## Lectures on Chevalley groups

Last update: 18 July 2013

## §9. The orders of the finite Chevalley groups

Presently we will prove:

Theorem 24: Let $W$ be a finite reflection group on a real space $V$ of finite dimension $\ell ,S$ the algebra of polynomials on $V,$ $I\left(S\right)$ the subalgebra of invariants under $W\text{.}$ Then:

 (a) $I\left(S\right)$ is generated by $\ell$ homogeneous algebraically independent elements ${I}_{1},\dots ,{I}_{\ell }\text{.}$ (b) The degrees of the ${I}_{j}\text{'s,}$ say ${d}_{1},\dots ,{d}_{\ell },$ are uniquely determined and satisfy $\underset{j}{\Sigma }\left({d}_{j}-1\right)=N,$ the number of positive roots. (c) For the irreducible Weyl groups the ${d}_{i}\text{'s}$ are as follows: $W di's Aℓ Bℓ,Cℓ Dℓ E6 E7 E8 F4 G2 2,3,…,ℓ+1 2,4,…,2ℓ 2,4,…,2ℓ-2,ℓ 2,5,6,8,9,12 2,6,8,10,12,14,18 2,8,12,14,18,20,24,30 2,6,8,12 2,6$

Our main goal is:

Theorem 25:

 (a) Let $G$ be a universal Chevalley group over a field $k$ of $q$ elements and the ${d}_{i}\text{'s}$ as in Theorem 24. Then $|G|={q}^{N}\underset{i}{\Pi }\left({q}^{{d}_{i}}-1\right)$ with $N=\Sigma \left({d}_{i}-1\right)=$ the number of positive roots. (b) If $G$ is simple instead, then we have to divide by $c=|\text{Hom}\left({L}_{1}/{L}_{0},{k}^{*}\right)|,$ given as follows: $G Aℓ Bℓ,Cℓ Dℓ E6 E7 E8 F4 G2 c (ℓ+1,q-1) (2,q-1) (4,qℓ-1) (3,q-1) (2,q-1) 111$

Remark: We see that the groups of type ${B}_{\ell }$ and ${C}_{\ell }$ have the same order. If $\ell =2$ the root systems are isomorphic so the groups are isomorphic. We will show later that if $\ell \ge 3$ the groups are isomorphic if and only if $q$ is even.

The proof of Theorem 25 depends on the following identity.

Theorem 26: Let $W$ and the ${d}_{i}\text{'s}$ be as in Theorem 24 and $t$ an indeterminate. Then $\underset{w\in W}{\Sigma }{t}^{N\left(w\right)}=\underset{i}{\Pi }\left(1-{t}^{{d}_{i}}\right)\left(1-t\right)\text{.}$

We show first that Theorem 25 is a consequence of Theorems 24 and 26.

Lemma 54: If $G$ is as in Theorem 25(a) then $|G|={q}^{N}{\left(q-1\right)}^{\ell }\underset{w\in W}{\Sigma }{q}^{N\left(w\right)}\text{.}$

 Proof. Recall that, by Theorems 4 and 4', $G=\bigcup _{w\in W}BwB$ (disjoint) and $BwB=UHw{U}_{w}$ with uniqueness of expression. Hence $|G|=|U||H|·\underset{w\in W}{\Sigma }|{U}_{w}|\text{.}$ Now by Corollary 1 to the proposition of §3, $|U|={q}^{N}$ and $|{U}_{w}|={q}^{N\left(w\right)}\text{.}$ By Lemma 28, $|H|={\left(q-1\right)}^{\ell }\text{.}$ $\square$

Corollary: $U$ is a $p\text{-Sylow}$ subgroup of $G,$ if $p$ denotes the characteristic of $k\text{.}$

 Proof. $p|{q}^{N\left(w\right)}$ unless $N\left(w\right)=0\text{.}$ Since $N\left(w\right)=0$ if and only if $w=1,$ $p\nmid \Sigma {q}^{N\left(w\right)}\text{.}$ $\square$

 Proof of Theorem 25. (a) follows from Lemma 54 and Theorem 26. (b) follows from the fact that the center of the universal group is isomorphic to $\text{Hom}\left({L}_{1}/{L}_{0},{k}^{*}\right)$ and the values of ${L}_{1}/{L}_{0}$ found in §3. $\square$

Before giving general proofs of Theorems 24 and 26 we give independent (case by case) verifications of Theorems 24 and 26 for the classical groups.

