Weyl's character formula

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 10 April 2010

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, $$\begin{array}{rcl}𝔱& \tilde{\to }& 𝔱*\\ \alpha & \mapsto & ⟨\alpha ,ċ⟩\end{array}.$$ For a root $\alpha$ define $${\alpha }^{\vee }=\frac{2\alpha }{⟨\alpha ,\alpha ⟩}\text{and}{H}_{\alpha }=\left\{x\in 𝔱|\alpha \left(x\right)=0\right\}.$$ 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 $$\begin{array}{rcll}{s}_{\alpha }:& 𝔱& \to & 𝔱\\ & \lambda & \mapsto & \lambda -⟨\lambda ,{\alpha }^{\vee }⟩\alpha .\end{array}$$

INSERT DIAGRAM OF HYPERPLANE WITH REFLECTION HERE LATER

So

1. $W$ acts on $𝔱$, and
2. $𝔱-\bigcup _{\alpha \in R}{H}_{\alpha }={ℝ}^n\backslash \left(\bigcup _{\alpha \in R}{H}_{\alpha }\right)$ 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 $$\begin{array}{rcl}R\left(T\right)& =& \text{representation ring of }T\\ & =& \text{Grothendieck ring of representations of }G,\text{ and}\\ R\left(G\right)& =& \text{representation ring of }G.\end{array}$$

This means that $R\left(G\right)=\mathrm{span}-\left\{\left[G^{\lambda }\right]|\lambda \in \hat{G}\right\}$ 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 $$\sum _{\lambda \in \hat{G}}{m}_{\lambda }\left[G^{\lambda }\right]\text{instead of}\underset{\lambda \in \hat{G}}{\oplus }{\left(G^{\lambda }\right)}^{\oplus m_{\lambda }}.$$ Define $$ℂP=\mathrm{span}-\left\{e^{\lambda }|\lambda \in P\right\}\text{with multiplication}{e}^{\lambda }{e}^{\mu }=e^{\lambda +\mu },$$ for $\lambda ,\mu \in P.$ Then $$ℂP\cong R\left(T\right),\text{since}R\left(T\right)=\mathrm{span}-\left\{\left[X^{\lambda }\right]|\lambda \in P\right\}.$$ The action of $W$ on $R\left(T\right)$ (see (???)) induces an action of $W$ on $ℂP$ given by $$w{e}^{\lambda }=e^{w\lambda },\text{for}w\in W,\lambda \in P.$$ Note that $$\epsilon \left(w\right)=\underset{𝔥}{\det }\left(w\right)=\pm 1$$ since the action of $w$ on $𝔥$ is by an orthogonal matrix. The vector spaces of symmetric and alternating functions are $$ℂ{\left[P\right]}^W=\left\{f\in ℂP|wf=f\text{ for all }w\in W\right\},$$ and $$𝒜=\left\{f\in ℂP|wf==\epsilon \left(w\right)f\text{ for all }w\in W\right\},$$ respectively. Note that $ℂ{\left[P\right]}^W$ is a ring but $𝒜$ is only a vector space.

Define $$P^{+}=P\cap \stackrel{-}{C}\text{and}{P}^{++}=P\cap C.$$ 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 $$m_{\lambda }=\sum _{\gamma \in W_{\lambda }}{e}^{\gamma },\lambda \in P^{+},$$ forms a basis of $ℂ{\left[P\right]}^W.$ Define $$a_{\mu }=\sum _{w\in W}\epsilon \left(w\right)e^{w\mu },$$ 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. $\left\{a_{\mu }|\mu \in P^{++}\right\}$ is a basis of $𝒜.$

The fundamental weights ${\omega }_1,\dots ,{\omega }_n$ in $𝔱$ are defined by $$⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij},$$ where $H_{{\alpha }_j}$ are the walls of $C.$ Write $$\alpha >0\text{if}⟨\lambda ,\alpha ⟩>0\text{for all}\lambda \in C.$$ Then $$\rho =\sum _{i=1}^n{\omega }_i=\frac{1}{2}\sum _{\alpha >0}\alpha ,$$ is the element of $𝔱$ defined by $$⟨\rho ,{\alpha }_i^{\vee }⟩=1,\text{for all}{\alpha }_1,\dots ,{\alpha }_n.$$

The map $$\begin{array}{rcl}{P}^{+}& \to & P^{++}\\ \lambda & \mapsto & \lambda +\rho \end{array}$$ is a bijection, and $$\begin{array}{rcl}ℂ{\left[P\right]}^W& \to & 𝒜\\ f& \mapsto & a_{\rho }f\end{array}$$ is a vector space isomorphism.

