reference request – Parabolic invariants of coinvariant algebras for reflection groups

Let $W$ be a finite reflection group and $V$ its reflection representation (over $mathbb{C}$). Let $S$ be the symmetric algebra on $V^*$, $I_Wsubseteq S$ the ideal generated by the non-constant $W$-invariants in $S$, and $R=S/I$ the coinvariant algebra.

It’s well known that $R$ is finite dimensional over $mathbb{C}$ of dimension $|W|$, that its top degree piece is one dimensional and spanned by (the image in $R$) of the product of the forms vanishing on the reflecting hyperplanes, and that the operator $mathrm{alt} = frac{sum_{win W} (-1)^w w}{|W|}$ projects an arbitrary element of the appropriate degree piece of $S$ onto its representative in $R$. A standard reference is the book Reflection groups and Coxeter groups by Humphreys.

Now let $P$ denote some subset of the Coxeter generators of $W$, and let $W_P$ and $W^P$ denote respectively the subgroup generated by $P$ and the set of minimal length coset representatives for $W/W_P$. Let $R^P$ denote the invariants of $W_P$ as a subring of $R$. I believe the following analogues of the above facts are true at least when $W$ is a Weyl group, for geometric reasons related to the cohomology rings of generalized flag manifolds:

  • $R^P$ has dimension $|W^P|=|W|/|W_P|$
  • The top degree piece of $R^P$ is one dimensional and spanned by the product of the hyperplanes corresponding to roots NOT in $W_P$.
  • The operator $frac{sum_{win W^P} (-1)^w w}{|W^P|}$ projects an arbitrary element of the appropriate degree piece of $S^{W_P}$ to its representative in $R^P$.

I am looking for a reference which actually proves these facts. I prefer an algebraic treatment that extends at least to the finite reflection groups that are not Weyl groups, and ideally one that extends (with appropriate modifications) to complex reflection groups.

(If this is in Kane or Hiller – apologies – library access to physical books requires jumping through more hoops given the pandemic. If it’s in Humphreys or Lehrer-Taylor, I’ll have to admit to illiteracy.)