Weights, roots and the Weyl integral formula

## Weights and roots

Let $G$ be a compact connected group. A maximal torus of $G$ is a maximal connected subgroup of $G$ isomorphic to ${\left({S}^{1}\right)}^{k}$ for some positive integer $k.$

Fix a maximal torus $T$ in $G.$ The group $T$ is a maximal connected abelian subgroup of $G.$ The Weyl group is $W= N G T /T,where N G T = g∈G| gT g -1 =T .$ The Weyl group $W$ acts on $T$ by conjugation. The map $G/T×T → φ G gT t ↦ gt g -1$ is surjective and $\mathrm{Card}\left({\phi }^{-1}\left(g\right)\right)=\left|W\right|$ for any $g\in G.$ It follows from this that

1. Every element $g\in G$ is in some maximal torus.
2. Any two maximal tori in $G$ are conjugate.
Thus, maximal tori, are unique up to conjugacy and cover the group $G.$

Let $P$ be an index set for the irreducible representations of $T.$ Since the irreducible representations of ${S}^{1}$ are indexed by $ℤ,P\cong {ℤ}^{k}.$ The set $P$ is called the weight lattice of $G.$ If $λ∈Pthen X λ:T→ℂ* ,$ denotes the corresponding irreducible representation of $T.$ The $W$-action on $T$ induces a $W$-action on $P$ via $X wλ t = X λ w -1 t ,for allt∈T.$ A representation $V$ of $G$ is a representation of $T,$ by restriction, and, as a $T$-module,

The vector space ${V}_{\lambda }$ is the ${X}^{\lambda }$ isotypic component of the $T$-module $V.$ The $W$-action on $T$ gives The vector space ${V}_{\lambda }$ is the $\lambda$-weight space of $V.$ A weight vector of weight $\lambda$ in $V$ is a vector $v$ in ${V}_{\lambda }.$

Let $G$ be a compact connected Lie group and let $𝔲=\mathrm{Lie}\left(G\right).$ The group $G$ acts on $𝔲$ the adjoint representation. Extend the adjoint representation of $G$ on the complex vector space $𝔤 ℂ =𝔲⊕i𝔲=ℂ⊗ℝ𝔲.$ By ??? this representation extends to a representation of the complex algebraic group ${G}_{ℂ}$ which is the complexification of $G.$ Since $G$ is compact, the adjoint representation of ${G}_{ℂ}$ of ${𝔤}_{ℂ}$, and thus the adjoint representation of ${𝔤}_{ℂ}$ on itself, is completely decomposable. This shows that ${𝔤}_{ℂ}$ is a complex semisimple Lie algebra.

The adjoint representation ${𝔤}_{ℂ}$ of $G$ has a weight decomposition $𝔤 ℂ = ⊕ α∈P 𝔤 α ,$ and the root system of $G$ is the set $R= α∈P| α≠0, 𝔤 α ≠0$ of nonzero weights of the adjoint representation. The roots are the elements of $R.$ Set $𝔥={𝔤}_{0}.$ Then $𝔤 ℂ =𝔥⊕ ⊕ α∈R 𝔤 α$ is the decomposition of ${𝔤}_{ℂ}$ into the Cartan subalgebra $𝔥$ and root spaces ${𝔤}_{\alpha }.$ (Note that the usual denotation is ${𝔥}_{ℝ}=i𝔥,{𝔥}_{ℂ}=𝔥\oplus i𝔥,$ where $𝔥$ is a Cartan subalgebra of $𝔤,$ ie a maximal abelian subspace of $𝔤.$ Also ${𝔤}_{0}.$ Also ${𝔤}_{0}={𝔥}_{ℂ}$ since $𝔥$ is maximal abelian in $𝔤.$ Also $𝔥=𝔱\oplus i𝔱$ where $𝔱$ is the Lie algebra of the maximal torus $T$ of $G,$ and the maximal abelian sublgebra in $𝔤.$ Don't forget to think of $X: T → ℂ* t ↦ X λ t e h ↦ e λ h$ and $λ: 𝔥 → ℂ h ↦ λ h$

The Weyl group $W$ is generated by ${s}_{\alpha },\alpha \in R.$ The action of $W$ on $𝔥*$ is generated by the transformations $s α : 𝔥* → 𝔥* λ ↦ λ- λ α ∨ where α ∨ = 2α αα ,$ and $⟨,⟩:𝔥*×𝔥*\to ℝ$ is a nondegenerate symmetric bilinear form.