Theorem 24: Type ${A}_{\ell }\text{:}$ Here $W\cong {S}_{\ell +1}$ permuting $\ell +1$ linear functions ${\omega }_{1},\dots ,{\omega }_{\ell +1}$ such that ${\sigma }_{1}=\Sigma {\omega }_{i}=0\text{.}$ In this case the elementary symmetric polynomials ${\sigma }_{2},\dots ,{\sigma }_{\ell +1}$ are invariant and generate all other polynomials invariant under $W\text{.}$

Types ${B}_{\ell },{C}_{\ell }\text{:}$ Here $W$ acts relative to a suitable basis ${\omega }_{1},\dots ,{\omega }_{\ell }$ by all permutations and sign changes. Here the elementary symmetric polynomials in ${\omega }_{1}^{2},\dots ,{\omega }_{\ell }^{2}$ are invariant and generate all other polynomials invariant under $W\text{.}$

Type ${D}_{\ell }\text{:}$ Here only an even number of sign changes can occur. Thus we can replace the last of the invariants for ${B}_{\ell },{\omega }_{1}^{2}\dots {\omega }_{\ell }^{2}$ by ${\omega }_{1}\dots {\omega }_{\ell }\text{.}$

Theorem 26: Type ${A}_{\ell }\text{:}$ Here $W\cong {S}_{\ell +1}$ and $N\left(w\right)$ is the number of inversions in the sequence $\left(w\left(1\right),\dots ,w\left(\ell +1\right)\right)\text{.}$ If we write ${P}_{\ell }\left(t\right)=\sum _{w\in W\cong {S}_{\ell +1}}{t}^{N\left(w\right)}$ then ${P}_{\ell +1}\left(t\right)={P}_{\ell }\left(t\right)\left(1+t+{t}^{2}+\dots +{t}^{\ell +1}\right),$ as we see by considering separately the $\ell +2$ values that $w\left(\ell +2\right)$ can take on. Hence the formula ${P}_{\ell }\left(t\right)=\underset{j=2}{\overset{\ell +1}{\Pi }}\left(1-{t}^{j}\right)/\left(1-t\right)$ follows by induction.

Exercise: Prove the corresponding formulas for types ${B}_{\ell },{C}_{\ell }$ and ${D}_{\ell }\text{.}$ Here the proof is similar, the induction step being a bit more complicated.

Part (a) of Theorem 24 follows from:

Theorem 27: Let $G$ be a finite group of automorphisms of a real vector space $V$ of finite dimension $\ell$ and $I$ the algebra of polynomials on $V$ invariant under $G\text{.}$ Then:

 (a) If $G$ is generated by reflections, then $I$ is generated by $\ell$ algebraically independent homogeneous elements (and 1). (b) Conversely, if $I$ is generated by $\ell$ algebraically independent homogeneous elements (and 1) then $G$ is generated by reflections.

Example: Let $\ell =2$ and $V$ have coordinates $x,y\text{.}$ If $G=\left\{±\text{id.}\right\},$ then $G$ is not a reflection group. $I$ is generated by ${x}^{2},xy,$ and ${y}^{2}$ and no smaller number of elements suffices.

Notation: Throughout the proof we let $S$ be the algebra of all polynomials on $V,$ ${S}_{0}$ the ideal in $S$ generated by the homogeneous elements of $I$ of positive degree, and $Av$ stand for average over $G$ (i.e. $AvP={|G|}^{-1}\underset{g\in G}{\Sigma }gP\text{).}$

Proof of (a). (Chevalley, Am. J. of Math. 1995.)

(1) Assume ${I}_{1},{I}_{2},\dots$ are elements of $I$ such that ${I}_{1}$ is not in the ideal in $I$ generated by the others and that ${P}_{1},{P}_{2},\dots$ are homogeneous elements of $S$ such that $\Sigma {P}_{i}{I}_{i}=0\text{.}$ Then ${P}_{1}\in {S}_{0}\text{.}$

 Proof. 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 $d=\text{deg} {P}_{1}\text{.}$ 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 $i,$ $L|\left({P}_{i}-g{P}_{i}\right)\text{.}$ Hence $\Sigma \left(\left({P}_{i}-g{P}_{i}\right)/L\right){I}_{i}=0,$ 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 ${P}_{1}\in {S}_{0}\text{.}$ $\square$

We choose a minimal finite basis ${I}_{1},\dots ,{I}_{n}$ for ${S}_{0}$ formed of homogeneous elements of $I\text{.}$ Such a basis exists by Hilbert's Theorem.

