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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
Keywords:
universal enveloping algebra
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References:
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