		Proof.
	Since $$w\left(a_{\rho }f\right)=\left(w{a}_{\rho }\right)\left(wf\right)=\epsilon \left(w\right)a_{\rho }f,$$ the second map is well defined. Let $$g=\sum _{\lambda \in P}{g}_{\lambda }{e}^{\lambda }\in 𝒜.$$ Then, for a positive root $\alpha ,$ $$-g=s_{\alpha }g=\sum _{\lambda \in P}{g}_{\lambda }{e}^{s_{\alpha }\lambda },$$ and so $$g=\sum _{\lambda ,⟨\lambda ,\alpha ⟩>0}{g}_{\lambda }\left(e^{\lambda }-s^{s_{\alpha }\lambda }\right).$$ Since $$e^{\lambda }-e^{s_{\alpha }\lambda }=\left(e^{\lambda -\alpha }+\dots +e^{\lambda -⟨\lambda ,{\alpha }^{\vee }⟩\alpha }\right)\left(e^{\alpha }-1\right),$$ 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 $$\prod _{\alpha >0}\left(e^{\alpha }-1\right)=e^{\alpha }\prod _{\alpha >0}\left(e^{\frac{\alpha }{2}}-e^{-\frac{\alpha }{2}}\right)=e^{\rho }{a}_{\rho },$$ 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 ${\chi }^{\lambda }=\frac{a_{\lambda +\rho }}{a_{\rho }},\text{for}\lambda \in P^{+},$ so that the $\left\{{\chi }^{\lambda }|\lambda \in P^{+}\right\}$ are the basis of $ℂ{\left[P\right]}^W$ obtained by taking the inverse image of the basis $\left\{a_{\lambda +\rho }|\lambda \in P^{+}\right\}$ of $𝒜.$ Extend these functions to all of $U$ by setting $${\chi }^{\lambda }\left(gt{g}^{-1}\right)={\chi }^{\lambda }\left(t\right),\text{for all}g\in U.$$ Since $${\int }_T{X}^{\lambda }\left(t\right)X^{\mu }\left(t\right)dt={\delta }_{\lambda \mu },$$ for $\lambda ,\mu \in P,$ then we have $${\int }_T{a}_{\lambda +\rho }\left(t\right)$$ a μ+ρ t dt= δ λμ W , and thus, by (???), $${\delta }_{\lambda \mu }={\int }_G{\chi }^{\lambda }\left(g\right)$$ 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 $${\mathrm{Tr}}_{U^{\lambda }}\left(g\right)=\sum _{i=1}^d{M}_{ii}^{\lambda }\left(g\right),\text{where}{M}_{ij}^{\lambda }=⟨v_i^{\lambda },g{v}_j^{\lambda }⟩,$$ for an orthonormal basis $v_1^{\lambda },\dots ,v_n^{\lambda }$ of $U^{\lambda }.$ Then $${\int }_G{\mathrm{Tr}}_{U^{\lambda }}\left(g\right)$$ 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 ${\chi }^{\lambda }=\pm {\mathrm{Tr}}_{U^{\lambda }}.$

