Characters and growth of admissible representations of reductive p-adic groups

We use coefficient systems on the affine Bruhat-Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.Comment: Changes in the third version: the proof of Theorem 7.2 contained some mistakes. We repaired it, but to make it work we had to adjust Definition 7.1 a littl

Modular representations of p-adic groups

I will survey some results in the theory of modular representations of a reductive $p$-adic group, in positive characteristic $\ell \neq p$ and $\ell=p$

On the existence of admissible supersingular representations of pp-adic reductive groups

Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_p$, and that $C$ is a field of characteristic $p$. We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $C$.Comment: 58 pages, with an appendix by Sug Woo Shin. This replaces arXiv:1712.10142 and arXiv:1808.08255. v2: Minor changes following referee report; to appear in Forum Math. Sigm

Eigenvarieties and invariant norms: Towards p-adic Langlands for U(n)

We give a proof of the Breuil-Schneider conjecture in a large number of cases, which complement the indecomposable case, which we dealt with earlier in [Sor]. In some sense, only the Steinberg representation lies at the intersection of the two approaches. In this paper, we view the conjecture from a broader global perspective. If $U_{/F}$ is any definite unitary group, which is an inner form of \GL(n) over \K, we point out how the eigenvariety \X(K^p) parametrizes a global $p$-adic Langlands correspondence between certain $n$-dimensional $p$-adic semisimple representations $\rho$ of \Gal(\bar{\Q}|\K) (or what amounts to the same, pseudo-representations) and certain Banach-Hecke modules $\mathcal{B}$ with an admissible unitary action of U(F\otimes \Q_p), when $p$ splits. We express the locally regular-algebraic vectors of $\mathcal{B}$ in terms of the Breuil-Schneider representation of $\rho$. Upon completion, this produces a candidate for the $p$-adic local Langlands correspondence in this context. As an application, we give a weak form of local-global compatibility in the crystalline case, showing that the Banach space representations $B_{\xi,\zeta}$ of Schneider-Teitelbaum [ScTe] fit the picture as predicted. There is a compatible global mod $p$ (semisimple) Langlands correspondence parametrized by \X(K^p). We introduce a natural notion of refined Serre weights, and link them to the existence of crystalline lifts of prescribed Hodge type and Frobenius eigenvalues. At the end, we give a rough candidate for a local mod $p$ correspondence, formulate a local-global compatibility conjecture, and explain how it implies the conjectural Ihara lemma in [CHT].Comment: Comments and suggestions are very welcom

Modulo pp representations of reductive pp-adic groups: functorial properties

Let $F$ be a local field with residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. In a previous paper, Florian Herzig and the authors classified irreducible admissible $C$-representations of $G=\mathbf{G}(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. Here, for a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an irreducible admissible $C$-representation $\tau$ of $M$, we determine the lattice of subrepresentations of $\mathrm{Ind}_P^G \tau$ and we show that $\mathrm{Ind}_P^G \chi \tau$ is irreducible for a general unramified character $\chi$ of $M$. In the reverse direction, we compute the image by the two adjoints of $\mathrm{Ind}_P^G$ of an irreducible admissible representation $\pi$ of $G$. On the way, we prove that the right adjoint of $\mathrm{Ind}_P^G$ respects admissibility, hence coincides with Emerton's ordinary part functor $\mathrm{Ord}_{\overline{P}}^G$ on admissible representations.Comment: 39 page