Let F be a finite extension of ${\mathbb{Q}}_p$ and let q be the cardinality of its residue field. The Breuil-Schneider conjecture for $G=G{L}_n(F)$ [BS07] gives a necessary and sufficient condition for the existence of an invariant norm on $\rho \otimes \pi$, where ρ is an irreducible algebraic representation of G and π is an irreducible smooth representation of G over $\bar{F}$. The conjecture is still open, even when $n=2$, if π is a principal series representation. In this case, assuming π is unramified and $\rho ={\mathrm{Sym}}^k\otimes {\det }^m$, it had been verified by Breuil [Bre03b] and De Ieso [DI13] when $k<q$. We extend their work to the range $k<q^2/2$, imposing some technical conditions on π.

令F为的有限扩展${\mathbb{问}}_p$令q为其残差字段的基数。Breuil-Schneider猜想$G=G{\mathrm{大号}}_{ñ}（F）$ [BS07]给出了存在不变性准则的必要和充分条件 $\rho \otimes \pi$，其中ρ是一个不可约代数表示ģ和π是一个不可约平滑表示ģ过$\stackrel{〜}{F}$。猜想仍然是开放的，即使$ñ=\mathrm{2个}$，如果π是主序列表示。在这种情况下，假设π是无分支的，并且$\rho ={\mathrm{象征}}^{ķ}\otimes {\det }^{米}$，它已由Breuil [Bre03b]和De Ieso [DI13]进行了验证，当时 $ķ<q$。我们将他们的工作扩展到范围$ķ<q^{\mathrm{2个}}/\mathrm{2个}$，对π施加一些技术条件。