arXiv subjects

Julien Hauseux

Publications and source records attributed to Julien Hauseux.

8 recordsLinked to original sources

arXiv2021

p-adic etale cohomology of period domains

We compute the p-torsion and p-adic etale cohomologies with compact support of period domains over local fields in the case of basic isocrystals for quasi-split reductive groups. For the p-torsion case, we follow the method used by Orlik in his computations of the l-torsion etale cohomology using as a key new ingredient the computation of Ext groups between mod p generalized Steinberg representations of p-adic groups. For the p-adic case, we don't use Huber's definition of etale cohomology with compact support as Orlik did since it seems to give spaces that are much too big; instead we use continuous etale cohomology with compact support.

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arXiv2018

Functorial properties of generalised Steinberg representations

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 $G$ containing $P$. We study the functor $\mathrm{St}_Q^G$ taking a smooth $R$-representation $σ$ of $L$ which extends to a representation $\mathrm{e}_G(σ)$ of $G$ trivial on $U$ to the smooth $R$-representation $\mathrm{e}_G(σ) \otimes_R \mathrm{St}_Q^G(R)$ of $G$ where $\mathrm{St}_Q^G(R)$ is the generalised Steinberg representation.

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arXiv2018

On the exactness of ordinary parts over a local field of characteristic $p$

Let $G$ be a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$, $P$ be a parabolic subgroup of $G$, and $R$ be a commutative ring. When $R$ is artinian, $p$ is nilpotent in $R$, and $\mathrm{char}(F)=p$, we prove that the ordinary part functor $\mathrm{Ord}_P$ is exact on the category of admissible smooth $R$-representations of $G$. We derive some results on Yoneda extensions between admissible smooth $R$-representations of $G$.

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arXiv2017

Deformation rings and parabolic induction

We study deformations of smooth mod $p$ representations (and their duals) of a $p$-adic reductive group $G$. Under some mild genericity condition, we prove that parabolic induction with respect to a parabolic subgroup $P=LN$ defines an isomorphism between the universal deformation rings of a supersingular representation $\barσ$ of $L$ and of its parabolic induction $\barπ$. As a consequence, we show that every Banach lift of $\barπ$ is induced from a unique Banach lift of $\barσ$.

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arXiv2017

Parabolic induction and extensions

Let $G$ be a $p$-adic reductive group. We determine the extensions between admissible smooth mod $p$ representations of $G$ parabolically induced from supersingular representations of Levi subgroups of $G$, in terms of extensions between representations of Levi subgroups of $G$ and parabolic induction. This proves for the most part a conjecture formulated by the author in a previous article and gives some strong evidence for the remaining part. In order to do so, we use the derived functors of the left and right adjoints of the parabolic induction functor, both related to Emerton's $δ$-functor of derived ordinary parts. We compute the latter on parabolically induced representations of $G$ by pushing to their limits the methods initiated and expanded by the author in previous articles.

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arXiv2017

Complements sur les extensions entre series principales p-adiques et modulo p de G(F)

We complete the results of a previous article. Let $G$ be a split connected reductive group over a finite extension $F$ of $\mathbb{Q}_p$. When $F=\mathbb{Q}_p$, we determine the extensions between unitary continuous $p$-adic and smooth mod $p$ principal series of $G(\mathbb{Q}_p)$ without assuming the centre of $G$ connected nor the derived group of $G$ simply connected. This shows a new phenomenon: there may exist several non-isomorphic non-split extensions between two distinct principal series. We also complete the computations of self-extensions of a principal series in the non-generic cases when the centre of $G$ is connected. Finally, we determine the extensions of a principal series of $G(F)$ by an "ordinary" representation of $G(F)$ (i.e. parabolically induced from a special representation twisted by a character). In order to do so, we compute Emerton's $δ$-functor $\mathrm{H^\bullet Ord}_{B(F)}$ of derived ordinary parts with respect to a Borel subgroup on an ordinary representation of $G(F)$.

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arXiv2016

Sur une conjecture de Breuil-Herzig

Let $G$ be a split $p$-adic reductive group with connected centre and simply connected derived subgroup. We show that certain "chains" of principal series of $G$ do not exist and we establish several properties of the Breuil-Herzig construction $Π(ρ)^\mathrm{ord}$. In particular, we obtain a natural characterization of the latter and we prove a conjecture of Breuil-Herzig. In order to do so, we partially compute Emerton's $δ$-functor $\mathrm{H^\bullet Ord}_P$ of derived ordinary parts with respect to a parabolic subgroup on a principal series. We formulate a new conjecture on the extensions between smooth mod $p$ representations of $G$ parabolically induced from supersingular representations of Levi subgroups of $G$ and we prove it in the case of extensions by a principal series.

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arXiv2014

Extensions entre series principales p-adiques et modulo p de G(F)

Let $G$ be a split connected reductive group over a finite extension $F$ of $Q_p$. We determine the extensions between unitary continuous $p$-adic and smooth mod $p$ principal series of $G(F)$ in the generic case. In order to do so, we compute Emerton's delta-functor $\mathrm{H^\bullet Ord}_{B(F)}$ of derived ordinary parts with respect to a Borel subgroup on certain induced representations of $G(F)$ using a Bruhat filtration. These extensions come into play in the $p$-adic and mod $p$ Langlands programs.

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