28 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 $V$ be a symplectic vector space and $G\subset Sp(V)$ a finite group. Let $C$ be the parameter space of $G$-invariant functions on the symplectic reflections in $G$, let…

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…

Single-relation conjecture. The coordinate rings satisfy

Let $W$ be a complex reflection group, let ${\mathcal{C}}$ be its Calogero–Moser parameter space, and let $\kappa:{\mathcal{C}}\rightarrow{\mathcal{K}}$ be the parameter correspond…

Let $W$ be a complex reflection group, let $\tau$ be a finite-order element of the normalizer of $W$ in ${\mathrm{GL}}_{\mathbb{C}}(V)$, and let $c\in{\mathcal{C}}$ satisfy…

Let $W$ be a complex reflection group, let ${\mathcal{C}}$ be its parameter space, and let $\kappa:{\mathcal{C}}\rightarrow{\mathcal{K}}$ identify Calogero–Moser parameters with He…

Regular-number character criterion. Recall that $d$ is a regular number. For every $p\in{\mathcal Z}_k^{{\mathbb C}^\times}$, $p$ belongs to…

Character-sum refinement. Assume that $\tau$ is regular and that the fixed-component Calogero–Moser isomorphism conjecture holds. If…

Fixed-component Calogero–Moser isomorphism conjecture. Assume that $\tau$ is regular. There exists a linear map

Harish–Chandra compatibility conjecture. The maps from irreducible characters of the normalizer to Calogero–Moser fixed points, from those characters to unipotent representations v…

Cuspidality conjecture. Then $p$ is $\tau_d$-cuspidal.

Unipotent-family conjecture. There exists a unique bijection

Cuspidal-family conjecture. The $\tau$-cuspidal Lusztig $k$-families coincide with the $\tau$-cuspidal Calogero–Moser $k$-families.

Gordon–Martino conjecture. If $W$ is a Coxeter group, then

Let $W$ act as a complex reflection group on $V$, let $\tau$ be a finite-order element of the normalizer of $W$ in $\operatorname{GL}_{\mathbb C}(V)$, and let…

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…

Embedding conjecture. There exists an $SL_2$-equivariant embedding

Symplectic-leaf conjecture. Each conjugacy class of parabolic subgroups $(W')$ labels at most one symplectic leaf, and the geometric ordering on leaves equals the algebraic orderin…

Connectedness conjecture for the stabilizers. The groups $G_n$ are connected and hence irreducible for all $n\geq 0$.

Collective infinite transitivity conjecture. The action of $G$ on $\mathcal{C}$ is collectively infinitely transitive.

Infinite transitivity conjecture. For each $n\geq 1$, the action of $G$ on $\mathcal{C}_n$ is infinitely transitive.

Family and character conjecture. The partition of $\operatorname{Irr}(W)$ into Calogero–Moser $c$-families coincides with the partition into Kazhdan–Lusztig $c$-families, as conjec…

Two-sided cell conjecture. There exists a choice of ${\bar{\mathfrak r}}_c$ such that the partition of $W$ into two-sided Calogero–Moser $c$-cells coincides with the partition into…

Martino's conjecture. For every $b\in {\mathrm{Idem_{pr}}}({\bar Z}_c)$, there exists a central idempotent $b^{\mathcal H}$ of…