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Injection de modules sphériques pour les espaces symétriques réductifs dans certaines représentations induites. (Injection of spherical modules for reductive symmetric spaces into certain induced representations). (French) Zbl 0658.22003

Non-commutative harmonic analysis and Lie groups, Proc. Int. Conf., Marseille-Luminy 1985, Lect. Notes Math. 1243, 108-143 (1987).
[For the entire collection see Zbl 0607.00006.]
Let G be a real reductive group in the Harish-Chandra class with Lie algebra \({\mathfrak g}\), and let H be an open subgroup of the group \(G^{\sigma}\) of fixed points of an involution \(\sigma\) of G. Let \(\theta\) be a Cartan involution of G which commutes with \(\sigma\) and let K be the group of fixed points of \(\theta\). In this article a \({\mathfrak g}\)-module is said to be H-spherical if it is a (\({\mathfrak g},K)\)- admissible submodule of \(C^{\infty}(G/H)\) of finite length. By using the asymptotic behavior of K-finite and Z(\({\mathfrak g})\)-finite functions in \(C^{\infty}(G/H)\) established by E. van den Ban, the author is able to prove an analogue of the Casselman subrepresentation theorem announced several years ago by Ōshima. Also, he proves that any irreducible H- spherical module can be realized as a submodule of an induced representation \(Ind_{MAN\uparrow G}\delta \otimes \epsilon^{\lambda}\otimes l_ N\) of a parabolic subgroup \(P=MAN\) of G, such that: the Levi subgroup \(L=MA\) of P is \(\sigma\)- and \(\theta\)- stable, the split component of A of L, relative to \(\theta\), consists of all elements g of L such that \(\theta (g)=g^{-1}\), \(\sigma (g)=g^{- 1}\); \(\delta\) is an irreducible subrepresentation of \(L^ 2(M/M\cap H)\) and \(\lambda\) is a linear form on \({\mathfrak a}={\mathcal L}(A)\) such that Re \(\lambda\) is in the closure of the negative Weyl chamber determined by roots of \({\mathfrak a}\) in the Lie algebra of the unipotent radical N of P. On the way, the author also obtains the result that an H-spherical irreducible tempered representation can be realized as a subrepresentation of an induced one, as above, but with \(\lambda\) unitary (a result also announced by Ōshima). A result on the extension of analytic functions on A to G due to van den Ban and the author is included in an appendix.

MSC:

22E30 Analysis on real and complex Lie groups
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.

Citations:

Zbl 0607.00006