17 problems

Let $\mathsf{C}$ denote the extended vertex operator algebra obtained as the order-$2$ simple current extension of $\mathsf{A}_1\brac{5,2}$. Let $\mathfrak{g}_2$ be the simple Lie…

Let ${\mathsf{H}}_c$ be the rational Cherednik algebra of type ${\mathfrak{gl}}_n$ with parameter $c\in\mathbb{C}$, and let ${\mathsf{e}}$ be the symmetrizer idempotent for $S_n$.…

Let $G$ be a connected reductive group, let $G({\mathbf O})$ act on the affine Grassmannian $\operatorname{Gr}_G$, and let $\mathbb G_m$ act by loop rotation. Write $\check G$ for…

Let $\mathfrak g$ be a simple Lie algebra, let $f=f_1+f_2$ be a decomposition into nilpotent elements, and suppose … For $i=1,2$, let $\mathfrak g_i$ be the centralizer of…

Let $V$ be an admissible affine vertex algebra, and let $\mathcal{O}$ be the unique nilpotent orbit that is open in the associated variety of $V$. Quantum Hamiltonian reduction ass…

Let $\widehat{\mathfrak g}^{\rm tw}$ be the twisted affine algebra, let $\mathcal O$ be its category of modules, and let $H_j$ denote the homology of the twisted quantum Hamiltonia…

Let $\mathfrak g$ be the underlying Lie superalgebra, let $V^k(\mathfrak g)$ be an affine vertex algebra at level $k$, and let $L(\widehat\Lambda)$ be an admissible highest weight…

Let $\lambda=(\lambda_1\leq\lambda_2\leq\dots\leq\lambda_n)\vdash N$, and set $\widehat{\lambda}_i=(1,\dots,1,\lambda_i)\vdash N_i$ with $N_i=N-\sum_{j>i}\lambda_j$. Write…

Spectral-flow conjecture. For every $\lambda\in\mathrm{Pr}_{\mathbb{Z}}^k$ and $0\leq\theta\leq\bar q$, there exists $\mu\in\mathfrak{h}^*$ such that

Let ${\mathfrak g}$ be a simple Lie algebra, let $({\mathfrak a}_1,\ldots,{\mathfrak a}_L)$ be ordered subalgebras whose direct sum is a subalgebra of ${\mathfrak g}$, and let…

Master chiral algebra conjecture. These objects can be given the structure of a one-parameter vertex superalgebra; for generic $\phi$, $A[\mathfrak g,\phi,f,f']$ is simple and exte…

For integers $p\geq 4$, let $\mathcal B_p$ denote the algebras introduced by T. Creutzig, D. Ridout and S. Wood. Creutzig–Ridout–Wood conjecture. Each algebra $\mathcal B_p$ can be…

Let $n,m$ be positive integers and let $p$ satisfy … Let $\mathcal{W}_{n,m}$ be the family of W-algebras defined by quantum Hamiltonian reduction, let…

Let a bow variety be described as a Hamiltonian reduction of a finite-dimensional symplectic manifold, and assume that the associated dimension data satisfy the balanced condition,…

Type A reduction-by-stages conjecture. The intermediate algebra produced by quantum Hamiltonian reduction is isomorphic to the W-algebra $U(\mathfrak{g},e_2)$.

Reduced W-algebra identification conjecture. The reduced algebra is isomorphic to the W-algebra $U(\mathfrak{g},e_2)$:

Let $\mathfrak{g}$ be a semisimple Lie algebra of type $A$, and let $e$ and $e'$ be nilpotent elements such that $e'$ covers $e$ in the dominance ordering. Let…