(2) The ${I}_{i}\text{'s}$ are algebraically independent.

 Proof. If the ${I}_{i}$ are not algebraically independent, let $H\left({I}_{1},\dots ,{I}_{n}\right)=0$ 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 ${H}_{i}=\partial H\left({I}_{1},\dots ,{I}_{n}\right)/\partial {I}_{i}\text{.}$ By the choice of $H$ not all ${H}_{i}$ are $0\text{.}$ Choose the notation so that $\left\{{H}_{1},\dots ,{H}_{m}\right\}$ $\left(m\le n\right)$ 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 $0=\partial H/\partial {x}_{k}=\underset{i=1}{\overset{n}{\Sigma }}{H}_{i}\partial {I}_{i}/\partial {x}_{k}=\underset{i=1}{\overset{m}{\Sigma }}{H}_{i}\left(\partial {I}_{i}/\partial {x}_{k}+\underset{j=m+1}{\overset{n}{\Sigma }}{V}_{j,i}\partial {I}_{j}/\partial {x}_{k}\right)\text{.}$ 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 ${d}_{j}=\text{deg} {I}_{j}$ 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. $\square$

(3) The ${I}_{i}\text{'s}$ generate $I$ as an algebra.

 Proof. 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 ${I}_{i}\text{'s.}$ $\square$

(4) $n=\ell \text{.}$

 Proof. By (2) $n\le \ell \text{.}$ By Galois theory $ℝ\left(I\right)$ is of finite index in $ℝ\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right),$ hence has transcendence degree $\ell$ over $ℝ,$ whence $n\ge \ell \text{.}$ $\square$

By (2), (3) and (4) (a) holds.

$\square$

Proof of (b). (Todd, Shephard Can. J. Math. 1954.)

Let ${I}_{1},\dots ,{I}_{\ell }$ be algebraically independent generators of $I$ of degrees ${d}_{1},\dots ,{d}_{\ell },$ respectively.

(5) $\underset{i=1}{\overset{\ell }{\Pi }}{\left(1-{t}^{{d}_{i}}\right)}^{-1}=\underset{g\in G}{Av} \text{det}{\left(1-gt\right)}^{-1},$ as a formal identity in $t\text{.}$

 Proof. Let ${\epsilon }_{1},\dots ,{\epsilon }_{\ell }$ be the eigenvalues of $g$ and ${x}_{1},\dots ,{x}_{\ell }$ the corresponding eigenfunctions. Then $\text{det}{\left(1-gt\right)}^{-1}=\underset{i}{\Pi }\left(1+{\epsilon }_{i}t+{\epsilon }_{i}^{2}{t}^{2}+\dots \right)\text{.}$ 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 $\underset{i=1}{\overset{\ell }{\Pi }}{\left(1-{t}^{{d}_{i}}\right)}^{-1}\text{.}$ $\square$

(6) $\Pi {d}_{i}=|G|$ and $\Sigma \left({d}_{i}-1\right)=N=$ number of reflections in $G\text{.}$

 Proof. We have $det(1-gt)= { (1-t)ℓ if g=1, (1-t)ℓ-1 (1+t) if g is a reflection, a polynomial not divisible by (1-t)ℓ-1 otherwise.$ Substituting this in (5) and multiplying by ${\left(1-t\right)}^{\ell },$ we have $\Pi {\left(1+t+\dots +{t}^{{d}_{i}-1}\right)}^{-1}={|G|}^{-1}\left(1+N\left(1-t\right)/\left(1+t\right)+{\left(1-t\right)}^{2}P\left(t\right)\right)$ where $P\left(t\right)$ is regular at $t=1\text{.}$ Setting $t=1$ we get $\Pi {d}_{i}^{-1}={|G|}^{-1}\text{.}$ Differentiating and setting $t=1$ we get $\left(\Pi {d}_{i}^{-1}\right)\Sigma \left(-\left({d}_{i}-1\right)/2\right)={|G|}^{-1}\left(-N/2\right),$ so $\Sigma \left({d}_{i}-1\right)=N\text{.}$ $\square$

(7) Let $G\prime$ be the subgroup of $G$ generated by its reflections. Then $G\prime =G$ and hence $G$ is a reflection group.

