U(g)-finite locally analytic representations

By P. Schneider, Jeremy Teitelbaum. 2000

88 citations recorded by the source corpus.

In this paper we continue the study of locally analytic representations of a $p$-adic Lie group $G$ in vector spaces over a spherically complete non-archimedean field $K$, building on the algebraic approach to such representations introduced in our paper "Locally analytic distributions and p-adic representation theory, with applications to GL_2." In that paper we associated to a representation $V$ a module $M$ over the ring $D(G,K)$ of locally analytic distributions on $G$ and described an admissibility condition on $V$ in terms of algebraic properties of $M$. In this paper we determine the relationship between our admissibility condition on locally analytic modules and the traditional admissibility of Langlands theory. We then analyze the class of locally analytic representations with the property that their associated modules are annihilated by an ideal of finite codimension in the universal enveloping algebra of G, showing under some hypotheses on G that they are sums of representations of the form $X\otimes Y$, with X finite dimensional and Y smooth. The irreducible representations of this type are obtained when X and Y are irreducible. We conclude by analyzing the reducible members of the locally analytic principal series of $SL_2(\Qp)$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cd2e675ec0647760f4d0a71df3f6e6283f9f5391

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Citations in the corpus

The emerging p-adic Langlands programme (2011, background): “Strangely, the terminology “p-adic Langlands correspondence/programme” started to spread (at least in the author’s memory) only shortly after preprints of [6], [7], [8], [9], [14], [19], [24], [53], [59], [60], [61], [70] were available (that is, around 2004), although of course p-adic consideration”
Locally analytic vectors in representations of locally p-adic analytic groups (2004, background): “These connections have already been treated in [24]; our goal is to recall some points from this reference, and to present some minor additions to its discussion.”
On irreducible representations of compact $p$-adic analytic groups (2011, methodology): “See Prasad’s appendix in [68] for parallel results for distribution algebras.”
Algebras of p-adic distributions and admissible representations (2002, background): “In a series of earlier papers, ([ ST1 -4]) we began a systematic study of locally analytic representations of a locally L-analytic group G, where L ⊆ Cp is a finite extension of Qp. Such a representation is given by a continuous action of G on a locally convex topological vector space V over a spher”
Jacquet modules of locally analytic representations of p-adic reductive groups I. Construction and first properties (2006, methodology): “We refer the reader to the papers [16, 17, 18, 20] of Schneider and Teitelbaum for the foundations of locally analytic representation theory.”