8 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…

Admissible-polarization conjecture. The character formula

The conjectural character formula. The character value is given by

Let $\mu$ be $p$-regular, let $x\in\Wa$ satisfy $x\pdot\mu$ dominant, and let $X_1^p$ denote the restricted region in the weight lattice. Let $\operatorname{LCF}$ denote Lusztig's…

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 $\widehat{\mathfrak{g}}$ be an affine Lie superalgebra, let $\varphi:\widehat{\mathfrak{g}}\to\widehat{\mathfrak{g}}$ be a compatible homomorphism, and let…

Let $\widehat{\mathfrak{g}}$ be an affine Lie superalgebra, let $L(\Lambda)$ be a partially integrable tame $\widehat{\mathfrak{g}}$-module, and let $L^\#$ be the relevant lattice…

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…