47 problems

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 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 $W$ 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…

Calogero–Moser cell coincidence conjecture. If $W$ is a Coxeter group, the Calogero–Moser notions coincide with the corresponding Hecke-algebra notions defined by Kazhdan and Luszt…

Collective infinite transitivity conjecture. The $G$-action on

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…

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…

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

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

Let $W={\mathfrak S}_n\ltimes({\mathbb Z}/N{\mathbb Z})^n$ act naturally on ${\mathfrak h}={\mathbb C}^n$, and let $p:{\mathcal C}_W\to{\mathfrak h}/W\times{\mathfrak h}/W$ be the…

Let ${\mathcal C}_n$ be the Calogero–Moser space, and let $p:{\mathcal C}_n\to{\mathbb A}^{(n)}\times{\mathbb A}^{(n)}$ send a conjugacy class of a pair of matrices to the unordere…

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

Springer-correspondence conjecture. The module $L_\chi$ corresponds to $\mathbb O_\chi^{(1)}$ via the Springer correspondence, and $\mathsf{sign}\otimes L_\chi$ corresponds to…

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.

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

Grothendieck-class conjecture. Their classes in the Grothendieck ring are

Single-relation conjecture. The coordinate rings satisfy

Let $W$ be a finite Coxeter group, viewed as a finite complex reflection group. Calogero–Moser theory associates to $W$ families, cells, and cellular characters, while Kazhdan–Lusz…

Let $W$ be a Coxeter group. The Calogero–Moser cellular characters are the characters obtained from the Calogero–Moser cellular-character construction, and Lusztig constructible ch…

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…

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