18 problems

In generalized Springer theory, consider cuspidal simple perverse sheaves with coefficients reduced modulo a prime $p$. Mautner's cleanness conjecture. These cuspidal perverse shea…

Invariant exterior-algebra conjecture. Except when $e=F_4(a_3)$ in type $F_4$ or $e=E_8(a_7)$ in type $E_8$, this algebra is an exterior algebra on the subspace

Cohomological boundedness conjecture. The coherent Springer sheaf $\mathcal{S}$ and each specialization $\mathcal{S}_v$ are honest coherent sheaves; equivalently, their cohomology…

Lagrangian conjecture. The subvariety $\Lambda_{{\mathscr{X}}}\subseteq {\mathscr{X}}$ is Lagrangian. This conjecture concerns the geometry of the moment-map fiber associated with…

Component-group–subspace conjecture. There exists an isomorphism

Birational-sheet degree conjecture. The degree of $\mu$ equals the order of the stabilizer:

Let $G$ be a connected reductive group, let $E$ be an elliptic curve, and let $\operatorname{Bun}_G^{ss,0}(E)$ denote the moduli stack of semistable degree-zero $G$-bundles on $E$.…

Let $G$ be a connected reductive group and let the Springer block be the block corresponding to the canonical Springer cuspidal datum $Spr=(H,\mathcal O_\mathtt{q}(H))$, where $H$…

Let $G$ be a connected reductive group. A $\mathtt{q}$-cuspidal datum is a $G$-conjugacy class of pairs $(L,C)$, where $L$ is an elliptic-pseudo Levi subgroup of $G$ and $C$ is a s…

Let $G$) be a connected reductive group, let $W$ be its Weyl group, and let $\rho$ denote the relevant quantum Springer constructions appearing in the isomorphism and theorems cite…

Let $G$ be the reductive group under consideration, let $\widetilde{G}$ denote its Grothendieck resolution, and let ${\mathcal N}$ be the nilpotent cone in its Lie algebra. Write…

Orbit-fibre duality conjecture. The resulting diagram has an orbit-fibre duality, generalizing the known bijections in the complete-flag case $\mu=(1,\ldots,1)$, such that the maps

Let $tau=rho{ u}^{(r)}$ denote the irreducible $mathfrak{S}r$-representation indexed by a partition $u$ of $r$, let $rho{ u}^{(r)}$ be the corresponding Specht representation, and…

Kraft's conjecture. The algebras $A(C_y\cap\mathfrak{t})$ and $H^*(\mathcal B_x)$ should be isomorphic both as $S_n$-modules and as graded $\mathbb{C}$-algebras.

Affine Weyl-group regular-representation conjecture. The semi-simplification of

Lehrer–Shoji conjecture. The occurrences of $\wedge^iV$ in $H^*({\mathcal B}_e)$ satisfy

Kazhdan–Lusztig basis conjecture. The matching basis is the Kazhdan–Lusztig basis in all degrees.

Khovanov's realization conjecture. $X_n$ could be realized as a subspace of $(S^2)^n$, with components homeomorphic to $(S^2)^{n/2}$ and indexed by noncrossing matchings.