The sum of basic Weyl invariant degrees minus one counts positive roots

Molien trace identity: for $\mathrm{\mid }t\mathrm{\mid }<1$, $\mathrm{Hilb}(S^W,t)={\sum }_{d\ge 0}{\dim }_{\mathbb{C}}(S^W{)}_d{t}^d=\frac{1}{\mathrm{\mid }W\mathrm{\mid }}{\sum }_{w\in W}\det (1-tw{\mathrm{\mid }}_E{)}^{-1}\mathrm{.}$ Indeed the $W$-averaging operator $\pi =(1\mathrm{/}\mathrm{\mid }W\mathrm{\mid }){\sum }_{w}w$ is a projection onto $S^W$ in each finite-dimensional degree $S_d$, so $\dim (S^W{)}_d=\mathrm{tr}\pi {\mathrm{\mid }}_{S_d}=(1\mathrm{/}\mathrm{\mid }W\mathrm{\mid }){\sum }_w\mathrm{tr}(w{\mathrm{\mid }}_{S_d})$; on the other hand, for an operator with eigenvalues ${\mu }_1,\dots ,{\mu }_r$ on $E$, and hence ${\mu }_1^{-1},\dots ,{\mu }_r^{-1}$ on $E^{\ast }$, the degree-$d$ trace $\mathrm{tr}(w{\mathrm{\mid }}_{S_d})$ is the complete homogeneous symmetric polynomial $h_d({\mu }_1^{-1},\dots ,{\mu }_r^{-1})$, whose generating function is ${\prod }_i(1-{\mu }_i^{-1}t{)}^{-1}=\det (1-t{w}^{-1}{\mathrm{\mid }}_E{)}^{-1}$; averaging over $w$ and reindexing by $w\mapsto w^{-1}$ gives the displayed formula.

For $w\in W$ let $f$ be the dimension of its fixed space in $E$. Then $\det (1-tw{\mathrm{\mid }}_E)=(1-t{)}^f{g}_w(t)$ with $g_w(t)={\prod }_{\lambda \mathrm{\ne }1}(1-t\lambda )$ over the remaining eigenvalues $\lambda$ of $w$ and $g_w(1)={\prod }_{\lambda \mathrm{\ne }1}(1-\lambda )\mathrm{\ne }0$; all eigenvalues lie on the unit circle. In particular if $w\mathrm{\ne }1$ fixes a hyperplane pointwise, then $f=r-1$, the remaining eigenvalue is $-1$, and $\det (1-tw{\mathrm{\mid }}_E)=(1-t{)}^{r-1}(1+t)$.

If $w\in W$, $w\mathrm{\ne }1$, fixes a hyperplane $H$ pointwise, then $H={\alpha }^{\perp }$ for some root $\alpha$. Suppose $H$ is different from every root hyperplane. Then all intersections $H\cap {\beta }^{\perp }$ are proper linear subspaces of $H$, whose finite union cannot cover it over the infinite field $\mathbb{R}$ (the product of nonzero defining linear forms is a nonzero polynomial). Thus choose $x\in H$ with $(x,\beta )\mathrm{\ne }0$ for every root $\beta$; then $x$ lies in an open chamber $C$, and $w(x)=x$ gives $x\in C\cap w(C)$. As $w$ permutes the chambers, $C\cap w(C)\mathrm{\ne }\mathrm{\varnothing }$ forces $w(C)=C$, so $w$ lies in the stabilizer of $C$, which is trivial by simple transitivity. Hence $H$ is contained in a root hyperplane, and since both are hyperplanes, $H={\alpha }^{\perp }$ for a root $\alpha$.

Put $u=1-t$. Since $1-t^{d_i}=1-(1-u{)}^{d_i}=d_{i}u-\frac{d_i(d_i-1)}{2}{u}^2+O(u^3)$, one has $(1-t^{d_i}{)}^{-1}=d_i^{-1}{u}^{-1}(1+\frac{d_i-1}{2}u+O(u^2))$, and therefore ${\prod }_{i=1}^r(1-t^{d_i}{)}^{-1}=({\prod }_i{d}_i{)}^{-1}{u}^{-r}(1+\frac{{\sum }_i(d_i-1)}{2}u+O(u^2))=\frac{1}{\mathrm{\mid }W\mathrm{\mid }}{u}^{-r}(1+\frac{{\sum }_i(d_i-1)}{2}u+O(u^2)),$ the last equality by [F1]; this product equals $\mathrm{Hilb}(S^W,t)$ for $\mathrm{\mid }t\mathrm{\mid }<1$ by [F1].

The map $\alpha \mapsto s_{\alpha }$ induces a bijection from the set ${\mathrm{\Phi }}^{+}$ of positive roots onto the set of reflections in $W$; consequently $W$ has exactly $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ reflections. Injectivity: $s_{\alpha }=s_{\beta }$ forces the two reflections to have the same fixed hyperplane, so $\beta =\pm \alpha$, and among positive roots $\beta =\alpha$. Surjectivity: every reflection of $W$ has a fixed hyperplane, which is ${\alpha }^{\perp }$ for some root $\alpha$ by step 1.3, and the reflection of $E$ with fixed hyperplane ${\alpha }^{\perp }$ is unique, namely $s_{\alpha }$. Since $s_{\alpha }=s_{-\alpha }$ and every root hyperplane is ${\alpha }^{\perp }$ for exactly one positive root by [F3], distinct positive roots give distinct reflections.

Expanding the right-hand side of step 1.1 at $t=1$ in the variable $u=1-t$ gives $\frac{1}{\mathrm{\mid }W\mathrm{\mid }}{\sum }_{w\in W}\det (1-tw{\mathrm{\mid }}_E{)}^{-1}=\frac{1}{\mathrm{\mid }W\mathrm{\mid }}(u^{-r}+\frac{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}{2}{u}^{-(r-1)}+O(u^{-(r-2)}))\mathrm{.}$ The identity $w=1$ contributes $u^{-r}$. A nonidentity $w$ with fixed space of dimension $r-1$ is a reflection, and there are exactly $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ of them by step 2.1; each contributes $\det (1-tw{)}^{-1}=(1-t{)}^{-(r-1)}(1+t{)}^{-1}=u^{-(r-1)}(\frac{1}{2}+O(u))$ by step 1.2, because $(1+t{)}^{-1}=(2-u{)}^{-1}=\frac{1}{2}+O(u)$. Every other nonidentity $w$ has $f\le r-2$, hence contributes $u^{-f}\cdot O(1)=O(u^{-(r-2)})$ by step 1.2.

Steps 1.4 and 3.1 compute the same meromorphic function of $u$ near $u=0$; comparing the coefficients of $u^{-(r-1)}$, which are finite, gives $\frac{1}{\mathrm{\mid }W\mathrm{\mid }}\cdot \frac{{\sum }_i(d_i-1)}{2}=\frac{1}{\mathrm{\mid }W\mathrm{\mid }}\cdot \frac{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}{2}$, hence ${\sum }_i(d_i-1)=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$.

With the page degree normalization $\mathrm{deg}E=2$ the basic invariant $w_i$ has page degree $2d_i$, so the coinvariant algebra has Hilbert series ${\prod }_i(1-t^{2d_i})\mathrm{/}(1-t^2{)}^r$ and its top degree is ${\sum }_{i}2d_i-2r=2{\sum }_i(d_i-1)=2\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=\mathrm{\mid }\mathrm{\Phi }\mathrm{\mid }$, by step 4.1 and the published degree computation for the coinvariant algebra.