28 problems

Premet's conjecture. In characteristic zero, every finite $W$-algebra affords a one-dimensional representation.

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 $\boldsymbol{\boldsymbol{\text?}}$ 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 $G$ be the gauge group with complexified Lie algebra $\mathfrak{g}_{\mathbb C}$, let $\varrho:\mathfrak{su}(2)\to\mathfrak{g}$ be the Nahm-pole embedding, and let…

Let $G\cdot f=\operatorname{Ind}^\mathfrak{g}_\mathfrak{l}(L\cdot f_\mathfrak{l})$ be an induced nilpotent orbit, with good gradings $\Gamma$ on $\mathfrak{g}$ and…

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…

Generating-matrix conjecture. There exists a unique set of generators $w_{ij;k}$, $1\leq i,j\leq r$, and $0\leq k\leq\min\{p_i,p_j\}-1$, of $W(\mathfrak{g},f)$ for which

Let $\mathfrak{g}$ be a classical Lie algebra, let $\mathcal{W}$ be the associated finite W-algebra, and write … for its abelianization. An irreducible component of…

Let $k>0$, let $n$ be a positive integer, and set … Let $e\in\mathfrak g$ be a nilpotent element with Jordan type $(n+2k,n,n)$, let $U(\mathfrak g,e)$ be the associated finite $W$-…

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…

Finiteness conjecture. The constancy of $q_I$ on coherent families should hold without any assumption on $\Gamma$, and the entire set

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…

Let $\mathfrak{g}$ be a Lie algebra of Cartan type $A$, let $e\in\mathfrak{g}$ be nilpotent, and let $U(\mathfrak{g},e)$ be the corresponding finite $W$-algebra. Write … for the su…

Let $G$ be a reductive group, let $P$ be a parabolic subgroup, and let $e_{\mathcal L}$ be the nilpotent element associated with the Levi subgroup determined by $P$. Let…

Let $G$ be a reductive group, let $P$ be a parabolic subgroup, and let $G/P$ be the corresponding partial flag variety. Consider the space of based quasi-maps from $\mathbb{P}^1$ t…

Premet's conjecture. The map

Let $\mathfrak g$ be a simple complex Lie algebra, let $e\in\mathfrak g$ be nilpotent, and let $U(\mathfrak g,e)$ be the associated finite $W$-algebra. For a $U(\mathfrak g,e)$-mod…

Let $\mathfrak g$ be a simple complex Lie algebra of exceptional type, let $e\in\mathfrak g$ be nilpotent, and let $U(\mathfrak g,e)$ be the associated finite $W$-algebra. Premet's…

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…