The Saito determinant for Coxeter discriminant strata

Abstract

Let $W$ be a finite Coxeter group and $V$ its reflection representation. The orbit space ${\mathcal{M}}_W=V/W$ has the remarkable Saito flat metric defined as a Lie derivative of the $W$-invariant bilinear form $g$. We find determinant of the Saito metric restricted to an arbitrary Coxeter discriminant stratum in ${\mathcal{M}}_W$. It is shown that this determinant is proportional to a product of linear factors in the flat coordinates of the form $g$ on the stratum. We also find multiplicities of these factors in terms of Coxeter geometry of the stratum. This result may be interpreted as a generalisation to discriminant strata of the Coxeter factorisation formula for the Jacobian of the group $W$. As another interpretation, we find determinant of the operator of multiplication by the Euler vector field in the natural Frobenius structure on the strata.

arXiv comment54 pages, 6 tables

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Cited by in corpus (1)

• Mirror symmetry for extended affine Weyl groups