3 problems

Let ${\sf S}_n$ be the symmetric group and let $\mathbb{F}$ be an algebraically closed field of characteristic $p\geq 0$. Let {\mathscr P}^{\text{\rm p-res}}_n denote the set o…

Let $F/\mathbb{Q}_p$ and $k/\mathbb{F}_p$ be finite extensions. Let $\mathbf{G}$ be a split connected reductive $F$-group with connected centre and simply connected derived subgrou…

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…