1. If $\alpha$ is a root then $-\alpha$ is a root and these are the only two multiples of $\alpha$ which are roots. (The thing that makes this work is that the root spaces are purely imaginary).
2. If $\alpha$ is a root the $\mathrm{dim}\left({𝔤}_{\alpha }\right)=1.$
3. The only connected compact Lie groups with $\mathrm{dim}\left(T\right)=1$ are $S{O}_{3}\left(ℝ\right)$ and the twofold simply connected cover of $S{O}_{3}\left(ℝ\right)$.

Proof 1) Suppose that $\alpha$ is a root and that $x\in {𝔤}_{\alpha }.$ $X α : T → ℂ* e h ↦ e α h$ and $X -α : T → ℂ* e h ↦$ e α h = e -α h since $\alpha \left(h\right)\in iℝ$ for $h\in 𝔱.$ Then, for all $h\in 𝔱,$ $h x - = h- x- =$ hx = α h x - =-α h x - , and so $\stackrel{-}{x}\in {𝔤}_{-\alpha }.$ Thus ${𝔤}_{-\alpha }\ne 0$ and $-\alpha$ is a root. Note that $\left[x,\stackrel{-}{x}\right]\in 𝔥$ since it has weight $0.$

2) Consider ${X}^{\alpha }:T\to {ℂ}^{*}.$ Then ${T}_{\alpha }=\mathrm{ker}{X}^{\alpha }$ is closed in $T$ and is of codimension 1. Let ${T}_{\alpha }^{\circ }$ be the connected component of the identity in ${T}_{\alpha }$ and let ${Z}_{\alpha }={Z}_{G}\left({T}_{\alpha }^{\circ }\right)$ be the centraliser of ${T}_{\alpha }^{\circ }$ in $U$ (this is connected.) Then $ℂ ⊗ ℝ Lie Z α =𝔱⊕i𝔱⊕ ⊕ h∈ T α ,β h =1 𝔤 β =𝔥⊕ ⊕ k∈ℤ 𝔤 kα.$

Now $Z α → Z α / T α ∘ ∪| ∪| T → T/ T α ∘$ So $T/{T}_{\alpha }^{\circ }$ is a maximal torus of ${Z}_{\alpha }/{T}_{\alpha }^{\circ }$ and $\mathrm{dim}T/{T}_{\alpha }^{\circ }=1.$ Then $ℂ ⊗ ℝ Lie Z α = 𝔥 α ⊕ℂ H α ⊕ ⊕ k∈ℤ 𝔤 kα .$

If ${X}_{\alpha }\in {𝔤}_{\alpha }$ then $\left[{X}_{\alpha },{X}_{-\alpha }\right]=\lambda {H}_{\alpha }$ and $\lambda \ne 0$ since $ℂH$ is maximal abelian in $Lie Z α / T α ∘ =ℂ⊕ ⊕ k∈ℤ 𝔤 kα .$ Now consider the action of ${H}_{\alpha }$ on $ℂH⊕ ⊕ k∈ ℤ >0 𝔤 kα ⊕ℂ X α .$ Then $Tr H = 1 λ Tr X α X -α = 1 λ ad X α ad X -α - ad X -α ad X α =0.$ But this implies $0=0+ ∑ k∈ ℤ >0 dim 𝔤 kα kα H α -α H α .$ So ${𝔤}_{k\alpha }=0$ for $k>1$ and ${𝔤}_{\alpha }=ℂ{X}_{\alpha }.$ So $\mathrm{span}\left\{{X}_{\alpha },{X}_{-\alpha },{H}_{\alpha }\right\}$ is a three dimensional subalgebra of $𝔤.$

If $U$ is a compact connected Lie group such that $\mathrm{dim}T=1$ then $U$ has Lie algebra $𝔤= span X α X -α H α =𝔲⊕i𝔲.$ Then the Weyl group of $U$ is $\left\{1,{s}_{\alpha }\right\}\cong {S}_{2}$ where ${s}_{\alpha }$ comes from conjugation by an element of ${Z}_{\alpha }$ and so ${s}_{\alpha }$ leaves ${T}_{\alpha }$ fixed.

