## Towards the Locally Analytic Socle for GLn

**Speaker(s): ** Christophe Breuil

**Time: ** 00:00-00:00 April 29, 2013 - May 11, 2013

**Venue: ** Room 82J04, Jiayibing Building, BICMR

## Title: Towards the Locally Analytic Socle for GLn

**Organizer**: Ruochuan Liu

**Venue**: Room 82J04, Jiayibing Building, BICMR

**Speaker**: Christophe Breuil

**Time**: 4.29 - 5.11

**Week 1**: 11am-12am on Wednesday, Thursday and Friday.

**Week 2**: 11am-12am on Tuesday, Wednesday and Thursday.

**General Abstract**:

Let n be a positive integer. To any Hodge filtration with distinct Hodge-Tate weights on a diagonalizable generic n-diml representation of the Weil group of Qp, I will associate a semi-simple finite length locally analytic representation of GLn(Qp). I will show several properties of this representation. For instance, if it has a lattice invariant under GLn(Qp), then the Hodge filtration is weakly admissible. The restriction to diagonalizable representations of the Weil group and to the base field Qp are only for simplicity.

**Detailed schedule**:

**Lecture 1**

1.1 Introduction and motivation

1.2 Quick review of locally analytic representations

We give a quick review of locally analytic representations of p-adic analytic groups and related material (mainly due to Schneider and Teitelbaum).

**Lecture 2**

2.1 Quick review of Verma modules

We review the BGG category O^p_alg and some of its most interesting objects: generalized Verma modules.

2.2 The representations F_P^G(M,pi_P) (after Orlik and Strauch)

We explain important results due to Orlik and Strauch which allow to understand the topological constituents of some locally analytic parabolic inductions.

**Lecture 3**

3.1 More on the representations F_P^G(M,pi_P)

We state, and sometimes prove, some useful statements about the representations F_P^G(M,pi_P).

3.2 Examples for GL2(Qp) and GL3(Qp)

We give the explicit socle fi ltration of some locally analytic principal series of GL2(Qp) and GL3(Qp).

**Lecture 4**

4.1 Necessary conditions for integrality

We give necessary conditions for the representations F_P^G(M,pi_P) to admit an invariant lattice (in the case of locally analytic parabolic inductions, these conditions are due to Emerton).

4.2 De finition of the representations pi(D,h)

We defi ne some locally algebraic representations pi(D,h) of GLn(Qp) and recall a (special case of a) conjecture of Schneider and myself predicting when they should admit an invariant lattice.

**Lecture 5**

5.1 Some preliminaries

We give several preliminaries which will be used afterwards to defi ne some semi-simple locally analytic representations pi(D,h,Fil ) containing pi(D,h).

5.2 De finition of the representations pi(D,h,Fil )

To any Hodge filtration Fil of Hodge-Tate weights h on a diagonalizable Deligne-Fontaine module D, we associate a fi nite length semi-simple locally analytic representation pi(D,h,Fil ) of GLn(Qp).

**Lecture 6**

6.1 The link with weak admissibility

We prove that if pi(D,h,Fil ) has an invariant lattice, then the Hodge filtration Fil on D is weakly admissible.

6.2 Examples for GL3(Qp) and open questions

We finish with the description of a few representations pi(D,h,Fil ) (with Fil weakly admissible) in the case of GL3(Qp), and with a few questions for possible future developments.