13 problems

Let $(m,n)\in\mathbb{Z}\times\mathbb{N}$ be coprime. Let $\mathcal{H}_{m,n}$ and $\mathcal{H}'_{m,n}$ be the derived objects obtained from the ordinary and eccentric flag commuting…

Let $n\in\mathbb{N}$, let $(m,n)\in\mathbb{Z}\times\mathbb{N}$ be coprime, and let $d\geq 1$. For the equivariant derived category…

Single-relation conjecture. The coordinate rings satisfy

Let $\alpha=(K(a_1),\dots,K(a_d))$ be a pure-$K$ leading term datum, and let $k$ be any field. Consider the affine variety … defined by … and by the vanishing conditions $X_{ij}=0$…

Let $\mathfrak{m}$ and $\mathfrak{m}'$ be multisegments satisfying the condition referred to in the source as … , (1) $\operatorname{SA}(\mathfrak{m},\mathfrak{m}')$ and…

Let $\mathfrak{m}$ be a multisegment, let $V$ be the graded vector space determined by its supercuspidal support, and let $\mathfrak{C}_{\mathfrak{m}}$ be the corresponding irreduc…

Let $G$ be a reductive group with Lie algebra ${\mathfrak g}$, let ${\mathfrak t}$ be a Cartan subalgebra, let $W$ be the Weyl group, and let ${\mathfrak C}^d_G$ be the scheme of c…

Chevalley restriction conjecture. This morphism is an isomorphism. Equivalently, the categorical quotient ${\mathfrak C}^d_G\sslash G$ is reduced and normal. The source records thi…

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.

Centraliser invariant-field conjecture. For any $e\in{\mathfrak g}_0\cap\mathcal N$,

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…