42 problems

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…

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…

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…

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 $m$ be even, let $\frac{m}{p}$ be odd, and set $k=\mathbb{Q}(\sqrt[m]{1})$. Let $\underline{\overline{H}}_{\underline{c}}(\mu(G(m,p,n)))$ and $\overline H_c(G(m,p,n))$ be the t…

Let $m,n$ be coprime positive integers with $m>n$, and let $\mathrm{L}_{\frac{m}{n}}$ be the finite-dimensional irreducible representation of the rational Cherednik algebra…

Let $m,n$ be coprime positive integers, let $\mathfrak{h}$ be the $(n-1)$-dimensional standard representation of $S_n$, and let $\mathrm{L}_{\frac{m}{n}}$ be the unique finite-dime…

Proposed structure. The quotient and its character and Hilbert polynomial are

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 $\sf{H}_W$ be the Hecke category, let $\sf{AH}:\sf{H}_W^{\operatorname{op}}\to\mathsf{Mod}_2^b(\bb{A}_W)$ be the annular Hecke functor, and let $\bb{A}_{W,\epsilon}$ and…

Let $W$ be a type $B$ complex reflection group, let $\mathcal{O}_c(W)$ be its rational Cherednik algebra category $\mathcal{O}$ at parameter $c$, and let $W'\subset W$ be a parabol…

Let $n=kp+r$ with $0\le r<p$. Define … and … Here $\mathcal{L}_{t,c}$ is the irreducible quotient of the polynomial representation of the rational Cherednik algebra, and $c$ is gen…

Let $\boldsymbol{c}$ denote the critical level, let $\mathsf{F}_{\mathbf{c}}$ be the Suzuki functor, and let $\mathbb{W}_{\mathbf{c}}(\lambda)$ and $\Delta_0(\lambda)$ be the corre…

Let $W$ be a finite real reflection group, let $c\in\mathfrak{p}_{\mathbb{R}}$, and let $\lambda\in\operatorname{Irr}(W)$. Let $L_c(\lambda)$ be the corresponding irreducible ratio…

Let $W$ be a finite real reflection group acting on $\mathfrak{h}$, let $c\in\mathfrak{p}_{\mathbb{R}}$ be real, and let $\lambda\in\operatorname{Irr}(W)$. For…

Let $W$ be a finite real reflection group with rational Cherednik algebra parameter space $\mathfrak{p}$, let $\mathfrak{p}_{\mathbb{R}}$ be its real form, and let…

Let $W$ be a finite complex reflection group, and let $W_a$ and $W_b$ be parabolic subgroups. Write ${}^{a}W^{b}$ for a complete set of double coset representatives of…

Let $W$ be a complex reflection group, let $c\in{\boldsymbol{\mathcal C}}$, and let ${\boldsymbol{\mathcal Z}}_c$ be the corresponding Calogero-Moser space. Let…

Harish-Chandra-type conjecture. There exists a subalgebra $H_{1,\mathbf c}'\subseteq H_{1,\mathbf c}$ such that

Let $\overline{\mathbfsf{H}}$ be the restricted rational Cherednik algebra, with $\operatorname{DecGen}(\overline{\mathbfsf{H}})$ and…