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Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula. (English) Zbl 0645.43009
Author’s introduction: Let G be a connected real semisimple Lie group with finite centre, and let \(\tau\) be an involutive automorphism of G. Put \(G^{\tau}=\{x\in G:\tau (x)=x\}\), and let H be a closed subgroup of G with \((G^{\tau})_ e\subset H\subset G^{\tau}\); here \((G^{\tau})_ e\) denotes the identity component of \(G^{\tau}.\)
In this paper we investigate some properties of the algebra D(X) of invariant differential operators on the semisimple symmetric space \(X=G/H\). Our main results are that the action of D(X) diagonalizes over the discrete part of L 2(X) (Theorem 1.5), and that the irreducible constituents of an abstract Plancherel formula for X occur with finite multiplicities (Theorem 3.1). Both results are proved by using techniques of Harish-Chandra adapted to the situation at hand.
Reviewer: M.Flensted-Jensen

43A85 Harmonic analysis on homogeneous spaces
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
53C35 Differential geometry of symmetric spaces
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