18 problems

Let $\mathcal{L}$ be a finite dimensional restricted Lie algebra over an algebraically closed field $\mathfrak{k}$ of characteristic $p>0$, and let $\mathcal{N}(\mathcal{L})$ denot…

Let $\mathfrak{g}$ be a semisimple Lie algebra with Weyl group $W$, and let $X$ be the corresponding nilpotent cone. For each irreducible representation $\chi$ of $W$, let…

Humphreys' support-variety conjecture. The map ${\mathcal H}$ coincides with Lusztig's bijection.

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…

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

Set $G=\operatorname{SL}_n(\mathbb C)$, let $T$ be the torus of diagonal matrices, let $B$ be the upper-triangular Borel subgroup, and let $U$ be its unipotent subgroup. Let…

Suppose that $G_{\mathbb{R}}$ is split modulo center, let $H_s$ be a maximal torus with $\Delta(G,H_s)=\Delta_{\mathbb{R}}(G,H_s)$, and set … Here $\mathcal{N}_{\theta}$ is the K-n…

Vanishing conjecture. For any dominant weight $\mu$,

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

Assume that $\theta$ is quasi-split. Let $\mathcal N_1$ be the relevant nilpotent cone, let $\widetilde{\mathcal N}_1$ be its resolution of singularities, and let … be the resoluti…

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, let $V_k(\mathfrak{g})$ be its simple affine vertex algebra at level $k$, and let…

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.

Analogous reducibility conjecture. The reducibility behaviors of $C_r(\mathfrak{gl}_n)$ and $C_r(\mathcal{N}_n)$ should be similar.

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…

Limit-symbol algorithm conjecture. The same limit-symbol algorithm computes stalks of intersection cohomology complexes on orbit closures in $\mathfrak{N}$.

Rational smoothness conjecture. The special pieces $cmathbb{S}_{\mu;\nu}$ of $cmathfrak{N}$ in any characteristic, and $cmathcal{S}^B_{\mu;\nu}$ and $cmathcal{S}^C_{\mu;\nu}$ in ch…