𝑈(𝔤)-finite locally analytic representations

Schneider, P.; Teitelbaum, J.; Prasad, Dipendra

doi:10.1090/S1088-4165-01-00109-1

$U (\mathfrak {g})$-finite locally analytic representations
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by P. Schneider, J. Teitelbaum and Dipendra Prasad

Represent. Theory 5 (2001), 111-128

DOI: https://doi.org/10.1090/S1088-4165-01-00109-1

Published electronically: May 18, 2001

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Abstract:

In this paper 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 $G=SL_2(\mathbb Q_p)$ in terms of such tensor products.

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