10 problems

Let $B$ be a perfect crystal, let $\{w^{(k)}\}$ be a sequence of Weyl group elements, and let $\lambda=l\Lambda_0$ be the level-$l$ dominant integral weight for which the sequence…

Let $\widehat{\mathfrak{g}}$ be an affine Kac–Moody algebra, let $\Lambda=m\Lambda_0+\lambda$ be a dominant integral weight, and let $V(\Lambda)$ and $V(\lambda)$ denote the corres…

Let ${\mathfrak g}$ be the simple Lie algebra under consideration, let $\lambda$ be a dominant weight, and let $D(1,\lambda)$ and $W^1(\lambda)$ denote respectively the level-one D…

Let $G$ be a semisimple algebraic group with Borel subgroup $B$, Weyl group $W$, and dominant weights $\lambda,\mu\in P^+$. For $v,w\in W$, let $V_v(\lambda)$ and $V_w(\mu)$ be Dem…

Cancellation-free identity. There exists some $m\leq l\leq n$ such that

Saturation conjecture. For every $w\in W$ and every dominant $\lambda$, $\operatorname{char}(V_\lambda^w)$ is saturated.

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 $P$ be the weight lattice, let $D_c$ and $\mathbb D_c$ be the thick and local Demazure modules, and define … with $\deg X_{i,j}=1$. Let $\widetilde{\mathfrak b}^-$ be the relev…

Cherednik–Orr conjecture. Assume that $\la$ is an antidominant weight. Then

Fix a dominant integral weight $\Lambda$ and a sequence $(w^{(k)})$ in $W^{\mathrm{aff}}$ such that $l(w^{(k)}) \to \infty$. Let…