6 problems

Let $G$ be a reductive group over a field $k$ of good characteristic, with Lie algebra ${\mathfrak g}$. Let ${\cal N}_1$ be the restricted nilpotent cone, let $e\in{\cal N}_1$, and…

Non-normality question. Is the dual nilpotent cone $\mathcal{N}^*$ not a normal variety?

Monotonicity conjecture. If $C_r(\mathcal{N}_n)$ is reducible, then $C_r(\mathcal{N}_{n+1})$ is reducible.

Parabolic component conjecture. The variety $V_P$ is an irreducible component of $C_r(\mathcal{N}_n)$ for all $n,r\ge 4$.

Irreducibility implication. If $C_r(\mathfrak{gl}_n)$ is irreducible, then $C_r(\mathcal{N}_n)$ is irreducible.

Let $G$ be a reductive algebraic group of type $A$ with Lie algebra $\mathfrak{g}$, let $B$ be a Borel subgroup with nilpotent radical $\mathfrak{u}$, and let $\mathcal{N}$ be the…