an:04006561
Zbl 0621.22014
Schmid, Wilfried
Boundary value problems for group invariant differential equations
EN
??lie Cartan et les math??matiques d'aujourd'hui, The mathematical heritage of Elie Cartan, Semin. Lyon 1984, Ast??risque, No.Hors S??r. 1985, 311-321 (1985).
1985
a
22E46 43A85 58J32
connected semisimple Lie group; maximal compact subgroup; globalization; Harish-Chandra modules; global representations; analytic vectors; space of hyperfunction vectors; Helgason's conjecture; invariant differential operators; Riemannian symmetric space
[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.
D.Mili??i??
Zbl 0573.00010