19 problems

Let $({\rm G},{\rm G}')$ be an irreducible dual pair in ${\rm Sp}(W)$ with ${\rm rk}({\rm G})\leq {\rm rk}({\rm G}')$. Let ${\rm Chc}^{*}$ be the map transferring invariant distrib…

Let $\mathfrak g=\mathrm D_{n+1}^{(2)}$, let $m$ be a nonnegative integer, and set $\Lambda=2m\Lambda_0$. Define … Let $V(\Lambda)$ be the integrable highest-weight representation…

Let $u$ be a positive integer and set … Let $y$ be an automorphism of the root lattice $\widehat{Q}$ such that all roots $\gamma_i=y((u-1)\delta_{i0}K+\alpha_i^{\vee})$ are positiv…

Character formula conjecture. The displayed formula holds for all symmetrizable Kac-Moody superalgebras. This extends the corresponding character formula known for symmetrizable Ka…

Character decomposition conjecture.

Let $\mathfrak{g}$ be a simple Lie algebra of type $D_4$, $E_6$, $E_7$, or $E_8$. Let $\Lambda$ satisfy conditions $(i)$ and $(ii)$ of the character-formula theorem, and let…

Conjecture. For $|m-n|$ even, one has

Let $W_{k,i}$ be the virtual subspaces associated with level $k$ standard modules for $A_2^{(2)}$, and let their characters be denoted by $0chi_{W_{k,i}}'(x,q)$. Recursion conjectu…

Admissible-polarization conjecture. The character formula

Let $\mathfrak{g}$ be one of the simply laced Lie algebras $D_4$, $E_6$, $E_7$, or $E_8$, and let $\Lambda$ be a weight of $\widehat{\mathfrak{g}}$ of negative integer level satisf…

The conjectural character formula. The character value is given by

McGovern's conjecture. For each nilpotent orbit $\mathcal{O}$, there are a fixed character $\mu$, rational numbers $c_w$, and a subset $W_{\mathcal{O}}$ of $W_G$ such that

Let $G_2$ be the rank-two simple Lie algebra, let $z_1$ and $z_2$ denote its fundamental characters, and let $t_1,t_2$ be the variables recording highest weights. The character gen…

Let $\mathfrak{g}=\operatorname{osp}(n+2\mid n)$ or $D(2,1;a)$, let $L(\Lambda)$ be an integrable highest weight affine module satisfying the conditions stated for the normalized c…

Let $chat{\mathfrak{g}}$ be an affine superalgebra, let $L(\Lambda)$ be a tame integrable highest weight module, let $T$ be a set of pairwise orthogonal isotropic roots as in the d…

Bernstein–Leites conjecture. The formula $$ works for all dominant weights.

Let $\mathbb{F}$ be the generalized quantum algebra associated with the Cartan datum, let $\mathcal{P}$ be its weight lattice, and let $\mathcal{O}^{\mathcal{P}}(\mathbb{F})$ be th…

Suppose that $\mathfrak{g}$ is of type $B_n$, $C_n$ or $D_{n+1}$ and that $\lambda\in P^+$ satisfies $\lambda(h_i)=0$ for $i\ge n$. Let $\Psi_\lambda$ be the root subset defined fr…

Let $\mathfrak{g}$ be of type $B_n$, $C_n$ or $D_{n+1}$, and let $\lambda\in P^+$ satisfy $\lambda(h_i)=0$ for $i\ge n$. Define $i_\lambda=\max\{i\in I:\lambda(h_i)>0\}$ and, for…