标题： 四元数商奇点的零纤维 显示英文标题

标题： Zero fibers of quaternionic quotient singularities

摘要： 我们提出对实反射群的对角不变量环的Haiman猜想的一个推广，以不可约四元数反射群（也称为辛反射群）的背景进行。 对于作用在四元数向量空间$V$上的反射群$W$，通过将$V$视为复向量空间，我们考虑商映射$\pi:V \to V/W$在零点处的方案论纤维。 对于$W$一个(四元数)秩至少为$6$的不可约反射群，我们证明该纤维上的函数环存在一个由辛反射代数的不可约表示产生的$(g+1)^n$维商环，其中$g=2N/n$与$N$为$W$和$n=\mathrm{dim}_\mathbf{H}(V)$中反射的数量，我们猜想这一结论在一般情况下也成立。 我们观察到实际上零纤维的次数对于秩一群（对应于克莱因奇点）正是$g+1$。 在附录中，我们给出一个证明，说明三种类型的考克斯数，包括$g$，都是整数。 显示英文摘要

摘要： We propose a generalization of Haiman's conjecture on the diagonal coinvariant rings of real reflection groups to the context of irreducible quaternionic reflection groups (also known as symplectic reflection groups). For a reflection group $W$ acting on a quaternionic vector space $V$, by regarding $V$ as a complex vector space we consider the scheme-theoretic fiber over zero of the quotient map $\pi:V \to V/W$. For $W$ an irreducible reflection group of (quaternionic) rank at least $6$, we show that the ring of functions on this fiber admits a $(g+1)^n$-dimensional quotient arising from an irreducible representation of a symplectic reflection algebra, where $g=2N/n$ with $N$ the number of reflections in $W$ and $n=\mathrm{dim}_\mathbf{H}(V)$, and we conjecture that this holds in general. We observe that in fact the degree of the zero fiber is precisely $g+1$ for the rank one groups (corresponding to the Kleinian singularities). In an appendix, we give a proof that three variants of the Coxeter number, including $g$, are integers.