Extensions of mod $p$ representations of division algebras

Keisling, Andrew; Pentland, Dylan

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 45-74

Let $F$ be a local field over ${\mathbf{Q}}_p$ or ${\mathbf{F}}_p\left(\left(t\right)\right)$, and let $D$ be a central simple division algebra over $F$ of degree $d$. In the $p$-adic case, we assume $p>de+1$ where $e$ is the ramification degree over ${\mathbf{Q}}_p$; otherwise, we need only assume $p$ and $d$ are coprime. For the subgroup $I_1=1+{\varpi }_D{𝒪}_D$ of $D^{\times }$ we determine the structure of ${\mathrm{H}}^1(I_1,\pi )$ as a representation of $D^{\times }/I_1$ for an arbitrary smooth irreducible ${\overline{\mathbf{F}}}_p$-representation $\pi$ of $D^{\times }$. We use this to compute the group ${\mathrm{Ext}}_{D^{\times }}^1(\pi ,{\pi }^{\prime })$ for arbitrary smooth irreducible representations $\pi$ and ${\pi }^{\prime }$ of $D^{\times }$. In the $p$-adic case, via Poincaré duality we can compute the top cohomology groups and compute the highest degree extensions.

Soit $F$ un corps local sur ${\mathbf{Q}}_p$ ou ${\mathbf{F}}_p\left(\left(t\right)\right)$ et soit $D$ une algèbre à division centrale simple sur $F$ de degré $d$. Dans le cas $p$-adique, on suppose que $p>de+1$ où $e$ est le degré de ramification sur ${\mathbf{Q}}_p$ ; sinon on suppose seulement que $p$ et $d$ sont premiers entre eux. Pour le sous-groupe $I_1=1+{\varpi }_D{𝒪}_D$ de $D^{\times }$, on détermine la structure de ${\mathrm{H}}^1(I_1,\pi )$ en tant que représentation de $D^{\times }/I_1$ pour une ${\overline{\mathbf{F}}}_p$-représentation lisse irréductible quelconque $\pi$ de $D^{\times }$. Nous utilisons ceci pour calculer le groupe ${\mathrm{Ext}}_{D^{\times }}^1(\pi ,{\pi }^{\prime })$ pour des représentations lisses irréductibles quelconques $\pi$ et ${\pi }^{\prime }$ de $D^{\times }$. Dans le cas $p$-adique, via la dualité de Poincaré, nous pouvons calculer les groupes de cohomologie supérieurs et les extensions de degré maximal.

Classification : 22E50, 20J06
Keywords: cohomology, representations, extensions, division algebras

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