13 problems

Let $\boldsymbol{\boldsymbol{\text?}}$ be a tensor product parameter with associated free abelian group $S^{\boldsymbol{\boldsymbol{\text?}}}(V_{\mathbb Z})$, and let…

Let $W_S$ be the Weyl subgroup associated to $\gf_S$, and let $W_S\setminus W$ be the corresponding coset with its Bruhat ordering. For $\sigma\leq\tau$ in this coset, let…

Consider a finite W-algebra associated to an ${\bf sl}_2$ embedding that is principal in $\gf_S\subseteq\gf$. Let $M(\Lambda)$ be a Verma module, let $\bar{\beta}$ be a position in…

Codimension-one ideal conjecture. Theorem $$ should hold for all nilpotent elements $e\in\mathfrak{g}$.

Let $\mathfrak{o}_N$ be the orthogonal Lie algebra with generators $F_{ij}=E_{ij}-E_{ji}$, and let $PfF_I$ denote the noncommutative pfaffian of the skew-symmetric submatrix…

Rank-one centralizer conjecture. The $e$-admissible pairs are equivalent to one another. In particular, the associated finite $W$-algebras are isomorphic.

Let $\frak g$ be a basic classical Lie superalgebra, let $\chi$ be regular, and let $W_\chi$ be the associated finite W-superalgebra. Let $\pi$ denote the relevant quotient or repr…

Let ${\frak {g}}$ be a Lie superalgebra with reductive even part ${\frak {g}}_{\bar 0}$, let $W_\chi$ be its finite W-superalgebra, and let $Gr_KW_\chi$ denote its Kazhdan-graded a…

Let $\mathbb{O}$ be a special nilpotent orbit, and let $A$ and $\mathcal W$ denote the groups and finite W-algebra associated with the orbit as in the paper. The three exceptional…

Premet's conjecture. Every W-algebra has a one-dimensional representation, equivalently a two-sided ideal of codimension $1$.

Premet's conjecture. The map

Let $ab$ be a $W_0$-orbit in $ab^*$, and let $M(\Lambda,\chi)$ and $L(\Lambda,\chi)$ be the standard and irreducible objects in the category $\mathcal O(\chi)$, while…

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…