112 problems

For $n\geq 0$, let $S_n$ be the symmetric group, let $\mathrm{Inv}_n$ be its set of involutions, and let $\mathbf{K}_n(w)=-$ mean that Kostant's problem for the simple highest weig…

Let $(i,a)\in I\times_2\mathbb Z$ and $w\in W$, and let $Q_{w\varpi_i,a}$ be the corresponding formal solution of the extended $QQ$-system. Let $\chi_{\ell}$ denote the $\ell$-char…

Let $V_1,V_2,V_3$ be objects of the shifted category $\mathcal{O}_{sh}$. Associativity conjecture. There exists an isomorphism … Thus, $k$-fold tensor products in…

Let $\mathfrak{g}$ be an arbitrary quasi-reductive Lie superalgebra and let $\mathfrak p$ be a parabolic subalgebra of $\mathfrak{g}$. Write $\mathcal{PI}^{\mathfrak p}$ for the fu…

Let $R$ be the deformation ring, let $\overline R$ be the corresponding deformation ring after replacing $e$ by $e+1$, and let $O^{\nu,\Delta}_{\mu,R}$ and…

Let $\mathfrak{k}\simeq \mathrm{sl}(2)$ be a subalgebra of a Lie algebra $\mathfrak{g}$, and let the truncated category of admissible $(\mathfrak{g},\mathfrak{k})$-modules consist…

Let $A$ be a local deformation algebra, let $\mathcal{J}$ be an open and bounded subset of $\widehat{\mathfrak{h}}^*$, and let $\lambda,\mu\in\mathcal{J}$ be of critical level. Wri…

Rouquier's conjecture. For an arbitrary multi charge, the multiplicity of a simple module in a standard module in $\mathcal{O}_{\boldsymbol{s}}(\ell,1,m)$ equals the corresponding…

Let $e$ be of standard Levi type in a classical Lie algebra, and let $\mathcal O(e;\mathfrak q)$ denote the corresponding category introduced above. A generalized Whittaker-module…

Let $A$ be the algebra governing the principal block of category $\mathcal{O}$, let $\Delta$ be the direct sum of the relevant Verma modules, and let $\mathtt{E}$ be the graded ext…

Let $\mathfrak{g}$ be a semi-simple complex finite-dimensional Lie algebra, let $W$ be its Weyl group with longest element $w_0$, and let $\mathfrak{t}(w)$ and $\mathfrak{i}(w)$ de…

Let $V_1$ be the two-dimensional fundamental representation of $obreakfrak{sl}2$, and consider the two categorifications of the $Uq(mathfrak{sl}2)$-action on $V1^{otimes n}$ const…

Arbitrary-charge Fock-space conjecture. For arbitrary $s_i$, the multiplicity of a simple module in a standard module in ${\mathcal O}$ equals the corresponding coefficient of the…

Shift-functor equivalence conjecture. For general $W$, the shift functor associated to $\zeta$ should give an equivalence whenever $h_\cdot$ and $h_\cdot+\tau$ define the same orde…

Rouquier's derived equivalence conjecture. For every $\tau\in{\mathfrak t}_{\mathbf Z}$, there should be an equivalence

Serre functor conjecture. The functor $S$ is right exact, and its left derived functor ${\mathcal L}S$ is a Serre functor. This is a proposed description of the Serre functor for c…

Let $\mathfrak p$ and $\mathfrak p'$ be parabolic subalgebras defining parabolic categories $\mathcal{O}_0(\mathfrak{sl}_n)^{\mathfrak p}$ and…

Let $S_q=S_q(n,n)$ be the $q$-Schur algebra, where $q=\exp(2\pi i c)$, and let $\mathcal{O}_c$ denote category $\mathcal{O}$ for the rational Cherednik algebra $H_c$. GGOR's equiva…

Duality compatibility conjecture. The two functors $\Phi$ and $\psi_*$ are isomorphic.

q-Schur algebra conjecture. The category ${\mathcal O}$ is equivalent to the category of finitely generated modules over the associated $q$-Schur algebra.

Projectivity criterion for the KZ image. One could conjecture that the semisimplicity assumption on ${\mathcal H}(W_H)$ can be replaced by the assumption that…

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 $\kappa$ be a decomposition of $n$, let $\mathcal{O}^{\kappa}$ be the corresponding parabolic category, and let $\mathcal{B}_{\kappa}$ be its Springer variety. For each wall, l…

Let $\kappa=(k_1,\dots,k_m)$ be a decomposition of $n$, let $\mathcal{O}^{\kappa}$ be the corresponding parabolic highest weight category for $\mathfrak{sl}_n$, and let…

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…