 Proof. 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 $\Sigma \left({\partial }_{i}-1\right)=N=N\prime =\Sigma \left({d}_{i}^{\prime }-1\right)$ by (6). Hence ${d}_{i}={d}_{i}^{\prime }$ for all $i,$ so, again by (6), $|G|=\Pi {d}_{i}=\Pi {d}_{i}^{\prime }=|G\prime |,$ so $G=G\prime \text{.}$ $\square$

Corollary: The degrees ${d}_{1},{d}_{2},\dots$ above are uniquely determined and satisfy the equations (6).

Thus Theorem 24(b) holds.

$\square$

Exercise: For each reflection in $G$ choose a root $\alpha \text{.}$ Then $\text{det} \frac{\partial \left({I}_{1},{I}_{2},\dots \right)}{\partial \left({x}_{1},{x}_{2},\dots \right)}=\Pi \alpha$ up to multiplication by a nonzero number.

Remark: The theorem remains true if $ℝ$ is replaced by any field of characteristic $0$ and "reflection" is replaced by "automorphism of $V$ with fixed point set a hyperplane".

For the proof of Theorem 24(c) (determination of the ${d}_{i}\text{)}$ we use:

Proposition: Let $G$ and the ${d}_{i}$ be as in Theorem 27 and $w={w}_{1}\dots {w}_{\ell },$ the product of the simple reflections (relative to an ordering of $V$ (see Appendix I.8)) in any fixed order. Let $h$ be the order of $w\text{.}$ Then:

 (a) $N=\ell h/2\text{.}$ (b) $w$ contains $\omega =\text{exp} 2\pi i/h$ as an eigenvalue, but not $1\text{.}$ (c) If the eigenvalues of $w$ are $\left\{{\omega }^{{m}_{i}} | 1\le {m}_{i}\le h-1\right\}$ then $\left\{{m}_{i}+1\right\}=\left\{{d}_{i}\right\}\text{.}$

 Proof. This was first proved by Coxeter (Duke Math. J. 1951), case by case, using the classification theory. For a proof not using the classification theory see Steinberg, T.A.M.S. 1959, for (a) and (b) and Coleman, Can. J. Math. 1958, for (c) using (a) and (b). This can be used to determine the ${d}_{i}$ for all the Chevalley groups. As an example we determine the ${d}_{i}$ for ${E}_{8}\text{.}$ Here $\ell =8,$ $N=120,$ so by (a) $h=30\text{.}$ Since $w$ acts rationally $\left\{{\omega }^{n} | \left(n,30\right)=1\right\}$ are all eigenvalues. Since $\phi \left(30\right)=8=\ell$ these are all the eigenvalues. Hence the ${d}_{i}$ are 1, 7, 11, 13, 17, 19, 23, 29 all increased by 1, as listed previously. The proofs for ${G}_{2}$ and ${F}_{4}$ are exactly the same. ${E}_{6}$ and ${E}_{7}$ require further argument. $\square$

Exercise: Argue further.

Remark: The ${d}_{i}\text{'s}$ also enter into the following results, related to Theorem 24:

Let $ℒ$ be the original Lie algebra, $k$ a field of characteristic 0, $G$ the corresponding adjoint Chevalley group. The algebra of polynomials on $ℒ$ invariant under $G$ is generated by $\ell$ algebraically independent elements of degree ${d}_{1},\dots ,{d}_{\ell },$ the ${d}_{i}\text{'s}$ as above.

This is proved by showing that under restriction from $ℒ$ to $ℋ$ the $G\text{-invariant}$ polynomials on $ℒ$ are mapped isomorphically onto the $W\text{-invariant}$ polynomials on $ℋ\text{.}$ The corresponding result for the universal enveloping algebra of $ℒ$ then follows easily.

(b) If $G$ acts on the exterior algebra on $ℒ,$ the algebra of invariants is an exterior algebra generated by $\ell$ independent homogeneous elements of degrees $\left\{2{d}_{i}-1\right\}\text{.}$

This is more difficult. It implies that the Poincaré polynomial (whose coefficients are the Betti numbers) of the corresponding compact semisimple Lie group (the group $K$ constructed from $ℂ$ in §8) is $\Pi \left(1+{t}^{2{d}_{i}-1}\right)\text{.}$

Proof of Theorem 26. (Solomon, Journal of Algebra, 1966.)