So the Weyl group of $G$ contains all the ${s}_{\alpha },\alpha \in R.\square$

Example There are only two compact connected groups of dimension 3, $SO 3 andSpin 3 .$

Proof $G$ acts on $𝔤$ and this gives an imbedding $\mathrm{Ad}:G\to \mathrm{SO}\left(𝔤\right)$ (with respect to an $\mathrm{Ad}$ invariant form on $𝔤.$) This is an immersion since everything is connected. So $G$ is a conver of $\mathrm{SO}\left(3\right).\square$

## Weyl's integral formula

Let $G$ be the compact connected Lie group. Let $T$ be a maximal torus of $G$ and let $W$ be the Weyl group. Let $R$ be the set of roots. Then $W ∫ G f x dx= ∫ T ∏ α∈R X α t -1 ∫ G f gt g -1 dgdt.$

Proof First note that the map $G/T×T\to G$ given by $\left(gT,t\right)↦gt,$ can be used to define a (left) $G$ invariant measure on $G/T$ so that $∫ G f g dg= ∫ G/T×T f gt dtd gT ,$ and thus, for $y\in T,$ $∫ G f gy g -1 dg = ∫ G/T×T f gty t -1 g -1 dtd gT = ∫ G/T f gy g -1 dtd gT = ∫ G/T×T f gy g -1 d gT .$

Then the map $\phi :G/T×T\to G$ given by $\left(gT,t\right)↦gt{g}^{-1}$ yields

 $W ∫ G f g dg= ∫ G/T×T f gt g -1 J gT t dtd gT ,$
where ${J}_{\left(gT,t\right)}$ os the determinant of the differential at $\left(gT,t\right)$ of the map $\phi .$ By translation, ${J}_{\left(gT,t\right)}$ is the same as the determinant of the differential at the identity, $Te$, of the map ${L}_{g{t}^{-1}{g}^{-1}}\circ \phi \circ {L}_{gt},$$G/T×T → G/T×T → G → G xT y ↦ gxT ty ↦ gx ty gx -1 ↦ g t -1 g -1 gx ty gx -1 .$ Since $\left(g{t}^{-1}{g}^{-1}\right)\left(gx\right)ty{\left(gx\right)}^{-1}=g{t}^{-1}xty{x}^{-1}{g}^{-1}$ this differential is $𝔤/𝔥 → 𝔤 XY ↦ Ad g Ad t -1 X +Y-X .$ So ${J}_{\left(gT,t\right)}$ is the determinant of the linear transformation of $𝔤$ given by $Ad 𝔤 g Ad 𝔤/𝔥 t -1 - id 𝔤/𝔥 0 0 id 𝔥 ,$ where the second factor is a block $2×2$ matrix with respect to the decomposition $𝔤/𝔥\oplus 𝔥$ and ${\mathrm{Ad}}_{𝔤/𝔥}$ is the adjoint action of $T$ restricted to the subspace $𝔤/𝔥$ in $𝔤.$ The element ${t}^{-1}$ acts on the root space ${𝔤}_{\alpha }$ by the value ${X}^{\alpha }\left({t}^{-1}\right)$ where ${X}^{\alpha }:T\to ℂ*$ is the character of $T$ associated to the root $\alpha .$ Since $G$ is unimodular, $\mathrm{det}\left({\mathrm{Ad}}_{g}\right)=1,$ and since $𝔤/𝔥=\underset{\alpha \in R}{\oplus }{𝔤}_{\alpha },$
 $J gT t = ∏ α∈R X α t -1 -1 = ∏ α∈R X α t -1 ,$
where the last equality follows from the fact that if $\alpha$ is a root then $-\alpha$ is also a root. Then combine the numbered equations above to prove the theorem.

It follows from this theorem that, if $\chi$ and $\eta$ are class functions on $G$ then

 $∫ G χ g$ η g dg = 1 W ∫ T ∏ α∈R X α t -1 ∫G χ gt g -1 η gt g -1 dgdt = 1 W ∫ T ∏ α>0 X α t -1 X -α t -1 ∫G χ t η t dgdt = 1 W ∫ T ∏ α>0 X α 2 t - X - α 2 t X - α 2 t - X α 2 t ∫G χ t η t dgdt = 1 W ∫ T ∏ α>0 a ρ η t dt.