18 problems

Let $G_n$ be embedded in $G_{n+1}$ by $g\mapsto \left(\begin{array}{cc}g&\\&1\end{array}\right)$. Let $\pi$ and $\sigma$ be irreducible smooth representations of $G_{n+1}$ and…

Let $S$ and $T$ be smooth complex submanifolds of $X$ with intersection $S\cap T$, and let ${\cal E}$ and ${\cal F}$ be twisted holomorphic bundles on $S$ and $T$. The paper constr…

Let $\omega\in H^2(X,\mathcal{O}_X^*)$ define the $B$ field, let $i:S\hookrightarrow X$ and $j:T\hookrightarrow X$ be smooth complex submanifolds with $T\subseteq S$, and let…

Let $S$ be a smooth complex submanifold of $X$, let $i:S\hookrightarrow X$, and let ${\cal E}$ and ${\cal F}$ be twisted holomorphic bundles on $S$. The spectral sequence relating…

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 $X\subset\operatorname{Gr}(2,V_6)$ be the Fano variety of lines of a smooth cubic fourfold, and let $\lambda=(\lambda_1,\lambda_2,\lambda_3,\lambda_4)$ be a partition. The nota…

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…

The Frobenius-extension conjecture.

Conjecture IIa. Under the hypothesis of Conjecture II, for all $\lambda,\mu\in\Gamma$,

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, and let $\pi_1$ and $\pi_2$ be irreducible admissible representations of ${\rm SO}_{n+1}(F)$ and ${\rm SO}_n(F)$, respectively. Let…

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 $F$ be a non-Archimedean local field, let $obreakpi1$ be an irreducible generic representation of ${rm GL}{n+1}(F)$, and let $pi2$ be an irreducible generic representation of…

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…