17 problems

Let $V$ be a ${\mathscr{A}}_{n,\lambda,\nu}$-module, and let $F_i$ be the functor defined in the paper. Write $H^r({\mathscr{C}}^\bullet(V))$ for the $r$-th cohomology of the compl…

Gamma annihilator conjecture. Then

Birational Calogero-Moser conjecture. For every $c\in {\sf C}$, the variety $\operatorname{Spec}{\sf Z}_{0,c}$ is birationally isomorphic to $V/\Gamma$ as a Poisson variety.

Gelfand-Kirillov conjecture. For every $(t,c)\in\mathbb{C}^*\oplus {\sf C}$, there is an isomorphism between ${\sf Q}\#\Gamma$ and the Goldie ring of fractions of…

Gamma spherical Harish-Chandra conjecture. There is a filtration-preserving algebra isomorphism

Let $\Gamma_1\subset \operatorname{SL}_2(\mathbb{C})$, let $\Gamma_n=\mathfrak{S}_n\ltimes\Gamma_1^n$, and let $Q$ be the affine Dynkin diagram corresponding to $\Gamma_1$. Let…

Null-vector characterization conjecture. Each ideal of $H_{1,\eta}(A_{n-1})$ is the set of null-vectors of the degenerate bilinear form $B_{sp}$ for some $\varkappa$-trace $sp$ on…

Bezrukavnikov–Okounkov's conjecture. The spaces $\operatorname{gr}_iK_0(\operatorname{Rep}\Gamma_n)_{\mathbb C}$ should be eigenspaces of the residue $C_H$.

Etingof's refined representation-count conjecture. The number of isomorphism classes of finite-dimensional irreducible representations of $H_{c,k}(\Gamma_n)$ is

Etingof's refined filtration conjecture. For either filtration, there exists an isomorphism of vector spaces

Etingof's representation-count conjecture. The number of isomorphism classes of finite-dimensional irreducible representations of $H_{c,k}(\Gamma_n)$ is

Etingof's irrational-parameter filtration conjecture. For either filtration, there exists an isomorphism of vector spaces

Etingof's aspherical-locus conjecture. The aspherical locus is the union of the hyperplanes $E_{m,N}$ for $1\le N\le m-1$ and the hyperplanes $H_{\alpha,m,N}$ satisfying the stated…

Etingof's simplicity conjecture. The algebra $H_{c,k}(\Gamma_n)$ is simple if and only if $(\lambda,k)$ belongs to none of the hyperplanes $E_{m,N}$ or $H_{\alpha,m,N}$.

Etingof's q-nondegeneracy conjecture. If $q$ is not a root of unity, then the form $(,)_q$ is nondegenerate.

Etingof's positivity conjecture. The inner product $(,)$ is symmetric and positive definite, and in particular nondegenerate.

Mazorchuk–Stroppel conjecture. The left derived functor of