28 problems

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, let $M(\lambda)$ and $M(\mu)$ be Verma modules, and let $M(\mu)\hookrightarrow M(\lambda)$ be an embedding. Let $\rho$ denot…

Irreducibility for general $m$. For all $m\geq 3$, the equivalence of Theorem$\textup{}$ extends to ${\mathcal{H}_m}$ with multiple cocycle components…

Finite-dimensional quotient conjecture. The module $\mathbb{M}(r+\kappa,r,[\iota;n],\zeta)$ has a finite-dimensional irreducible quotient $L$ if and only if this condition holds. I…

Let a good Verma module mean a Verma module with a weight space decomposition, and let an exceptional Verma module be a module of the form…

Mixed tensor product conjecture. There is an isomorphism of $U(\mathfrak{gl}(m+1|n))$-modules

Let $M_A(\lambda,\eta)$ be a deformed standard Whittaker module over $A=\mathcal{O}(\mathbb{C}\gamma)_{(s)}$, and let $M(\lambda,\eta)$ denote its specialization at $s=0$. The Jant…

Let $(W,S)$ be a Weyl group, let $\Delta_e$ be the antidominant Verma module, and let $\mathbf{JI}$ denote the set of join-irreducible elements of $W$. For $x\in\mathbf{JI}$, write…

Let $(W,S)$ be a Weyl group with Bruhat order. For $w\in W$, let $\mathbf{JM}(w)$ be the distinguished set of Bruhat-maximal join-irreducible elements whose join is $w$, and let…

Let $\mathrm{Tr}^C_\zeta$ and $\mathrm{Tr}^H_m$ be the twisted traces on the quantum Coulomb and Higgs branch algebras, respectively. For Coulomb and Higgs branch operators…

Let $chi$ be semisimple and lie in a Cartan subalgebra $\mathfrak{h}$, and let $T$ be the corresponding torus. The $T$-action gives graded module categories…

Let $\mathfrak{g}$ be a simple Lie algebra and let $\mathscr{W}_\kappa$ be the affine $\mathscr{W}$-algebra associated to a principal nilpotent element of $\mathfrak{g}$ at level…

Dual-morphism conjecture. The linear map $\psi:M(W^*)\rightarrow M(V^*)$ associated to

Dual-morphism conjecture. The construction provides a morphism of $L$-modules $\psi:M(W^*)\rightarrow M(V^*)$ starting from a morphism $\varphi:M(V)\rightarrow M(W)$ in all degrees…

Kac–Rudakov conjecture. If $a,b,c,d\geq 0$ and $M(a,b,c,d)$ is degenerate, then

Rudakov's conjecture. There exists no morphism of higher degree, and Rudakov's list exhausts all the examples.

Let $\mathfrak g$ be a basic Lie superalgebra, let $\beta$ be an isotropic positive odd root, and let $\lambda$ be a weight that is $\beta$-atypical. Odd Verma-module homomorphism…

Singular vanishing conjecture. Then

Equality conjecture. The extension multiplicities satisfy

Gabber–Joseph conjecture. The dimensions of extensions of Verma modules are given by the $R$-polynomials; equivalently, the coefficients of $R_{x,y}(q)$ determine the dimensions…

Let $\rho$ be half the sum of the positive roots of $\mathfrak{gl}(n)$, and let $M(-\rho)$ be the irreducible Verma module of index $n-1$. Its Gelfand–Tsetlin degree is denoted by…

Rank-$r$ vertex-operator existence and uniqueness conjecture. Such a vertex operator exists uniquely with the stated parameters and relations. The conjecture would provide the miss…

Let $M(\boldsymbol a)$ be the Verma module, let ${}^{\mathbf R}M(\boldsymbol a)$ be its relevant $\mathbf R_T$-form, and let $\widehat W[\boldsymbol\lambda]$ be the operators index…

Let ${}_\mathbf A M(\boldsymbol a)$ be the $\mathbf A_T$-form of the Verma module generated by $|\boldsymbol a\rangle$, and let ${}_\mathbf A F(\boldsymbol a)$ be the $\mathbf A_T$…

Let $p\in\mathbb{N}$, and let $c,h,h_W,r$ be the parameters of the module $L^{\prime}(c,h,h_W)$. Assume … Reducibility criterion. The module $L^{\prime}(c,h,h_W)$ is reducible if a…

Let $V(c,\frac{1-r}{2},0)$ be the module in question, let $J^{\prime}(c,\frac{1-r}{2},0)$ denote the specified submodule, and let $u$ be a subsingular vector in…