Short version: if $G$ is a Coxeter group and $H \subset G$ is a parabolic subgroup, both acting on a space $V$, is it true that the invariant-coinvariant algebra $(S(V)_G)^H$ has a natural bilinear form induced by taking coefficients in the top component?
Now the long, detailed version. Let $G$ be a Coxeter group acting on a real vector space $V$ with a fixed generating set of reflections $S$. We can form the symmetric algebra $S(V)$, the invariant subalgebra $S(V)^G \subset S(V)$, and the ideal $I_G \subset S(V)$ generated by elements of $S(V)^G$ of positive degree. The coinvariant algebra is defined as $S(V)_G = S(V)/I_G$. By the Cehvalley-Shephard-Todd theorem it is isomorphic to the regular representation of $G$ as a $G$-module. If $H \subset G$ is a parabolic subgroup then the $H$-invariants $(S(V)_G)^H$ form a subalgebra of $S(V)_G$, which is finite dimensional and whose top nonzero component is $1$-dimensional over $\mathbb R$, say spanned by $s$. Then $(S(V)_G)^H$ has a nice bilinear pairing given by taking $(f,g)$ to be the coefficient of $s$ in the product $fg$ for all $f, g \in (S(V)_G)^H$.
If $G$ is a Weyl group, the algebra $(S(V)_G)^H$ is isomorphic to the cohomology ring of a generalized flag variety by a famous theorem of Borel. Plus the bilinear form described above is the pullback of the Poincaré pairing in said cohomology, and in particular it is nondegenerate. Questions:
Can the non-degeneracy of the bilinear form be proved by purely combinatorial methods, without appealing to Borel's theorem?
Can it be extended to all pairs $(G,H)$ where $G$ is just a finite Coxeter group?