Let $\Pi$ be the set of simple roots. If $\pi \subseteq \Pi$ let ${W}_{\pi }$ be the subgroup generated by all ${w}_{\alpha },$ $\alpha \in \Pi \text{.}$

(1) If $w\in {W}_{\pi }$ then $w$ permutes the positive roots with support not in $\pi \text{.}$

 Proof. 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 $w\beta >0\text{.}$ $\square$

(2) Corollary: If $w\in {W}_{\pi }$ then $N\left(w\right)$ is unambiguous (i.e. it is the same whether we consider $w\in W$ or $w\in {W}_{\pi }\text{).}$

(3) For $\pi \subseteq \Pi$ define ${W}_{\pi }^{\prime }=\left\{w\in W | w\pi >0\right\}\text{.}$ Then:

 (a) Every $w\in W$ can we written uniquely $w=w\prime {w}^{\prime \prime }$ with $w\prime \in {W}_{\pi }^{\prime }$ and ${w}^{\prime \prime }\in {W}_{\pi }\text{.}$ (b) In (a) $N\left(w\right)=N\left(w\prime \right)+N\left({w}^{\prime \prime }\right)\text{.}$

 Proof. (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 $w\prime \left(-\pi \right)<0$ 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). $\square$

(4) Let $W\left(t\right)=\underset{w\in W}{\Sigma }{t}^{N\left(w\right)},$ ${W}_{\pi }\left(t\right)=\underset{w\in {W}_{\pi }}{\Sigma }{t}^{N\left(w\right)}\text{.}$ Then $\underset{\pi \subseteq \Pi }{\Sigma }{\left(-1\right)}^{\pi }W\left(t\right)/{W}_{\pi }\left(t\right)={t}^{N},$ where $N$ is the number of positive roots and ${\left(-1\right)}^{\pi }={\left(-1\right)}^{|\pi |}\text{.}$

 Proof. 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 ${c}_{w}=\underset{\underset{w\pi >0}{\pi \subset \Pi }}{\Sigma }{\left(-1\right)}^{\pi }\text{.}$ If $w$ keeps positive exactly $k$ elements of $\Pi$ then $cw= { (1-1)k=0 if k≠0 1 if k=0.$ 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. $\square$

Corollary: $\Sigma {\left(-1\right)}^{\pi }|W|/|{W}_{\pi }|=1\text{.}$

Exercise: Deduce from (4) that if $\alpha$ and $\beta$ are complementary subsets of $\Pi$ then $\underset{\pi \supseteq \alpha }{\Sigma }{\left(-1\right)}^{\pi -\alpha }/{W}_{\pi }\left(t\right)=\underset{\pi \supseteq \beta }{\Sigma }{\left(-1\right)}^{\pi -\beta }/{W}_{\pi }\left({t}^{-1}\right)\text{.}$

Set $D=\left\{v\in V |$ $\left(v,\alpha \right)\ge 0$ for all $\alpha \in \Pi \right\},$ and for each $\pi \subseteq \Pi$ set ${D}_{\pi }=\left\{v\in V |$ for all $\alpha \in \pi ,\left(v,\beta \right)>0$ for all $\beta \in \Pi -\pi \right\}\text{.}$ ${D}_{\pi }$ is an open face of $D\text{.}$

(5) The following subgroups of $W$ are equal:

 (a) ${W}_{\pi }\text{.}$ (b) The stabilizer of ${D}_{\pi }\text{.}$ (c) The point stabilizer of ${D}_{\pi }\text{.}$ (d) The stabilizer of any point of ${D}_{\pi }\text{.}$

 Proof. (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. $\square$

(6) In the complex cut on real $k\text{-space}$ by a finite number of hyperplanes let ${n}_{i}$ be the number of $i\text{-cells.}$ Then $\Sigma {\left(-1\right)}^{i}{n}_{i}={\left(-1\right)}^{k}\text{.}$

 Proof. 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 $\left(i-1\right)\text{-cell}$ separating the two parts from each other, so that $\Sigma {\left(-1\right)}^{i}{n}_{i}$ remains unchanged. $\square$

(7) In the complex $K$ cut from $V$ by the reflecting hyperplanes let ${n}_{\pi }\left(w\right)$ $\left(\pi \subseteq \Pi ,w\in W\right)$ denote the number of cells $W\text{-congruent}$ to ${D}_{\pi }$ and $w\text{-fixed.}$ Then $\underset{\pi \subseteq \Pi }{\Sigma }{\left(-1\right)}^{\pi }{n}_{\pi }\left(w\right)=\text{det} w\text{.}$

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

If $\chi$ is a character on ${W}_{1},$ a subgroup of $W,$ then ${\chi }^{W}$ denotes the induced character defined by $\left(*\right)$ ${\chi }^{W}\left(w\right)={|{W}_{1}|}^{-1}\underset{\underset{xw{x}^{-1}\epsilon {W}_{1}}{x\in W}}{\Sigma }\chi \left(xw{x}^{-1}\right)\text{.}$ (See, e.g., W. Feit, Characters of finite groups.)

