25 problems

Let $c$ be a parameter for which the Heckman–Opdam functor $F$ is an equivalence between the highest weight categories $\mathcal{O}_c$ and $\mathcal{O}_{\sigma(c)}$. Let…

Let $A_{\kappa}$ be a finite-dimensional algebra with $A_{\kappa}\text{-mod}\cong\mathcal{O}^{\kappa}$, and let $e\in A_{\kappa}$ be the maximal idempotent such that the left…

Let $\mathcal{O}^{n,n}$ be the parabolic highest weight category for $\mathfrak{sl}_{2n}$, and let $A_{n,n}$ be its finite-dimensional algebra model. Let $e\in A_{n,n}$ be the idem…

Let ${\bf G}$ be a connected reductive algebraic group defined over the finite field $\mathbb{F}_q$, and let $\mathscr{O}({\bf G})$ be its principal representation category over a…

Let $J(r)$ be the free Jordan algebra of rank $r$, and let $\mathcal{C}^{fin}$ be the category of finite-dimensional $\mathbb{Z}$-graded $\mathfrak{sl}_2(J(r))$-modules. For each d…

Let the matching blocks be as in the intersection-of-series block bijection conjecture, and let $\vec{\nu}_e$ and $\vec{\nu}_m$ be charges for the corresponding specializations. Le…

Let $\mathbb{G}$ be a generic finite reductive group, and let $e,m>0$ with corresponding $\Phi_e$- and $\Phi_m$-cuspidal pairs. Consider the intersection of the two Harish–Chandra…

Highest weight category conjecture. The category $\mathscr{O}({\bf G})$ is a highest weight category.

Let $P$ be the weight set, and for each $\la\in P$ let $\Delta_\la$, $\overline\nabla_\la^{\vee o}$, $\mathbb{P}_\la$, and $L_\la$ denote the corresponding standard, dual costandar…

Let $(W,P)$ and $(W',P')$ be parabolic Coxeter systems, and let $\boldsymbol{\beta}$ denote an isomorphism between closed subsets $\Sigma\subseteq W$ and $\Sigma'\subseteq W'$ that…

Highest weight conjecture. The category $\mathscr{O}({\bf G})$ is a highest weight category in the sense of Cline, Parshall and Scott.

Let $\widehat{\mathcal O}_{\overline\Lambda}^{{\operatorname{res}}}$ be a regular (in the sense of Feigin–Frenkel linkage) integral critical block. Let … be the full subcategory co…

Principal-block equivalence conjecture. The principal blocks $\mathcal B_0$ for all parameters $\zeta$ are equivalent as highest weight categories.

Let $\mathcal{P}_\ell(n)$ be the poset of $\ell$-tuples of diagrams, embedded in $\mathbb{Z}^{\mathbf{s}}$ as described, and let $\mathcal{O}_{\kappa,\mathbf{s}}(n)$ and…

Assume that $X_0$ is a singular Poisson variety, $X$ is a smooth symplectic variety, and $\rho:X\rightarrow X_0$ is a Poisson resolution. Suppose that $T_1\times T_2$ acts on $X$ a…

Let $\mathfrak{g}=\mathbf{g}\otimes\mathbb{C}[x,\xi]$, and let $\lambda,\mu\in\mathsf{P}_{\Delta}$ be incomparable integral dominant weights. Let $\overline{\imath}_{\lambda}$ and…

Let $L_0(n)$ be the Lie algebra of vector fields appearing in the paper's vector-fields examples, and let $\mathcal{I}(L_0(n))$ be its associated category of graded modules.…

Let $\hat{\mathbf{g}}$ be the affine Lie algebra associated with $\mathbf{g}$, and let $\mathcal{M}_{k,\lambda}$ be an integrable $\hat{\mathbf{g}}$-module of level $k$ and highest…

Let $\mathbf{g}$ be a finite-dimensional Lie algebra, and let $\mathcal{I}(\mathbf{g}\otimes\mathbb{C}[x])$ denote the category of graded finitely generated $\mathbf{g}$-locally fi…

Let $\mathcal{I}(\mathfrak{g})$ be the highest weight category associated with a graded Lie algebra $\mathfrak{g}$ admitting an anti-involution. Suppose properties $(f1)$--$(f3)$ h…

Let $\mathcal{O}$ be the category associated with the cyclotomic rational double affine Hecke algebra, and let $\mathbf{A}$ be the subcategory of an affine parabolic category…

Multiplicity invariance conjecture. These numbers depend only on the combinatorial type of $\mathcal{C}$.

Fix $e,\ell,m>0$ and let $\nu\in\mathbb{N}^{\ell}(m)$. Let $\mathcal{O}^{\nu}(n)$ be the affine category block decomposition introduced in the source, let $\mathbf{S}^{\nu}(n)$ be…

Let $q\in k^\times$, and let $p$ be the least natural number such that … Let $n,w$ be natural numbers with $n\geq w$. Let ${\cal Q}_p(n,w)$ be the quasi-hereditary subquotient of…

Let ${\cal D}_{A_\infty}(w,w)$ be the algebra defined in chapter 8, and let $k{\bf B}_{\rho,w}^{{\cal S}_q}$ denote a Rock block of a $q$-Schur algebra of weight $w$. Schur–Rock co…