200 problems

Let $\mathfrak{g}$ be the Lie algebra of an exceptional simple algebraic group, let $\0$ be a special nilpotent orbit, and let ${\mathcal P}(\0)$ be its special piece. Let $H$ be t…

Let $G_\mathbb{F}$ be the reductive group over the field $\mathbb{F}$ considered in the paper, and let $T$ be an indecomposable tilting $G_\mathbb{F}$-module. The support variety o…

Jiang Conjecture. The following hold: (1) if a partition $\underline p$ is not related to $\eta_{\widehat{\mathfrak g}_n,\mathfrak g_n}(\underline p(\psi))$, then…

Lusztig's special pieces conjecture. Every special piece ${\mathcal P}(\0)$ is rationally smooth.

Weak Arthur packet decomposition conjecture. There is a set $\mathcal T$ and an appropriate notion of Arthur packet such that

Let $G_{d53d}$ be a connected reductive complex Lie group, let $G$ be a real form of $G_{d53d}$, let $\check G$ be its Langlands dual group, and let $\check{\mathfrak g}$ be the Li…

Vogan's quantization conjecture. For each such cover $\widetilde{\mathcal{O}}$, there is a canonically defined completely prime primitive ideal $I_0(\widetilde{\mathcal{O}})$ in th…

Let $R=R(\eta,\gamma)$ be a dominant sequence of rectangles, and let $R'$ be obtained from $R$ by replacing a subsequence … by … where $|\alpha|=|\beta|$ and $\alpha^+$ dominates…

Let $L_k(\mathfrak g)$ be a non-admissible simple affine vertex algebra associated with a complex finite-dimensional simple Lie algebra $\mathfrak g$, and let…

Let $G$ be a complex reductive Lie group, and let $X=G\cdot\lambda=G/G(\lambda)$ be a nilpotent coadjoint orbit. Suppose an irreducible $G$-equivariant local system on $X$ is given…

Let $F$ be either ${\mathbb R}$ or a finite extension of ${\mathbb Q}_p$, let $G$ be a finite central extension of the group of $F$-points of a connected reductive algebraic group…

Achar–Sommers' conjecture. The maximal element in this expression is $h^{\vee}$, the semisimple element of a Jacobson–Morozov triple of $\mathcal{O}^{\vee}$. The paper states that…

Let $G$ be a reductive group with nilpotent cone $\mathfrak N$, let ${\mathfrak O}$ be a $G$-orbit in $\mathfrak N$, and let $\Omega$ be the corresponding two-sided Kazhdan–Lusztig…

Let ${\mathcal O}$ be a nilpotent orbit with associated parameter $\lambda_{\mathcal O}$, and let $R_{+0}$ and $R_{+1}$ be the positive roots of gradation $0$ and $1$, respectively…

Let ${\mathcal O}$ be a nilpotent orbit and let $d_{\mathcal O}$ be its distinguished involution. Let $T(d_{\mathcal O}\cdot 0)$ be the corresponding tilting module and let…

Let ${\mathcal O}$ be a nilpotent orbit, let $a({\mathcal O})=\frac12\operatorname{codim}_{\mathcal N}{\mathcal O}$, and let $j:{\mathcal O}\to {\mathcal N}$ be the inclusion of th…

Let $\lambda$ range over dominant weights, and let $({\mathcal O}_\lambda,\rho_\lambda)$ be the nilpotent orbit and irreducible representation of…

Let ${\mathcal O}_\lambda$ be the nilpotent orbit whose closure is the support of $C(\lambda)$, and let $j_{{\mathcal O}_\lambda}:{\mathcal O}_\lambda\hookrightarrow {\mathcal N}$…

Let $G$ be a simple simply connected algebraic group, let ${\mathcal N}$ be its nilpotent cone, and let $K_{\mathcal G}({\mathcal N})$ have canonical basis elements $C(\lambda)$ in…

Let $G$ be a simple simply connected algebraic group, let ${\mathcal N}$ be its nilpotent cone, and let ${\mathcal G}=G\times {\mathbb C}^*$ act on ${\mathcal N}$ as above. Let…

Let $G$ be a connected reductive algebraic group over ${\mathbb{C}}$ with simply-connected derived group, let $\tilde{W}_a$ be its extended affine Weyl group, and let…

Characteristic-free filtration conjecture. There is a characteristic-free isomorphism of $SL(E)$-modules

Let ${\mathfrak g}={\mathfrak{sl}}_n$ and let ${\mathcal V}$ be an orbital variety, that is, an irreducible component of ${\mathcal O}\cap{\mathfrak n}$ for a nilpotent orbit…

Namikawa's conjecture. For any two symplectic resolutions $f_i:X_i\to W$, $i=1,2$, there are deformations

Union conjecture. In ${\mathfrak s}{\mathfrak l}_n$, the closure $\overline{\mathcal U}$ is a union of orbital varieties.