86 problems

Let $W$ be a Coxeter group. The Calogero–Moser cells are the equivalence classes in $W$ defined using the $\mathbb{C}^*$-fixed points of the Calogero–Moser space, while Lusztig's t…

Let $n$ be a positive integer, let $\frac{r}{s}$ be a term of the $n$-th Farey sequence, and let $\tilde{M}_{r/s}$ and $\tilde{M}^{\prime}_{r/s}$ be the compositions of generalized…

Let $c$ be a parameter for which the Heckman–Opdam functor $F$ is an equivalence between the highest weight categories $\mathcal{O}_c$ and $\mathcal{O}_{\sigma(c)}$. Let…

Let $V$ be a finite-dimensional complex vector space, let $W\subset \operatorname{GL}_{\mathbb{C}}(V)$ be a finite reflection group, and let $c$ be a complex-valued function on the…

Let $W$ be a complex reflection group, let $c$ be a parameter, and assume the stated freeness and symmetrizing-form hypothesis for the generic Hecke algebra. Write ${\bar Z}_c$ for…

Rouquier's conjecture. For an arbitrary multi charge, the multiplicity of a simple module in a standard module in $\mathcal{O}_{\boldsymbol{s}}(\ell,1,m)$ equals the corresponding…

Gordon–Martino conjecture. There is a natural identification of the $CM_{\bf c}$-partition and the $KL_{\bf L}$-partition, induced by attaching a $KL_{\bf L}$-cell to an irreducibl…

Serre functor conjecture. The functor $S$ is right exact, and its left derived functor ${\mathcal L}S$ is a Serre functor. This is a proposed description of the Serre functor for c…

Let $W$ be a real reflection group with reflection representation $V$. A positive integer $r$ is $W$-good if, for every parameter $c$ with $|c|=r$, the Cherednik algebra $H_c(W)$ a…

Let $W$ be a real reflection group of rank $\ell$, with Coxeter number $h$ and basic invariant degrees $d_1,\dots,d_\ell$. Let $r$ satisfy $r\equiv d_i-1\pmod h$ for some…

Let $W$ be a real reflection group with reflection representation $V$, let $H(W)$ be the universal rational Cherednik algebra, and let $\mathcal{QH}\subset H(W)$ be the space of un…

Singularity conjecture. The space ${\mathcal{C}}{\mathcal{M}}_c$ is singular for every value of $c$.

Let $\operatorname{Hilb^n\mathbb{C}^2}$ be the Hilbert scheme of $n$ points in $\mathbb{C}^2$, and let $\mathcal{B}_1$ denote its tautological rank $n$ bundle and $\mathcal{P}_1$ i…

Let ${\mathsf{H}}_c$ be the rational Cherednik algebra of type ${\mathfrak{gl}}_n$ with parameter $c\in\mathbb{C}$, and let ${\mathsf{e}}$ be the symmetrizer idempotent for $S_n$.…

Let $S_q=S_q(n,n)$ be the $q$-Schur algebra, where $q=\exp(2\pi i c)$, and let $\mathcal{O}_c$ denote category $\mathcal{O}$ for the rational Cherednik algebra $H_c$. GGOR's equiva…

Duality compatibility conjecture. The two functors $\Phi$ and $\psi_*$ are isomorphic.

q-Schur algebra conjecture. The category ${\mathcal O}$ is equivalent to the category of finitely generated modules over the associated $q$-Schur algebra.

Projectivity criterion for the KZ image. One could conjecture that the semisimplicity assumption on ${\mathcal H}(W_H)$ can be replaced by the assumption that…

The $(q,t)$ Catalan character conjecture. In particular, $\chi_{\mathbf e\cdot L(\mathsf{triv})}(q,t)=C_n^{(k)}(q,t)$, the $(q,t)$ Catalan number. This conjecture refines the corre…

Finite-dimensional representation conjecture. The algebra ${\mathsf{H}}_c$ has a nonzero finite-dimensional representation $V$ if and only if $r$ is coprime to $n$; moreover, in th…

Zero-fiber multiplicity conjecture. The scheme-theoretic multiplicity of $\Upsilon^{-1}(0)$ at $\chi$ equals

Type-A zero-fiber conjecture. For each $\chi\in\Upsilon^{-1}(0)$, there is a unique, up to $W$-conjugacy, pair of subsets $S_1,S_2\subset R$ such that

Let $\mathfrak{g}$ be a reductive Lie algebra and let $\mathcal{CM}_c=\operatorname{Spec}Z(H^{\mathrm{rat}}_{c,0})$ be its Calogero–Moser space, with $c$ as in the source. Let…

Let $\mathfrak{g}$ be a reductive Lie algebra. Choose a well-chosen $\mathbb{Q}$-factorial terminalization … and let $X_{\mathfrak{g},\mathrm{Losev}}$ be the intermediate symplecti…

Let $\mathfrak{g}$ be a complex reductive Lie algebra, and let $X_{\mathfrak{g},\mathrm{symb}}$ denote its symbolic replacement variety. Let the rational Cherednik algebra of…