Weights, roots and the Weyl integral 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

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 $$W=N_G\left(T\right)/T,\text{where}{N}_G\left(T\right)=\left\{g\in G|gT{g}^{-1}=T\right\}.$$ The Weyl group $W$ acts on $T$ by conjugation. The map $$\begin{array}{rcl}G/T\times T& \stackrel{\phi }{\to }& G\\ \left(gT,t\right)& \mapsto & \left(gt{g}^{-1}\right)\end{array}$$ 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 $\mathbb{Z},P\cong {\mathbb{Z}}^k.$ The set $P$ is called the weight lattice of $G.$ If $$\lambda \in P\text{then}{X}^{\lambda }:T\to ℂ*,$$ denotes the corresponding irreducible representation of $T.$ The $W$-action on $T$ induces a $W$-action on $P$ via $$X^{w\lambda }\left(t\right)=X^{\lambda }\left(w^{-1}t\right),\text{for all}t\in T.$$ A representation $V$ of $G$ is a representation of $T,$ by restriction, and, as a $T$-module, $$V=\underset{\lambda \in P}{\oplus }{V}_{\lambda },\text{where}{V}_{\lambda }=\left\{v\in V|tv=X^{\lambda }\left(t\right)v\text{ for all }t\in T.\right\}$$

The vector space $V_{\lambda }$ is the $X^{\lambda }$ isotypic component of the $T$-module $V.$ The $W$-action on $T$ gives $$\dim \left(V_{\lambda }\right)=\dim \left(V_{w\lambda }\right),\text{for all}w\in W\text{ and }\lambda \in P.$$ 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 $${𝔤}_{ℂ}=𝔲\oplus i𝔲=ℂ\otimes ℝ𝔲.$$ 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 $${𝔤}_{ℂ}=\underset{\alpha \in P}{\oplus }{𝔤}_{\alpha },$$ and the root system of $G$ is the set $$R=\left\{\alpha \in P|\alpha \ne 0,{𝔤}_{\alpha }\ne 0\right\}$$ of nonzero weights of the adjoint representation. The roots are the elements of $R.$ Set $𝔥={𝔤}_0.$ Then $${𝔤}_{ℂ}=𝔥\oplus \left(\underset{\alpha \in R}{\oplus }{𝔤}_{\alpha }\right)$$ 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 $$\begin{array}{rcll}X:& T& \to & ℂ*\\ & t& \mapsto & X^{\lambda }\left(t\right)\\ & e^h& \mapsto & e^{\lambda \left(h\right)}\end{array}$$ and $$\begin{array}{rcll}\lambda :& 𝔥& \to & ℂ\\ & h& \mapsto & \lambda \left(h\right)\end{array}$$

The Weyl group $W$ is generated by $s_{\alpha },\alpha \in R.$ The action of $W$ on $𝔥*$ is generated by the transformations $$\begin{array}{rcll}{s}_{\alpha }:& 𝔥*& \to & 𝔥*\\ & \lambda & \mapsto & \lambda -⟨\lambda ,{\alpha }^{\vee }⟩\end{array}\text{where}{\alpha }^{\vee }=\frac{2\alpha }{⟨\alpha ,\alpha ⟩},$$ and $⟨,⟩:𝔥*\times 𝔥*\to ℝ$ 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 $\dim \left({𝔤}_{\alpha }\right)=1.$
3. The only connected compact Lie groups with $\dim \left(T\right)=1$ 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 $x\in {𝔤}_{\alpha }.$ $$\begin{array}{rcll}{X}^{\alpha }:& T& \to & ℂ*\\ & e^h& \mapsto & e^{\alpha \left(h\right)}\end{array}$$ and $$\begin{array}{rcll}{X}^{-\alpha }:& T& \to & ℂ*\\ & e^h& \mapsto & \end{array}$$ e α h = e -α h since $\alpha \left(h\right)\in iℝ$ for $h\in 𝔱.$ Then, for all $h\in 𝔱,$ $$\left[h\stackrel{-}{x}\right]=\left[\stackrel{-}{h},\stackrel{-}{x}\right]=$$ 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 $T_{\alpha }=\ker X^{\alpha }$ 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 $$ℂ{\otimes }_{ℝ}\mathrm{Lie}\left(Z_{\alpha }\right)=𝔱\oplus i𝔱\oplus \left({\oplus }_{h\in T_{\alpha },\beta \left(h\right)=1}{𝔤}_{\beta }\right)=𝔥\oplus \underset{k\in \mathbb{Z}}{\oplus }{𝔤}_{k\alpha .}$$

