11 problems

Let $\mathbb H$ be the graded Hecke algebra, let $\Delta$ be its set of simple roots, and for $I\subset\Delta$ let $\widetilde{\mathbb H}_I$ be the corresponding graded Hecke algeb…

Let $F$ be a non-archimedean local field, let $\Delta\operatorname{SL}_2$ denote the diagonally embedded copy of $\operatorname{SL}_2$ in…

Let $F$ be a non-archimedean local field. Let $\pi_1$ and $\pi_2$ be Arthur type representations of $\operatorname{GL}_n(F)$ and $\operatorname{GL}_{n-1}(F)$, respectively. Say tha…

Breuil's conjecture. There is a natural $E$-linear bijection

Let $G_n=\mathrm{GL}_n(F)$, and let $\pi_M$ and $\pi_N$ be Arthur type representations of $G_{n+1}$ and $G_n$, respectively. For $k\geq 1$, write $\pi_M^{[k]}$ and…

Let $\Lambda$ be a level-one fundamental weight, let $b,c\in P$, let $m\in\mathbf Q$, and let $p>0$. Let $D_b$ and $D_c$ be thick Demazure modules, let $\mathring{\mathbb D}_b$ and…

Let $F$ be the underlying nonarchimedean local field, let $n$ be a positive integer, let $\pi_1$ be an irreducible generic representation of $GL(n+1,F)$, and let $\pi_2$ be an irre…

The Frobenius-extension conjecture.

Conjecture II. For all $\lambda,\mu\in\Gamma$ and all $n\geq 0$, the following equalities hold:

Let $F$ be a non-Archimedean local field, let $\pi_1$ and $\pi_2$ be irreducible admissible representations of ${\rm GL}_{n+1}(F)$ and ${\rm GL}_n(F)$, respectively, and let…

Let $k$ be a field of characteristic $p>2$, let $\lambda,\mu\vdash d$ be partitions, and write $D^{p\lambda}$ and $D^{p\mu}$ for the corresponding simple modules of symmetric group…