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. $$\begin{array}{rcl}{\chi }^{\lambda }\left(1\right)& =& \underset{t\to 0}{\lim }{\chi }^{\lambda }\left(e^{t\rho }\right)\\ & =& \underset{t\to 0}{\lim }\frac{\sum _{w\in W}\epsilon \left(w\right)X^{w\left(\lambda +\rho \right)}\left(e^{t\rho }\right)}{\sum _{w\in W}\epsilon \left(w\right)X^{w\rho }\left(e^{t\rho }\right)}\\ & =& \underset{t\to 0}{\lim }\frac{\sum _{w\in W}\epsilon \left(w\right)e^{⟨w\left(\lambda +\rho \right),t\rho ⟩}}{\sum _{w\in W}\epsilon \left(w\right)e^{⟨w\rho ,t\rho ⟩}}\\ & =& \underset{t\to 0}{\lim }\frac{\sum _{w\in W}\epsilon \left(w\right)e^{t⟨\lambda +\rho ,w^{-1}\rho ⟩}}{\sum _{w\in W}\epsilon \left(w\right)e^{t⟨\rho ,w^{-1}\rho ⟩}}\\ & =& \underset{t\to 0}{\lim }\frac{a_{\rho }\left(e^{t\left(\lambda +\rho \right)}\right)}{a_{\rho }\left(e^{t\rho }\right)}\\ & =& \underset{t\to 0}{\lim }\frac{\prod _{\alpha >0}\left(X^{\frac{\alpha }{2}}-X^{-\frac{\alpha }{2}}\right)\left(e^{t\left(\lambda +\rho \right)}\right)}{\prod _{\alpha >0}\left(X^{\frac{\alpha }{2}}-X^{-\frac{\alpha }{2}}\right)\left(e^{t\rho }\right)}\\ & =& \underset{t\to 0}{\lim }\frac{\prod _{\alpha >0}\left(e^{t⟨\lambda +\rho ,\frac{\alpha }{2}⟩}-e^{-t⟨\lambda +\rho ,\frac{\alpha }{2}⟩}\right)}{\prod _{\alpha >0}\left(e^{t⟨\rho ,\frac{\alpha }{2}⟩}-e^{-t⟨\rho ,\frac{\alpha }{2}⟩}\right)}\\ & =& \underset{t\to 0}{\lim }\frac{\sinh \left(t⟨\lambda +\rho ,\frac{\alpha }{2}⟩\right)}{\sinh \left(t⟨\rho ,\frac{\alpha }{2}⟩\right)}\\ & =& \prod _{\alpha >0}\frac{⟨\lambda +\rho ,\frac{\alpha }{2}⟩}{⟨\rho ,\frac{\alpha }{2}⟩}\\ & =& \prod _{\alpha >0}\frac{⟨\lambda +\rho ,{\alpha }^{\vee }⟩}{⟨\rho ,{\alpha }^{\vee }⟩}\end{array}.$$

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 $$\begin{array}{rcl}\text{irreducible representations}& \stackrel{1-1}{\to }& P^{+}\\ V^{\lambda }& \mapsto & \text{highest weight of }{V}^{\lambda }\end{array}$$
2. The character of $V^{\lambda }$ is $${\chi }^{\lambda }=\frac{\sum _{w\in W}\epsilon \left(w\right)e^{w\left(\lambda +\rho \right)}}{\sum _{w\in W}\epsilon \left(w\right)e^{w\rho }}$$ where $\rho \in P^{+}$ is defined by $⟨\rho ,{\alpha }_i^{\vee }⟩=1$ for $1\le i\le n$ and $\epsilon \left(w\right)=\det \left(w\right).$
3. The dimension of $V^{\lambda }$ is $$d_{\lambda }=\frac{\prod _{\alpha >0}⟨\lambda +\rho ,{\alpha }^{\vee }⟩}{\prod _{\alpha >0}⟨\rho ,{\alpha }^{\vee }⟩}.$$
4. $${\chi }^{\lambda }=\sum _{p\in {𝒫}_{\lambda }}{e}^{p\left(1\right)},$$ where ${𝒫}_{\lambda }$ is the set of all paths obtained by acting on $p_{\lambda }$ by root operators.

Remark: By part (d) $$\dim \left({V^{\lambda }}_{\mu }\right)=\text{number of paths in }{𝒫}_{\lambda }$$ which end at $\mu .$ (For the path model some copying can be done from the Barcelona abstract.)

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

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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