21 problems

Irreducibility conjecture. The intersection in codimension one of two irreducible components of a Springer fiber of type $A$ is irreducible.

For a partition $\lambda\vdash n$, let $O_\lambda$ be the nilpotent conjugacy class in $\mathfrak{gl}_n$ with Jordan block sizes given by $\lambda$, and let $\overline{O}_\lambda$…

Let Springer fibers be associated with two-row Young shapes, and let their irreducible components be described by the corresponding tableau and cup-diagram data. An iterated…

Let $\mathcal{B}_{\boldsymbol{\rho}}$ be the component of the two-row Springer fiber indexed by a standard tableau $\boldsymbol{\rho}$ of shape $(\rho,\rho)$, with cup diagram…

Let $\mathcal{B}_{\boldsymbol{\rho}}$ be the component of the two-row Springer fiber indexed by a standard tableau $\boldsymbol{\rho}$ of shape $(\rho,\rho)$. Let…

Let $n$ be a positive integer, let $\boldsymbol{\rho}$ be a standard Young tableau of shape $(n-\rho,\rho)$, and let $\mathcal{I}_{\boldsymbol{\rho}}$ be the associated ideal. Prim…

Let $\fg$ be a classical Lie algebra of type $B$, $C$, or $D$, with dual Lie algebra $\fg^\vee$, and let $e^\vee\in\fg^\vee$ be nilpotent with partition $\lambda^\vee$. Let…

Canonical-quotient conjecture. The $\bar{A}(e)$-sets $\Irr(\spr)/K_e$ and $Y$ are isomorphic.

Let $e$ be a nilpotent element in a classical Lie algebra, let $\operatorname{Fix}$ denote the fixed-point variety associated with the Springer fiber, and let $S_e$ be the specifie…

Let $(e,h,f)$ be an $\mathfrak{sl}_2$ triple in $\mathfrak{g}$, let $T_e$ be the torus with Lie algebra $\mathbb{C}h$, and let $\operatorname{Spr}$ be the Springer fiber. Set…

Connectedness and equidimensionality conjecture. The fiber ${{}^{\textsf{e}}\hskip-2pt\pi}_{\hskip-2pt\lambda[q]}^{-1}(Y,u)$ is connected, and all of its irreducible components hav…

Xi's fixed-point Springer-fiber conjecture. There is a ring isomorphism

Let $G$ be the symplectic group in characteristic $2$, let $N$ be the nilpotent element under consideration, let $\mathcal{B}$ be the flag variety of $G$, let $\mathcal{B}_N$ be th…

Let $\mathbf{k}$ be the ground field, and let $N\in\operatorname{Lie}(SO_{2n+1})(\mathbf{k})$ and $N'\in\operatorname{Lie}(Sp_{2n})(\mathbf{k})$ lie in special nilpotent orbits rel…

Let $G$ be a simple algebraic group over an algebraically closed field, let $\mathfrak{g}=\operatorname{Lie}G$, and let $N\in\mathfrak{g}$ lie in a special nilpotent orbit. Let…

Let the topological models introduced by Khovanov for Springer fibers of type $A$ corresponding to nilpotent endomorphisms with two equally sized Jordan blocks be given. Khovanov's…

The algebras $A_{n,k}$ are graded algebras arising from diagrammatic categories related to category $\mathcal{O}$. They are connected with the cohomology rings of closed attracting…

Let $\mathcal{S}_{n-k,k}$ be the Slodowy slice, let $Y$ be the corresponding Springer fiber, and let $A$ range over all irreducible components of $Y$. Write…

Let $Y$ be the two-block Springer fiber, and let $\mathcal{S}$ and $\mathcal{S}'$ be standard tableaux indexing Springer fiber components $\mathcal{C}_{\mathcal{S}}$ and…

Let $\mathcal{C}$ be the heart of the $t$-structure on coherent sheaves on the smooth compactification $Z_n$, and let $A$ be the corresponding highest-weight algebra. Coherent-shea…

Let $N$ be a nilpotent endomorphism of $\mathbb{C}^n$ with two Jordan blocks, and let the closures of cells in the Biaynicki-Birula paving of its Springer fiber be given. Fung's co…