Weyl's character formula

## Weyl's character formula

The adjoint representation $𝔤$ is a unitary representation of $G.$ So the Weyl group $W$ acts on $𝔥$ by unitary operators. So $W$ acts on $𝔱$ by orthonormal matrices. Identify $𝔱$ and $𝔱*=\mathrm{Hom}\left(𝔱,ℝ\right)=\left\{\alpha :𝔱\to ℝ\right\}$ with the innder product, $𝔱 → ~ 𝔱* α ↦ αċ .$ For a root $\alpha$ define $α∨ = 2α αα and H α = x∈𝔱| α x =0 .$ Then the reflection ${s}_{\alpha }$ in the hyperplane ${H}_{\alpha },$ which comes from ${Z}_{\alpha }={Z}_{G}\left({T}_{\alpha }^{\circ }\right)/{T}_{\alpha }^{\circ },$ is $s α : 𝔱 → 𝔱 λ ↦ λ- λ α ∨ α.$

INSERT DIAGRAM OF HYPERPLANE WITH REFLECTION HERE LATER

So

1. $W$ acts on $𝔱$, and
2. $𝔱-\bigcup _{\alpha \in R}{H}_{\alpha }={ℝ}^{n}\\left(\bigcup _{\alpha \in R}{H}_{\alpha }\right)$ is a union of chambers (these are the connected components).

PICTURE OF CHAMBERS AND WEIGHT LATTICE

The Weyl group $W$ permutes these chambers and if we fix a choice of chamber $C$ then we can identify the chambers $wC,w\in C.$ (See Brocker-tom Dieck V (2.3iv) and the Claim at the bottom of p 193.

PICTURE OF CHAMBERS LABELLED BY wC

Let

This means that $R\left(G\right)=\mathrm{span}-\left\{\left[{G}^{\lambda }\right]\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\lambda \in \stackrel{^}{G}\right\}$ with

1. addition given by $\left[{G}^{\lambda }\right]+\left[{G}^{\mu }\right]=\left[{G}^{\lambda }\oplus {G}^{\mu }\right],$ and
2. multiplication given by $\left[{G}^{\lambda }\right]\left[{G}^{\mu }\right]=\left[{G}^{\lambda }\otimes {G}^{\mu }\right].$
Thus, in $R\left(G\right)$ it makes sense to write $∑ λ∈ G ^ m λ G λ instead of ⊕ λ∈ G ^ G λ ⊕ m λ .$ Define $ℂP=span- e λ | λ∈P with multiplication e λ e μ = e λ+μ ,$ for $\lambda ,\mu \in P.$ Then $ℂP≅R T ,sinceR T =span- X λ | λ∈P .$ The action of $W$ on $R\left(T\right)$ (see (???)) induces an action of $W$ on $ℂP$ given by $w e λ = e wλ ,forw∈W,λ∈P.$ Note that $ε w = det 𝔥 w =±1$ since the action of $w$ on $𝔥$ is by an orthogonal matrix. The vector spaces of symmetric and alternating functions are and respectively. Note that $ℂ{\left[P\right]}^{W}$ is a ring but $𝒜$ is only a vector space.

Define $P + =P∩ C - and P ++ =P∩C.$ The set ${P}^{+}$ is the set of dominant weights.

Every $W$-orbit on $P$ contains a unique element of ${P}^{+}$ and so the set of monomial symmetric functions $m λ = ∑ γ∈ W λ e γ ,λ∈ P + ,$ forms a basis of $ℂ{\left[P\right]}^{W}.$ Define $a μ = ∑ w∈W ε w e wμ ,$ for $\mu \in P.$ Then
1. $w{a}_{\mu }=\epsilon \left(w\right){a}_{\mu },$ for all $w\in W$ and all $\mu \in P,$
2. ${a}_{\mu }=0,$ if $\mu \in {H}_{\alpha }$ for some $\alpha ,$ and
3. $\left\{{a}_{\mu }\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\mu \in {P}^{++}\right\}$ is a basis of $𝒜.$
The fundamental weights ${\omega }_{1},\dots ,{\omega }_{n}$ in $𝔱$ are defined by $ω i α j ∨ = δ ij ,$ where ${H}_{{\alpha }_{j}}$ are the walls of $C.$ Write $α>0if λα >0for allλ∈C.$ Then $ρ= ∑ i=1 n ω i = 1 2 ∑ α>0 α,$ is the element of $𝔱$ defined by $ρ α i ∨ =1,for all α 1 ,…, αn .$

The map $P + → P ++ λ ↦ λ+ρ$ is a bijection, and $ℂ P W → 𝒜 f ↦ a ρ f$ is a vector space isomorphism.

