Suppose $I_1\in$ ideal in $S$ generated by $I_2,\dots \text{.}$ Then $I_1=\underset{i\ge 2}{\Sigma }{R}_i{I}_i$ for some $R_2,\dots \in S$ so that $I_1=Av{I}_1=\underset{i\ge 2}{\Sigma }\left(Av{R}_i\right)I_i$ belongs to the ideal in $I$ generated by $I_2,\dots ,$ a contradiction. Hence $I_1$ does not belong to the ideal in $S$ generated by $I_2,\dots \text{.}$

We now prove (1) by induction on $d=\text{deg}\hspace{0.17em}{P}_1\text{.}$ If $d=0,$ $P_1=0\in S_0\text{.}$ Assume $d>0$ and let $g\in G$ be a reflection in a hyperplane $L=0\text{.}$ Then for each $i,$ $L|(P_i-g{P}_i)\text{.}$ Hence $\Sigma ((P_i-g{P}_i)/L)I_i=0,$ so by the induction assumption $P_1-g{P}_1\in S_0,$ i.e. $P_1\equiv g{P}_1$ (mod $S_0\text{).}$ Since $G$ is generated by reflections this holds for all $g\in G$ and hence $P_1\equiv Av{P}_1$ (mod $S_0\text{).}$ But $Av{P}_1\in S_0$ so $P_1\in S_0\text{.}$

$\square$

If the $I_i$ are not algebraically independent, let $H(I_1,\dots ,I_n)=0$ be a nontrivial relation with all monomials in the $I_i\text{'s}$ of the same minimal degree in the underlying coordinates $x_1,\dots ,x_{\ell }\text{.}$ Let $H_i=\partial H(I_1,\dots ,I_n)/\partial I_i\text{.}$ By the choice of $H$ not all $H_i$ are $0\text{.}$ Choose the notation so that $\{H_1,\dots ,H_m\}$ $(m\le n)$ but no subset of it generates the ideal in $I$ generated by all the $H_i\text{.}$ Let $H_j=\underset{i=1}{\overset{m}{\Sigma }}{V}_{j,i}{H}_i$ for $j=m+1,\dots ,n$ where $V_{j,i}\in I$ and all terms in the equation are homogeneous of the same degree. Then for $k=1,2,\dots ,\ell$ we have $0=\partial H/\partial x_k=\underset{i=1}{\overset{n}{\Sigma }}{H}_i\partial I_i/\partial x_k=\underset{i=1}{\overset{m}{\Sigma }}{H}_i(\partial I_i/\partial x_k+\underset{j=m+1}{\overset{n}{\Sigma }}{V}_{j,i}\partial I_j/\partial x_k)\text{.}$ By (1) $\partial I_1/\partial x_k+\underset{j=m+1}{\overset{n}{\Sigma }}{V}_{j,1}\partial I_j/\partial x_k\in S_0\text{.}$ Multiplying by $x_k,$ summing over $k,$ using Euler's formula, and writing $d_j=\text{deg}\hspace{0.17em}{I}_j$ we get $d_1{I}_1+\underset{j=m+1}{\overset{n}{\Sigma }}{V}_{j,1}{d}_j{I}_j=\underset{i=1}{\overset{n}{\Sigma }}{A}_i{I}_i$ where $A_i$ belongs to the ideal in $S$ generated by the $x_k\text{.}$ By homogeneity $A_1=0\text{.}$ Thus $I_1$ is in the ideal generated by $I_2,\dots ,I_n,$ a contradiction.

$\square$

Assume $P\in I$ is homogeneous of positive degree. Then $P=\Sigma P_i{I}_i,$ $P_i\in S\text{.}$ By averaging we can assume that each $P_i\in I\text{.}$ Each $P_i$ is of degree less than the degree of $P,$ so by induction on its degree $P$ is a polynomial in the $I_i\text{'s.}$

$\square$

By (2) $n\le \ell \text{.}$ By Galois theory $ℝ\left(I\right)$ is of finite index in $ℝ(x_1,x_2,\dots ,x_n),$ hence has transcendence degree $\ell$ over $ℝ,$ whence $n\ge \ell \text{.}$

