B(P_ 2:P_ 1:\xi:-{\bar \lambda})\), adjoint with respect to a natural unitary structure on V(\(\xi\)). The proof involves a lengthy reduction to \(\sigma\)-split-rank one.
Reviewer: W.Rossmann

MSC:

22E30 Analysis on real and complex Lie groups
22E46 Semisimple Lie groups and their representations
43A85 Harmonic analysis on homogeneous spaces
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