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Sur une conjecture de Breuil-Herzig
Representation Theory · Submitted 25 May 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In particular, we obtain a natural characterization of the latter and we prove a conjecture of Breuil-Herzig.
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The abstract, sentence by sentence
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 $\Pi(\rho)^\mathrm{ord}$. 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 $\delta$-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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $G$ be a split $p$-adic reductive group with connected centre and simply connected derived subgroup. | background 1.00 | 0.05 | 0.02 | asserted 0.76 |
| s1 | We show that certain "chains" of principal series of $G$ do not exist and we establish several properties of the Breuil-Herzig construction $\Pi(\rho)^\mathrm{ord}$. | result 0.66 | 0.97 | 0.48 | asserted 1.00 |
| s2 | In particular, we obtain a natural characterization of the latter and we prove a conjecture of Breuil-Herzig. | result 0.91 | 0.96 | 0.73 | asserted 1.00 |
| s3 | In order to do so, we partially compute Emerton's $\delta$-functor $\mathrm{H^\bullet Ord}_P$ of derived ordinary parts with respect to a parabolic subgroup on a principal series. | method 0.98 | 0.93 | 0.04 | asserted 0.98 |
| s4 | 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. | contribution 0.71 | 0.95 | 0.14 | asserted 0.86 |
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Original abstract
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 $\Pi(\rho)^\mathrm{ord}$. 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 $\delta$-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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