Abstract: Let $G$ be a finite, complex
reflection group acting on a complex vector space $V$, and $\delta$ its disciminant
polynomial. The fibres of $\delta$ admit commuting actions of $G$ and a cyclic
group $C_m$. The virtual $G\times C_m$ character given by the Euler
characteristic of a fibre is a refinement of the zeta function of the geometric
monodromy, calculated by Denef and Loeser in 1995. Recent enhancements
to Springer's theory of regular elements make it possible to describe this
virtual character explicitly, in terms of a certain poset of subgroups of
$G$. This poset is a diagram invariant $G$, in the sense of Brou\'e, Malle,
and Rouquier. I will attempt to justify its interest and significance.
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