Combinatorics of the irreducible components of $\mathcal{H}_n^Γ$ in type $D$ and $E$

Abstract: In this article, we give a combinatorial model in terms of symmetric cores of the indexing set of the irreducible components of $\mathcal{H}_n^{\Gamma}$ (the $\Gamma$-fixed points of the Hilbert scheme of $n$ points in $\mathbb{C}^2$) containing a monomial ideal, whenever $\Gamma$ is a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ isomorphic to the binary dihedral group. Moreover, we show that if $\Gamma$ is a subgroup of $\mathrm{SL}_2(\mathbb{C})$ isomorphic to the binary tetrahedral group, to the binary octahedral group or to the binary icosahedral group, then the $\Gamma$-fixed points of $\mathcal{H}_n$ which are also fixed under $\mathbb{T}_1$, the maximal diagonal torus of $\mathrm{SL}_2(\mathbb{C})$, are in fact $\mathrm{SL}_2(\mathbb{C})$-fixed points. Finally, we prove that in that case, the irreducible components of $\mathcal{H}_n^{\Gamma}$ containing a $\mathbb{T}_1$-fixed point are of dimension $0$.