23 problems

Polynomiality conjecture. For $d>0$, $\dim W^d(\lambda)^{\lambda-\mu}$ is a polynomial in $\lambda_1,\dots,\lambda_{{\rm rk}({\mathfrak g})}$, of degree $\mu^i d$ in the variable…

Let $\mathfrak g$ be a finite-dimensional simple Lie algebra, let $\lambda\in P^+$, and let $N\in\mathbb Z_{>0}$. Write $W_N(\lambda)$ for the graded truncated Weyl module for…

Cherednik–Orr conjecture. The PBW character satisfies

Let ${\mathfrak{g}}$ be a simple Lie algebra, let $\lambda=\lambda_1+\cdots+\lambda_n$ be a sum of dominant integral weights, and let $c_1,\ldots,c_n$ be pairwise distinct complex…

Let ${\mathfrak{g}}$ be a complex simple Lie algebra. For an $n$-tuple $\pi$ of polynomials defining a Weyl module $W(\pi)$, let $\lambda_\pi$ be its associated dominant weight; si…

Let ${\mathfrak g}={\mathfrak gl}_r$, let $\xi$ be a partition, let $W^2(\xi)$ be the two-dimensional Weyl module, and let $V_\xi$ be the…

Let $\lambda^1,\dots,\lambda^n$ be weights, let $W^1(\lambda^i)$ be the level-one Weyl modules, and let $W^1(\lambda^i)^{[k]}$ denote their level-$k$ constructions. Let $*$ denote…

Let ${\mathfrak g}$ be a simple Lie algebra, let $\lambda^1,\dots,\lambda^n$ be weights, let $V(\lambda^i)$ be the corresponding finite-dimensional ${\mathfrak g}$-modules, and let…

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…

Characteristic-free filtration conjecture. There is a characteristic-free isomorphism of $SL(E)$-modules

The inclusion conjecture for $\alpha_{n,s}$. The map $\alpha_{n,s}$ is an inclusion.

Let $W_{\mathbb C^d}(\{0\}_{n\omega})$ be the indicated local Weyl module for $sl_2$, and let $W_{\mathbb C^d}(\{0\}_{n\omega})^{n-2i}$ denote its weight space of weight $n-2i$. Hi…

Let ${\mathfrak g}$ be a simple Lie algebra with fundamental representations $V^{(i)}$, $i=1,\dots,r$, and highest weight vectors $v^{(i)}\in V^{(i)}$. Let…

Let $\mathfrak g$ be a finite-dimensional simple Lie algebra, let $\hat{\mathfrak g}$ be its affine Lie algebra, and let $\bpi$ be an $n$-tuple of polynomials parametrizing a finit…

Let $\mathfrak a$ and $\mathfrak b$ be such that $U_{\mathbb Z}(\mathfrak a)$ and $U_{\mathbb Z}(\mathfrak b)$ have been defined. Let $\mathbf a=(a_1,\dots,a_n)\in\mathbb F^n$, and…

Let $\mathfrak g$ be a finite-dimensional simple Lie algebra, let $i\in I$, and let $m\geq 0$. For each $\boldsymbol m\in\mathscr P_m$, let $T_i(\boldsymbol m)$ be the quotient of…

Let $\mathfrak{sl}_n$ be the special linear Lie algebra and let $A=\mathbb{C}[t]/(t^K)$ be a truncated current algebra, where $K$ is positive. Consider local Weyl modules for…

Let $G=\operatorname{SO}(2n+1,\mathbb{F})$ over a perfect field $\mathbb{F}$ of characteristic $2$, and let $V_\mathbb{F}(\lambda_k^\circ)$ be the Weyl module with submodules … whe…

Let $G=\operatorname{SO}(2n+1,\mathbb{F})$ over a field $\mathbb{F}$ of characteristic $2$, and let $V_\mathbb{F}(\lambda_k^\circ)$ be the Weyl module containing the submodules … w…

Let $\mathfrak{g}$ be a simple Lie algebra, let $\mathfrak{a}\subset\mathfrak{g}$ be a simple Levi subalgebra, and let $\lambda$ be a dominant integral weight for $\mathfrak{g}$ su…

Let $\lambda\in P_0^+$, where $P_0^+$ denotes the set of dominant weights relevant to the twisted current algebra, and let a graded local Weyl module and the associated graded modu…

BGG reciprocity conjecture. The projective object $P(\lambda,r)$ admits this Weyl-module filtration, and the multiplicity of each global Weyl module is the corresponding compositio…

The reducibility criterion. Let $F$ be a weight element of $\mathcal U^-$ and $\omega\in X^+(T)$. Then