The extension conjecture for parabolically induced supersingular representations

Let $P,P'\subset G$ be standard parabolic subgroups with standard Levi subgroups $L\subset P$ and $L'\subset P'$, and let $\pi$ and $\pi'$ be supersingular representations of $L(F)$ and $L'(F)$ over $k_E$, respectively. Assume that $\operatorname{Ind}^{G(F)}_{P^-(F)}\pi$ and $\operatorname{Ind}^{G(F)}_{P'^-(F)}\pi'$ are irreducible. For $\alpha\in\Delta_L^\perp$, let $\pi^\alpha$ denote the twist of $\pi$ by the action of $s_\alpha$, and let $\omega:F^\times\to k_E^\times$ be the mod-$p$ cyclotomic character. The extension conjecture. The following assertions hold: (i) if $P'\not\subset P$ and $P\not\subset P'$, then

$$\operatorname{Ext}^1_{G(F)}\left(\operatorname{Ind}^{G(F)}_{P'^-(F)}\pi',\operatorname{Ind}^{G(F)}_{P^-(F)}\pi\right)=0;$$

(ii) if $F=\mathbb{Q}_p$, $P'=P$, and the central characters of $\pi$ and $\pi'$ are distinct, then

$$\dim_{k_E}\operatorname{Ext}^1_{G(\mathbb{Q}_p)}\left(\operatorname{Ind}^{G(\mathbb{Q}_p)}_{P^-(\mathbb{Q}_p)}\pi',\operatorname{Ind}^{G(\mathbb{Q}_p)}_{P^-(\mathbb{Q}_p)}\pi\right)=\operatorname{card}\left\{\alpha\in\Delta_L^\perp\mid\pi'\cong\pi^\alpha\otimes(\omega^{-1}\circ\alpha)\right\};$$

(iii) if $P'\subset P$, parabolic induction from $P^-$ induces an isomorphism

$$\operatorname{Ext}^1_{L(F)}\left(\operatorname{Ind}^{L(F)}_{(P'^-\cap L)(F)}\pi',\pi\right)\overset{\sim}{\longrightarrow}\operatorname{Ext}^1_{G(F)}\left(\operatorname{Ind}^{G(F)}_{P'^-(F)}\pi',\operatorname{Ind}^{G(F)}_{P^-(F)}\pi\right);$$

and (iv) if $P\subset P'$, induction from $P'^-$ induces an isomorphism

$$\operatorname{Ext}^1_{L'(F)}\left(\pi',\operatorname{Ind}^{L'(F)}_{(P^-\cap L')(F)}\pi\right)\overset{\sim}{\longrightarrow}\operatorname{Ext}^1_{G(F)}\left(\operatorname{Ind}^{G(F)}_{P'^-(F)}\pi',\operatorname{Ind}^{G(F)}_{P^-(F)}\pi\right).$$

The conjecture describes extensions between irreducible parabolic inductions from supersingular Levi representations and was suggested by Breuil for $G=\mathrm{GL}_n$. The source states that parts of the conjecture are proved in specified cases, but does not establish the full four-part statement.