$\square$

Let ${\epsilon }_1,\dots ,{\epsilon }_{\ell }$ be the eigenvalues of $g$ and $x_1,\dots ,x_{\ell }$ the corresponding eigenfunctions. Then $\text{det}{(1-gt)}^{-1}=\underset{i}{\Pi }(1+{\epsilon }_{i}t+{\epsilon }_i^2{t}^2+\dots )\text{.}$ The coefficient of $t^n$ is $\underset{p_1+p_2+\dots =n}{\Sigma }{\epsilon }_1^{p_1}{\epsilon }_2^{p_2}\dots ,$ i.e. the trace of $g$ acting on the space of homogeneous polynomials in $x_1,\dots ,x_{\ell }$ of degree $n,$ since the monomials $x_1^{p_1}{x}_2^{p_2}\dots$ form a basis for this space. By averaging we get the dimension of the space of invariant homogeneous polynomials of degree $n\text{.}$ This dimension is the number of monomials $I_1^{p_1}{I}_2^{p_2}\dots$ of degree $n,$ i.e., the number of solutions of $p_1{d}_1+p_2{d}_2+\dots =n,$ i.e. the coefficient of $t^n$ in $\underset{i=1}{\overset{\ell }{\Pi }}{(1-t^{d_i})}^{-1}\text{.}$

$\square$

We have

Substituting this in (5) and multiplying by ${(1-t)}^{\ell },$ we have $\Pi {(1+t+\dots +t^{d_i-1})}^{-1}={\left|G\right|}^{-1}(1+N(1-t)/(1+t)+{(1-t)}^{2}P\left(t\right))$ where $P\left(t\right)$ is regular at $t=1\text{.}$ Setting $t=1$ we get $\Pi d_i^{-1}={\left|G\right|}^{-1}\text{.}$ Differentiating and setting $t=1$ we get $\left(\Pi d_i^{-1}\right)\Sigma (-(d_i-1)/2)={\left|G\right|}^{-1}(-N/2),$ so $\Sigma (d_i-1)=N\text{.}$

$\square$

Let $I_i^{\prime },d_i^{\prime },$ and $N\prime$ refer to $G\prime \text{.}$ The $I_i^{\prime }$ can be expressed as polynomials in the $I_i$ with the determinant of the corresponding Jacobian not $0\text{.}$ Hence after a rearrangement of the $I_i,$ $\partial I_i/\partial I_i^{\prime }\ne 0$ for all $i\text{.}$ Hence $d_i\ge d_i^{\prime }\text{.}$ But $\Sigma ({\partial }_i-1)=N=N\prime =\Sigma (d_i^{\prime }-1)$ by (6). Hence $d_i=d_i^{\prime }$ for all $i,$ so, again by (6), $\left|G\right|=\Pi d_i=\Pi d_i^{\prime }=\left|G\prime \right|,$ so $G=G\prime \text{.}$

$\square$

If $\beta$ is a positive root and supp $\beta \not\subset \pi$ then $\beta =\underset{\alpha \in \Pi }{\Sigma }{e}_{\alpha }\alpha$ with some $e_{\alpha }>0,$ $\alpha \notin \pi \text{.}$ Now $w\beta$ is $\beta$ plus a vector with support in $\pi ,$ hence its coefficient of $\alpha$ is positive, so $w\beta >0\text{.}$

$\square$

(a) For any $w\in W$ let $w\prime \in W_{\pi }w$ be such that $N\left(w\prime \right)$ is minimal. Then $w\prime \alpha >0$ for all $\alpha \in \pi$ by Appendix II.19(a'). Hence $w\prime \in W_{\pi }^{\prime }$ so that $w\in W_{\pi }^{\prime }{W}_{\pi }\text{.}$ Suppose now $w=w\prime w^{\prime \prime }=u\prime u^{\prime \prime }$ with $w\prime ,u\prime \in W_{\pi }^{\prime }$ and $w^{\prime \prime },u^{\prime \prime }\in W_{\pi }\text{.}$ Then $w\prime w^{\prime \prime }{u^{\prime \prime }}^{-1}=u\prime \text{.}$ Hence $w\prime w^{\prime \prime }{u^{\prime \prime }}^{-1}\pi >0\text{.}$ Now $w\prime (-\pi )<0$ so $w^{\prime \prime }{u^{\prime \prime }}^{-1}\pi$ has support $\pi \text{.}$ Hence $w^{\prime \prime }{u^{\prime \prime }}^{-1}\pi \subseteq \pi$ so by Appendix 11.23 (applied to $W_{\pi }\text{)}$ $w^{\prime \prime }{u^{\prime \prime }}^{-1}=1\text{.}$ Hence $w\prime =u\prime ,$ $w^{\prime \prime }=u^{\prime \prime }\text{.}$

(b) follows from (a) and (1).

