16 problems

Let $n\geq 2$, and let $V(T(\bar{v}))$ be the $1$-singular $\mathfrak{gl}(n)$-module associated with a tableau $T(\bar{v})$. Conjecture. If $V(T(\bar{v}))$ is irreducible, then the…

Let $\mathfrak{g}$ be a simple Lie algebra of type $A$, with Cartan subalgebra $\mathfrak{h}$, Borel subalgebra $\mathfrak{b}$, simple roots $\Pi$, and Weyl group $W$. Let…

Let $\mathcal{GT}_{T(v)}(n)$ be the full subcategory of Gelfand–Tsetlin modules admitting a generalized tableaux realization with respect to the Gelfand–Tsetlin subalgebra $\Gamma$…

Futorny–Ovsienko sharpness conjecture. There is a simple module $M$ for which equality holds in this inequality for some $z$ with trivial stabilizer. Futorny and Ovsienko proposed…

Let $m$ be a maximal ideal of the Gelfand–Tsetlin subalgebra $Gamma$, let $v=v_{m}$ be the corresponding point of $mathbb C^{mu}$ up to permutation of coordinates, and let…

Let $L$ be a $1$-singular tableau, meaning that it contains exactly one singular pair, and let ${\mathsf m}_{L}$ be the associated maximal ideal of the Gelfand–Tsetlin subalgebra.…

Classification conjecture. If $V$ is a tableaux Gelfand–Tsetlin $\mathfrak{gl}_n$-module, then $V$ is isomorphic to some module

Classification conjecture. If $V$ is an irreducible Gelfand–Tsetlin $\mathfrak{gl}_n$-module with Gelfand–Tsetlin multiplicities $1$, having a basis consisting of noncritical table…

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…

The Gelfand–Tsetlin character conjecture. For any singular $v$ of index $2$ and any two irreducible subquotients $V$ and $W$ of $V(T(v))$, one has $V\simeq W$ if and only if…

Let $v$ be singular of index $2$, let ${\mathsf m}_v$ be the corresponding maximal ideal of the Gelfand–Tsetlin subalgebra, and let $V(T(v))$ be the associated Gelfand–Tsetlin modu…

Subquotient realization conjecture. Both $N_1$ and $N_2$ should appear as subquotients of $V(T(\bar{v}))$.

Irreducibility conjecture. The condition

Realization conjecture. Even in that single case, the irreducible Gelfand–Tsetlin modules should be realized as subquotients of universal derivative tableaux modules.

Let $U_q(\mathfrak{gl}_N)$ be the quantum enveloping algebra and let $\Gamma_q$ be its Gelfand–Tsetlin subalgebra. For any $\mathbf{m}\in \operatorname{Specm}\Gamma_q$, the fiber o…

Let $\pi$ be a pyramid with row lengths $(p_1,\ldots,p_n)$, let $W(\pi)$ be its finite $W$-algebra, and let $\Gamma$ be the Gelfand–Tsetlin subalgebra. For…