# Continuous representation theory of p-adic Lie groups

@inproceedings{Schneider2006ContinuousRT, title={Continuous representation theory of p-adic Lie groups}, author={Peter Schneider}, year={2006} }

In this paper we give an overview over the basic features of the continuous representation
theory of p-adic Lie groups as it has emerged during the last five years. Themain motivation
for developing such a theory is a possible extension of the local Langlands program to p-adic
Galois representations. This is still very much in its infancy. But in the last section we will
describe a first approximation to an extended Langlands functoriality principle for crystalline
Galois representations.

## 27 Citations

### Analytic vectors in continuous p-adic representations

- MathematicsCompositio Mathematica
- 2009

Abstract Given a compact p-adic Lie group G over a finite unramified extension L/ℚp let GL/ℚp be the product over all Galois conjugates of G. We construct an exact and faithful functor from…

### p-adic representations and arithmetic D-modules

- Mathematics
- 2021

The aim of the seminar is to give an introduction to the theory of representations of p-adic locally analytic groups, and the theory of Ù D-modules introduced in [2, 4]. We also aim to study…

### ON REDUCIBILITY OF p-ADIC PRINCIPAL SERIES REPRESENTATIONS OF p-ADIC GROUPS

- Mathematics
- 2015

We study the continuous principal series representations of split connected reductive p-adic groups over p-adic fields. We show that such representations are irreducible when the inducing character…

### INTERTWINING MAPS BETWEEN p-ADIC PRINCIPAL SERIES OF p-ADIC GROUPS

- Mathematics
- 2021

In this paper we study p-adic principal series representation of a padic group G as a module over the maximal compact subgroup G0. We show that there are no non-trivial G0-intertwining maps between…

### On reducibility of -adic principal series representations of -adic groups

- Mathematics
- 2016

We study the continuous principal series representations of split connected reductive padic groups over p-adic fields. We show that such representations are irreducible when the inducing character…

### Rigid vectors in $p$-adic principal series representations

- Mathematics
- 2022

In this paper we view pro-p Iwahori subgroups I as rigid analytic groups I for large enough p. This is done by endowing I with a natural p-valuation, and thereby generalizing results of Lazard for…

### Dimensions of some locally analytic representations

- Mathematics
- 2014

Let $G$ be the group of points of a split reductive group over a finite extension of ${\mathbb Q}_p$. In this paper, we compute the dimensions of certain classes of locally analytic…

### On unitary deformations of smooth modular representations

- Mathematics
- 2009

Let G be a lo/cally ℚp-analytic group and K a finite extension of ℚp with residue field k. Adapting a strategy of B. Mazur (cf. [Maz89]) we use deformation theory to study the possible liftings of a…

### LOCALLY ALGEBRAIC VECTORS IN THE BREUIL-HERZIG CONSTRUCTION

- Mathematics
- 2014

For a fairly general reductive group G/Qp , we explicitly compute the space of locally algebraic vectors in the Breuil-Herzig construction Π(ρ)ord, for a potentially semistable Borel-valued…

### Filtrations of smooth principal series and Iwasawa modules

- Mathematics
- 2017

Let $G$ be a reductive $p$-adic group. We consider the general question of whether the reducibility of an induced representation can be detected in a ``co-rank one" situation. For smooth…

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Let p be a prime number, L a finite extension of the field Qp of p-adic numbers, K a spherically complete extension field of L and G the group of L-rational points of a split reductive group over L.…

### Banach space representations and Iwasawa theory

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We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group…

### Modular Representations of p-adic Groups and of Affine Hecke Algebras

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Introduction The congruences between automorphic forms and their appli cations to number theory are a motivation to study the smooth representations of a reductive p-adic group G over an…

### First steps towards p-adic Langlands functoriality

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Abstract By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an…

### Banach-Hecke algebras and p-adic Galois Representations

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We take some initial steps towards illuminating the (hypothetical) $p$-adic local Langlands functoriality principle relating Galois representations of a $p$-adic field $L$ and admissible unitary…

### Duality for admissible locally analytic representations

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We study the problem of constructing a contragredient functor on the category of admissible locally analytic representations of a p-adic analytic group G. A naive contragredient does not exist. As a…

### Algebras of p-adic distributions and admissible representations

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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…