$\square$

We have, by (3), $W\left(t\right)/W_{\pi }\left(t\right)=\underset{w\in W_{\pi }^{\prime }}{\Sigma }{t}^{N\left(w\right)}\text{.}$ Therefore the contribution of the term for $w$ to the sum in (4) is $c_w{t}^{N\left(w\right)}$ where $c_w=\underset{\underset{w\pi >0}{\pi \subset \Pi }}{\Sigma }{(-1)}^{\pi }\text{.}$ If $w$ keeps positive exactly $k$ elements of $\Pi$ then

Therefore the only contribution is made by $w_0,$ the element of $w$ which makes all positive roots negative, so the sum in (4) is equal to $t^N$ as required.

$\square$

(a) $\subseteq$ (b) because $\pi$ is orthogonal to $D_{\pi }\text{.}$ (b) $\subseteq$ (c) because $D$ is a fundamental domain for $W$ by Appendix III.33. Clearly (c) $\subseteq$ (d). (d) $\subseteq$ (a) by Appendix III.32.

$\square$

This follows from Euler's formula, but may be proved directly by induction. In fact, if an extra hyperplane $H$ is added to the configuration, each original $i\text{-cell}$ cut in two by $H$ has corresponding to it in $H$ an $(i-1)\text{-cell}$ separating the two parts from each other, so that $\Sigma {(-1)}^i{n}_i$ remains unchanged.

$\square$

Each cell of $K$ is $W\text{-congruent}$ to exactly one $D_{\pi }\text{.}$ By (5) every cell fixed by $w$ lies in $V_w$ $(V_w=\{v\in V\hspace{0.17em}|\hspace{0.17em}wv=v\})\text{.}$ Applying (6) to $V_w$ and using $\text{dim}\hspace{0.17em}{D}_{\pi }=\ell -\left|\pi \right|$ we get $\underset{\pi \subset \Pi }{\Sigma }{(-1)}^{\pi }{n}_{\pi }\left(w\right)={(-1)}^{\ell -k},$ where $k=\text{dim}\hspace{0.17em}{V}_w\text{.}$ But $w$ is orthogonal, so that its possible eigenvalues in $V$ are $+1,$ $-1$ and pairs of conjugate complex numbers. Hence ${(-1)}^{\ell -k}=\text{det}\hspace{0.17em}w\text{.}$

$\square$

Assume first that $\chi \equiv 1\text{.}$ Now $xw{x}^{-1}\in W_{\pi }$ if and only if $xw{x}^{-1}$ fixes $D_{\pi }$ (by (5)) which happens if and only if $w$ fixes $x^{-1}{D}_{\pi }\text{.}$ Therefore $1_{\pi }\left(w\right)=n_{\pi }\left(w\right)$ by $(*)\text{.}$ By (7) this gives the result for $\chi \equiv 1\text{.}$ If $\chi$ is any character then ${\chi }_{\pi }=\chi ·1_{\pi }$ so (8) holds.

$\square$

In (8) take $\chi$ to be the character of $M,$ average over $w\in W,$ and use $(*)\text{.}$

$\square$

We have $w_{\alpha }p=-p=\left(\text{det}\hspace{0.17em}{w}_{\alpha }\right)p$ if $\alpha$ is a simple root by Appendix I.11. Since $W$ is generated by simple reflections $p$ is skew. If $f$ is skew and $\alpha$ a root then $w_{\alpha }f=\left(\text{det}\hspace{0.17em}{w}_{\alpha }\right)f=-f$ so $\alpha |f\text{.}$ By unique factorization $p|f\text{.}$

$\square$

We must show $(*)$ $\underset{\pi \subseteq \Pi }{\Sigma }{(-1)}^{\pi }\underset{i}{\Pi }{(1-t^{d_{\pi i}})}^{-1}=t^N\underset{i}{\Pi }{(1-t^{d_i})}^{-1}\text{.}$ Let $S=\underset{k=0}{\overset{\infty }{\Sigma }}{S}_k$ be the algebra of polynomials on $V,$ graded as usual. As in (5) of the proof of Theorem 27 the coefficient of $t^k$ on the left hand side of $(*)$ is $\underset{\pi \subseteq \Pi }{\Sigma }{(-1)}^{\pi }\text{dim}\hspace{0.17em}{I}_{\pi }\left(S_k\right)\text{.}$ Similarly, using (10), the coefficient of of $t^k$ on the right hand side of $(*)$ is $\text{dim}\hspace{0.17em}\hat{I}\left(S_k\right)\text{.}$ These are equal by (9).

$\square$