8 problems

Lusztig's crystal-graph conjecture. The operation $*$ on irreducible components of the nilpotent varieties gives rise to Kashiwara's crystal graph.

Let $\mathfrak{N}_{2k}$ and $\mathfrak{N}_{2k+1}$ denote the varieties indexed by the corresponding nilpotency parameters. The author's and de Mello's conjecture. The varieties…

Let $C_i\in\operatorname{Comp}(\bm{d}_i)$ for $i=1,\dots,k$, let $V=V^1\oplus\cdots\oplus V^k$, and let $C\in\operatorname{Comp}(V)$. Let $\pi(C)$ denote the irreducible representa…

Let $C$ be an irreducible component of a type $A$ nilpotent variety, and let $\pi(C)$ be the corresponding irreducible representation. Call a representation square-irreducible when…

Let $C_i\in\operatorname{Comp}(\bm{d}_i)$ for $i=1,2$, and suppose that at least one of $C_1,C_2$ is rigid, meaning that it contains an open orbit. Let $\pi(C_i)$ be the correspond…

Let $C_i\in\operatorname{Comp}(\bm{d}_i)$ for $i=1,2$, and let $\pi(C_i)$ be the corresponding irreducible representations. Let $\operatorname{Ext}^1_\Pi$ denote the first extensio…

Let $k$ be a field, let $n\geq 4$ and $r\geq 4$, and write $\mathfrak{gl}_n$ for the Lie algebra of $n\times n$ matrices. Let $C_r(\mathcal{N}(\mathfrak{gl}_n))$ denote the variety…

Let $\mathfrak g$ be a complex semisimple Lie algebra, and consider the variety of commuting nilpotent pairs in $\mathfrak g$. Baranovsky's equidimensionality conjecture. This vari…