 Proof. Since $w a ρ f = w a ρ wf =ε w a ρ f,$ the second map is well defined. Let $g= ∑ λ∈P g λ e λ ∈𝒜.$ Then, for a positive root $\alpha ,$ $-g= s α g= ∑ λ∈P g λ e s α λ ,$ and so $g= ∑ λ, λα >0 g λ e λ - s s α λ .$ Since $e λ - e s α λ = e λ-α +…+ e λ- λ α ∨ α e α -1 ,$ the element $g$ is divisible by ${e}^{\alpha }-1.$ Thus, since all the factors in the product are coprime in $ℂP,$ $g$is divisible by $∏ α>0 e α -1 = e α ∏ α>0 e α 2 - e - α 2 = e ρ a ρ ,$ where the last equality follows from the fact that ${a}_{\rho }$ is divisible by the product $\prod _{\alpha >0}\left({e}^{\frac{\alpha }{2}}-{e}^{-\frac{\alpha }{2}}\right)$ and these two expressions have the same top monomial, ${e}^{\rho }.$ Since $g\in 𝒜$ divisble by ${a}_{\rho }$ the map $ℂP\to 𝒜$ is invertible. $\square$

Define ${\chi }^{\lambda }=\frac{{a}_{\lambda +\rho }}{{a}_{\rho }},\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{2em}{0ex}}\lambda \in {P}^{+},$ so that the $\left\{{\chi }^{\lambda }\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\lambda \in {P}^{+}\right\}$ are the basis of $ℂ{\left[P\right]}^{W}$ obtained by taking the inverse image of the basis $\left\{{a}_{\lambda +\rho }\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\lambda \in {P}^{+}\right\}$ of $𝒜.$ Extend these functions to all of $U$ by setting $χ λ gt g -1 = χ λ t ,for allg∈U.$ Since $∫ T X λ t X μ t dt= δ λμ ,$ for $\lambda ,\mu \in P,$ then we have $∫ T a λ+ρ t$ a μ+ρ t dt= δ λμ W , and thus, by (???), $δ λμ = ∫ G χ λ g$ χ μ g dg,for allλ,μ P+ . Thus the ${\chi }^{\lambda },\lambda \in {P}^{+}$ are an orthonormal basis of the set of class functions in $C{\left(G\right)}^{\mathrm{rep}}.$ If ${U}^{\lambda }$ is an irreducible representation of $U$ then $Tr U λ g = ∑ i=1 d M ii λ g ,where M ij λ = v i λ g v j λ ,$ for an orthonormal basis ${v}_{1}^{\lambda },\dots ,{v}_{n}^{\lambda }$ of ${U}^{\lambda }.$ Then $∫ G Tr U λ g$ Tr U μ g dg= δ λμ , and so the functions ${\mathrm{Tr}}_{{U}^{\lambda }}$ are another orthonormal basis of the set of class functions in $C{\left(G\right)}^{\mathrm{rep}}.$ It follows that ${\chi }^{\lambda }=±{\mathrm{Tr}}_{{U}^{\lambda }}.$

It only remains to check that the sign is positive to show that the ${\chi }^{\lambda }$ are the irreducible characters of $U.$ This follows from the following computation. $χ λ 1 = lim t→0 χ λ e tρ = lim t→0 ∑ w∈W ε w X w λ+ρ e tρ ∑ w∈W ε w X wρ e tρ = lim t→0 ∑ w∈W ε w e w λ+ρ tρ ∑ w∈W ε w e wρ tρ = lim t→0 ∑ w∈W ε w e t λ+ρ w -1 ρ ∑ w∈W ε w e t ρ w -1 ρ = lim t→0 a ρ e t λ+ρ a ρ e tρ = lim t→0 ∏ α>0 X α 2 - X - α 2 e t λ+ρ ∏ α>0 X α 2 - X - α 2 e tρ = lim t→0 ∏ α>0 e t λ+ρ α 2 - e -t λ+ρ α 2 ∏ α>0 e t ρ α 2 - e -t ρ α 2 = lim t→0 sinh t λ+ρ α 2 sinh t ρ α 2 = ∏ α>0 λ+ρ α 2 ρ α 2 = ∏ α>0 λ+ρ α ∨ ρ α ∨ .$

Let U be a compact connected Lie group and let T be the maximal torus and L the corresponding lattice.

1. The irreducible representations of $U$ are indexed by dominant integral weights $\lambda \in {L}^{+}$ under the correspondence
2. The character of ${V}^{\lambda }$ is $χ λ = ∑ w∈W ε w e w λ+ρ ∑ w∈W ε w e wρ$ where $\rho \in {P}^{+}$ is defined by $⟨\rho ,{\alpha }_{i}^{\vee }⟩=1$ for $1\le i\le n$ and $\epsilon \left(w\right)=\mathrm{det}\left(w\right).$
3. The dimension of ${V}^{\lambda }$ is $d λ = ∏ α>0 λ+ρ α ∨ ∏ α>0 ρ α ∨ .$
4. $χ λ = ∑ p∈ 𝒫 λ e p 1 ,$ where ${𝒫}_{\lambda }$ is the set of all paths obtained by acting on ${p}_{\lambda }$ by root operators.

Remark: By part (d) which end at $\mu .$ (For the path model some copying can be done from the Barcelona abstract.)

Remark: Point out that $R\left(T\right)=ℤL,$ where $L$ is the lattice corresponding to $T.$ Also point out that $R\left(U\right)=R{\left(T\right)}^{W}\cong {\left(ℤL\right)}^{W}.$