Restricting invariants of unitary reflection groups

Amend, Nils; Berardinelli, Angela; Douglass, J.; Röhrle, Gerhard

doi:10.1090/tran/7129

Restricting invariants of unitary reflection groups
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by Nils Amend, Angela Berardinelli, J. Matthew Douglass and Gerhard Röhrle PDF

Trans. Amer. Math. Soc. 370 (2018), 5401-5424 Request permission

Abstract:

Suppose that $G$ is a finite, unitary reflection group acting on a complex vector space $V$ and $X$ is the fixed point subspace of an element of $G$. Define $N$ to be the setwise stabilizer of $X$ in $G$, $Z$ to be the pointwise stabilizer, and $C=N/Z$. Then restriction defines a homomorphism from the algebra of $G$-invariant polynomial functions on $V$ to the algebra of $C$-invariant functions on $X$. Extending earlier work by Douglass and Röhrle for Coxeter groups, we characterize when the restriction mapping is surjective for arbitrary unitary reflection groups $G$ in terms of the exponents of $G$ and $C$ and their reflection arrangements. A consequence of our main result is that the variety of $G$-orbits in the $G$-saturation of $X$ is smooth if and only if it is normal.

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