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Boundary value problems for group invariant differential equations. (English) Zbl 0621.22014
Élie Cartan et les mathématiques d’aujourd’hui, The mathematical heritage of Elie Cartan, Semin. Lyon 1984, Astérisque, No.Hors Sér. 1985, 311-321 (1985).
[For the entire collection see Zbl 0573.00010.]
Let G be a connected semisimple Lie group with finite center and K its maximal compact subgroup. Let \({\mathfrak g}\) be the complexified Lie algebra of G. The author defines two exact functors \(V\to V_{\min}\) and \(V\to V_{\max}\), called minimal and maximal globalization, respectively, from the category of Harish-Chandra modules for (\({\mathfrak g},K)\) into the category of global representations of G. If a Harish-Chandra module is the module of K-finite vectors in a Banach representation \((\pi,V_{\pi})\) of G, the natural inclusion of the minimal globalization into the space \(V_{\pi}^{\omega}\) of analytic vectors in \(V_{\pi}\) is an isomorphism of topological vector spaces. Dually, the space of hyperfunction vectors \(V_{\pi}^{-\omega}\) of \(V_{\pi}\) is topologically isomorphic to the maximal globalization.
A number of interesting consequences of these results (including a new proof of Helgason’s conjecture about the spaces of joint eigenfunctions of G-invariant differential operators on the Riemannian symmetric space G/K) is discussed. Full details will appear elsewhere.
Reviewer: D.Miličić

MSC:
22E46 Semisimple Lie groups and their representations
43A85 Harmonic analysis on homogeneous spaces
58J32 Boundary value problems on manifolds