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\(U({\mathfrak g})\)-finite locally analytic representations. (English) Zbl 1028.17007
Summary: We continue our algebraic approach to the study of locally analytic representations of a \(p\)-adic Lie group \(G\) in vector spaces over a non-Archimedean complete field \(K\). We characterize the smooth representations of Langlands theory which are contained in the new category. More generally, we completely determine the structure of the representations on which the universal enveloping algebra \(U(\mathfrak g)\) of the Lie algebra \(\mathfrak g\) of \(G\) acts through a finite dimensional quotient. They are direct sums of tensor products of smooth and rational \(G\)-representations. Finally we analyze the reducible members of the principal series of the group \(G= \text{SL}_2(\mathbb Q_p)\) in terms of such tensor products.

MSC:
17B15 Representations of Lie algebras and Lie superalgebras, analytic theory
22E50 Representations of Lie and linear algebraic groups over local fields
17B35 Universal enveloping (super)algebras
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