(8) Let $\chi$ be a character on $W$ and ${\chi }_{\pi }={\left(\chi |{W}_{\pi }\right)}^{W}$ $\left(\pi \subseteq \Pi \right)\text{.}$ then $\underset{\pi \subseteq \Pi }{\Sigma }{\left(-1\right)}^{\pi }{\chi }_{\pi }\left(w\right)=\chi \left(w\right)$ $\text{det} w$ for all $w\in W\text{.}$

 Proof. 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 $\left(*\right)\text{.}$ 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. $\square$

(9) Let $M$ be a finite dimensional real $W\text{-module,}$ ${I}_{\pi }\left(M\right)$ be the subspace of ${W}_{\pi }\text{-invariants,}$ and $\stackrel{ˆ}{I}\left(M\right)$ be the space of $W\text{-skew-invariants}$ (i.e. $\stackrel{ˆ}{I}\left(M\right)=\left\{m\in M | wm=\left(\text{det} w\right)m$ for all $w\in W\right\}\text{).}$ Then $\underset{\pi \subseteq \Pi }{\Sigma }{\left(-1\right)}^{\pi }\text{dim} {I}_{\pi }\left(M\right)=\text{dim} \stackrel{ˆ}{I}\left(M\right)\text{.}$

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

(10) If $p=\Pi \alpha ,$ the product of the positive roots, then $p$ is skew and $p$ divides every skew polynomial on $V\text{.}$

 Proof. We have ${w}_{\alpha }p=-p=\left(\text{det} {w}_{\alpha }\right)p$ 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 ${w}_{\alpha }f=\left(\text{det} {w}_{\alpha }\right)f=-f$ so $\alpha |f\text{.}$ By unique factorization $p|f\text{.}$ $\square$

(11) Let $P\left(t\right)=\Pi \left(1-{t}^{{d}_{i}}\right)/\left(1-t\right)$ and for $\pi \subseteq \Pi$ let $\left\{{d}_{\pi i}\right\}$ and ${P}_{\pi }$ be defined for ${W}_{\pi }$ as $\left\{{d}_{i}\right\}$ and $P$ are for $W\text{.}$ Then $\underset{\pi \subseteq \Pi }{\Sigma }{\left(-1\right)}^{\pi }P\left(t\right)/{P}_{\pi }\left(t\right)={t}^{N}\text{.}$

 Proof. We must show $\left(*\right)$ $\underset{\pi \subseteq \Pi }{\Sigma }{\left(-1\right)}^{\pi }\underset{i}{\Pi }{\left(1-{t}^{{d}_{\pi i}}\right)}^{-1}={t}^{N}\underset{i}{\Pi }{\left(1-{t}^{{d}_{i}}\right)}^{-1}\text{.}$ 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 $\left(*\right)$ is $\underset{\pi \subseteq \Pi }{\Sigma }{\left(-1\right)}^{\pi }\text{dim} {I}_{\pi }\left({S}_{k}\right)\text{.}$ Similarly, using (10), the coefficient of of ${t}^{k}$ on the right hand side of $\left(*\right)$ is $\text{dim} \stackrel{ˆ}{I}\left({S}_{k}\right)\text{.}$ These are equal by (9). $\square$

(12) Proof of Theorem 26. We write (11) as $\left({t}^{N}-{\left(-1\right)}^{\Pi }\right)/P\left(t\right)=\underset{\pi ⫅̸\Pi }{\Sigma }{\left(-1\right)}^{\pi }/{P}_{\pi }\left(t\right)$ and (4) as $\left({t}^{N}-{\left(-1\right)}^{\Pi }\right)/W\left(t\right)=\underset{\pi ⫅̸\Pi }{\Sigma }{\left(-1\right)}^{\pi }/{W}_{\pi }\left(t\right)\text{.}$ Then, by induction on $|\Pi |,$ $W\left(t\right)=P\left(t\right)\text{.}$

$\square$

Remark. Step (7), the geometric step, represents the only simplification of Solomon's original proof.

## Notes and References

This is a typed excerpt of Lectures on Chevalley groups by Robert Steinberg, Yale University, 1967. Notes prepared by John Faulkner and Robert Wilson. This work was partially supported by Contract ARO-D-336-8230-31-43033.