7 problems

Let $\mathfrak{g}$ be a complex reductive finite-dimensional Lie algebra, and let $M$ be a simple highest weight $\mathfrak{g}$-module. Kostant's problem asks whether the natural i…

Let $\mathfrak{g} = \oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$ be a direct sum of special linear Lie algebras, with positive roots $\Delta$ as above, and let…

Small-module realization conjecture. For a suitable choice of $e\in\mathcal{O}_{\mathfrak{k}}(I,J)$, every small $U_\chi(\mathfrak{g}_{\mathfrak{k}})$-module appears as a compositi…

Type A irreducibility conjecture. All associated varieties $V(L_w)$ are irreducible.

Let $I$ be an index set, let $\mathcal{F}$ be the generalized quantum algebra defined by the source, and let $\mathcal{P}$ be its weight lattice. For each $\Lambda\in\mathcal{P}^+$…

Let $\lambda'$ be an integral weight, and suppose that $L(\lambda)$ is a subquotient of $\mathcal D_{\alpha_i}L(\lambda')$. Let $JH(\lambda)$ denote the relevant Jordan–Hölder cons…

Let $ab ab$ be a $W_0$-orbit in $ab^*$, and choose $ab 'tinab$ such that … for every $ab\in\Phi_0^+$. Let $L(\Lambda,e)$ be the irreducible highest weight $U(\mathfrak g,e)$-module…