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We investigate the homogeneous $2$-local representations of the twin group $T_n$ for all integers $n\geqslant 2$. A complete classification is obtained, yielding three distinct families of representations. We show that each of these…

We study a family of complex representations of the group GL(n,O), where O is the ring of integers of a non-archimedean local field F. These representations occur in the restriction of the Grassmann representation of GL(n,F) to its maximal…

Let $G$ be a $p$-adic reductive group and $R$ be a noetherian Jacobson $\mathbb{Z}[1/p]$-algebra. In this note, we show that every smooth irreducible $R$-linear representation of $G$ is admissible using the finiteness result of Dat, Helm,…

Let $F$ be a $p$-adic field, and $K$ a quadratic extension of $F$. Using Tadic's classification of the unitary dual of $GL(n,K)$, we give the list of irreducible unitary representations of this group distinguished by $GL(n,F)$, in terms of…

Let $G$ be the $F$-points of a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$ and $R$ be a commutative ring. Let $P=LU$ be a parabolic subgroup of $G$ and $Q$ be a parabolic subgroup of $G$…

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

Let O be a complete discrete valuation domain with finite residue field. In this paper we describe the irreducible representations of the groups Aut(M) for any finite O-module M of rank two. The main emphasis is on the interaction between…

To a finite quadratic module, that is, a finite abelian group D together with a non-singular quadratic form Q:D --> Q/Z, it is possible to associate a representation of either the modular group, SL(2,Z), or its metaplectic cover, Mp(2,Z),…

In all forms of the local Langlands program the abelian category of smooth representations of p-adic groups G in vector spaces over a field k plays a central role. Of particular interest are its finiteness properties. If the field k has…

In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside \'etale cohomology of certain algebraic varieties. Recently, a $p$-adic version of this theory started to emerge: there are $p$-adic…

Let $G$ be the unramified unitary group $U(2, 1)(E/F)$ defined over a non-archimedean local field $F$ of odd residue characteristic $p$, and $B$ be the standard Borel subgroup of $G$. In this note, we study the problem of the restriction of…

Without using the $p$-adic Langlands correspondence, we prove that for many finite length smooth representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ on $p$-torsion modules the $\mathrm{GL}_2(\mathbf{Q}_p)$-linear morphisms coincide with the…

Let F be a local non-archimedean field and let G be the group of F-valued points of a reductive algebraic group over F. In this paper we compute the Ext-groups of generalized Steinberg representations in the category of smooth…

We prove Breuil's conjecture concerning the reduction modulo $p$ of trianguline representations $V$ and of the representations $\Pi(V)$ of $\mathrm{GL}_2(\mathbf{Q}_p)$ associated to them by the $p$-adic Langlands correspondence. The main…

We study the representation theory of a generalized graded Hecke algebra associated to a complex reflection group of type G(r,1,n), defined by Ram and Shepler. We use a realization of this algebra in the corresponding symplectic reflection…

In the case of p-adic general linear groups, each irreducible representation is parabolically induced by a tensor product of irreducible representations supported by cuspidal lines. One gets in this way a parameterization of the irreducible…

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let's mention : (1) the control of the image of the Galois representation modulo $p$, (2) Hida's…

We classify the automorphic representations (over number fields) and the irreducible admissible representations (over local fields) of unitary groups which are not quasi-split, under the assumption that the same is known for quasi-split…

We propose in this paper an approach to Breuil's conjecture on a Langlands correspondence between $p$-adic Galois representations and representations of $p$-adic Lie groups in $p$-adic topological vector spaces. We suggest that Berthelot's…

Higher Deligne-Lusztig representations are virtual smooth representations of parahoric subgroups in a $p$-adic group. They are natural analogs of classical Deligne-Lusztig representations of reductive groups over finite fields. The most…