Algebras of p-adic distributions and admissible representations

By P. Schneider, Jeremy Teitelbaum. 2002

304 citations recorded by the source corpus.

Let G be a compact, locally L-analytic group, where L is a finite extension of Qp. Let K be a discretely valued extension field of L. We study the algebra D(G,K) of K-valued locally analytic distributions on G, and apply our results to the locally analytic representation theory of G in vector spaces over K. Our objective is to lay a useful and powerful foundation for the further study of such representations. We show that the noncommutative, nonnoetherian ring D(G,K) "behaves" like the ring of functions on a rigid Stein space, and that (at least when G is Qp-analytic) it is a faithfully flat extension of its subring K\otimes Zp[[G]], where Zp[[G]] is the completed group ring of G. We use this point of view to describe an abelian subcategory of D(G,K) modules that we call coadmissible. We say that a locally analytic representation V of G is admissible if its strong dual is coadmissible as D(G,K)-module. For noncompact G, we say V is admissible if its strong dual is coadmissible as D(H,K) module for some compact open subgroup H. In this way we obtain an abelian category of admissible locally analytic representations. These methods allow us to answer a number of questions raised in our earlier papers on p-adic representations; for example we show the existence of analytic vectors in the admissible Banach space representations of G that we studied in "Banach space representations ...", Israel J. Math. 127, 359-380 (2002). Finally we construct a dimension theory for D(G,K), which behaves for coadmissible modules like a regular ring, and show that smooth admissible representations are zero dimensional.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/08a77145aafb80d7c5e0e085b5cc1e5808135c36

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Citations in the corpus

On irreducible representations of compact $p$-adic analytic groups (2011, methodology): “In a series of papers including [68, 69, 70, 71] Schneider and Teitelbaum study a class of rings that they call the distribution algebras D(G,K) of a p-adic analytic group G.”
The p-adic local Langlands correspondence for GL_2(Q_p) (2013, background): “Also, let Π an be the space of locally analytic vectors of Π, for which we refer the reader to [58] and [33].”
Cohomology of arithmetic families of (,Γ)-modules (2012, methodology): “Structure of the paper In §2, we carry out some preliminary arguments relating sheaves on relative annuli to coadmissible modules in the sense of Schneider and Teitelbaum [46].”
Coleman maps and the p-adic regulator (2010, background): “Note that H(0) and φ(B+rig,Qp) are both Fréchet– Stein algebras in the sense of [Schneider and Teitelbaum 2003] (by Theorem 5.”
On Jordan-H\"older series of some locally analytic representations (2010, background): “The dual space D(H) = D(H,K) = Can(H,K)′ is a topological K-algebra which has the structure of a FréchetStein algebra when H is compact [35].”
Duality for admissible locally analytic representations (2004, methodology): “mbeds naturally and fully into the topological category Mtop G . Finally Rep a K(G) is deﬁned to be the preimage of CG under the anti-equivalence (∗). For the sake of completeness we also recall from [ST2] Prop. 6.4 that: – Repa K(G) is an abelian category; kernel and image of a morphism in Rep a K(”
A Local-Global Compatibility Conjecture in the p-adic Langlands Programme for GL 2/Q (2006, background): “In [51] another topology on Ban is considered, which realizes it as a closed subspace of the space of locally analytic B-valued functions on GL2(Qp).”
Locally analytic vectors in representations of locally p-adic analytic groups (2004, methodology): “1 we present our alternative definition of the category of admissible locally analytic G-representations, and prove that it is equivalent to the definition presented in [27].”
The cohomology of locally analytic representations (2010, background): “Assuming H to be a locally L-analytic group we denote by CH the abelian category of all admissible locally analytic representations of H in the sense of [36], section 6.”
The emerging p-adic Langlands programme (2011, background): “In general one has B = 0, but if B is admissible as a representation of the compact group GL2(Zp) it is a major result due to Schneider and Teitelbaum (which holds in much greater generality) that the subspace B is never 0 and is even dense in B ([62]).”
On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms (2006, methodology): “If F is a coherent rigid analytic sheaf on T̂ , then the space of global sections Γ(T̂ ,F) is naturally a topological Fréchet module over C(T̂ , E), and the theory of quasi-Stein rigid analytic spaces shows that taking global sections induces an equivalence of categories between the category of cohe”
Relative P-adic Hodge Theory: Foundations (2013, unknown): “As such, it behaves much like a Fréchet-Stein algebra in the sense of Schneider and Teitelbaum [110]; in particular, it enjoys some cohomological properties more typical of Banach algebras than of general Fréchet algebras.”
Jacquet modules of locally analytic representations of p-adic reductive groups I. Construction and first properties (2006, methodology): “In the case of admissible locally analytic representations, this is proved in [20] (see also [8, cor.”