Now $$\begin{array}{rcl}{Z}_{\alpha }& \to & Z_{\alpha }/T_{\alpha }^{\circ }\\ \cup |& & \cup |\\ T& \to & T/T_{\alpha }^{\circ }\end{array}$$ So $T/T_{\alpha }^{\circ }$ is a maximal torus of $Z_{\alpha }/T_{\alpha }^{\circ }$ and $\dim T/T_{\alpha }^{\circ }=1.$ Then $$ℂ{\otimes }_{ℝ}\mathrm{Lie}\left(Z_{\alpha }\right)={𝔥}_{\alpha }\oplus ℂH_{\alpha }\oplus \left(\underset{k\in \mathbb{Z}}{\oplus }{𝔤}_{k\alpha }\right).$$

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 $$\mathrm{Lie}\left(Z_{\alpha }/T_{\alpha }^{\circ }\right)=ℂ\oplus \left(\underset{k\in \mathbb{Z}}{\oplus }{𝔤}_{k\alpha }\right).$$ Now consider the action of $H_{\alpha }$ on $$ℂH\oplus \left(\underset{k\in {\mathbb{Z}}_{>0}}{\oplus }{𝔤}_{k\alpha }\right)\oplus ℂX_{\alpha }.$$ Then $$\mathrm{Tr}\left(H\right)=\frac{1}{\lambda }\mathrm{Tr}\left(\left[X_{\alpha },X_{-\alpha }\right]\right)=\frac{1}{\lambda }\left({\mathrm{ad}}_{X_{\alpha }}{\mathrm{ad}}_{X_{-\alpha }}-{\mathrm{ad}}_{X_{-\alpha }}{\mathrm{ad}}_{X_{\alpha }}\right)=0.$$ But this implies $$0=0+\sum _{k\in {\mathbb{Z}}_{>0}}\dim \left({𝔤}_{k\alpha }\right)k\alpha \left(H_{\alpha }\right)-\alpha \left(H_{\alpha }\right).$$ So ${𝔤}_{k\alpha }=0$ for $k>1$ and ${𝔤}_{\alpha }=ℂX_{\alpha }.$ 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 $\dim T=1$ then $U$ has Lie algebra $$𝔤=\mathrm{span}\left\{X_{\alpha },X_{-\alpha },H_{\alpha }\right\}=𝔲\oplus i𝔲.$$ 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, $$\mathrm{SO}\left(3\right)\text{and}\mathrm{Spin}\left(3\right).$$

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 $$\left|W\right|{\int }_{G}f\left(x\right)dx={\int }_T\prod _{\alpha \in R}\left(X^{\alpha }\left(t\right)-1\right){\int }_{G}f\left(gt{g}^{-1}\right)dgdt.$$

Proof First note that the map $G/T\times T\to G$ given by $\left(gT,t\right)\mapsto gt,$ can be used to define a (left) $G$ invariant measure on $G/T$ so that $${\int }_{G}f\left(g\right)dg={\int }_{G/T\times T}f\left(gt\right)dtd\left(gT\right),$$ and thus, for $y\in T,$ $$\begin{array}{rcl}{\int }_{G}f\left(gy{g}^{-1}\right)dg& =& {\int }_{G/T\times T}f\left(gty{t}^{-1}{g}^{-1}\right)dtd\left(gT\right)\\ & =& {\int }_{G/T}f\left(gy{g}^{-1}\right)dtd\left(gT\right)={\int }_{G/T\times T}f\left(gy{g}^{-1}\right)d\left(gT\right).\end{array}$$

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

$$\left|W\right|{\int }_{G}f\left(g\right)dg={\int }_{G/T\times T}f\left(gt{g}^{-1}\right)J_{\left(gT,t\right)}dtd\left(gT\right),$$

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},$$$\begin{array}{rclllll}G/T\times T& \to & G/T\times T& \to & G& \to & G\\ \left(xT,y\right)& \mapsto & \left(gxT,ty\right)& \mapsto & \left(gx\right)ty{\left(gx\right)}^{-1}& \mapsto & \left(g{t}^{-1}{g}^{-1}\right)\left(gx\right)ty{\left(gx\right)}^{-1}.\end{array}$$ 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 $$\begin{array}{rcl}𝔤/𝔥& \to & 𝔤\\ \left(X,Y\right)& \mapsto & {\mathrm{Ad}}_g\left({\mathrm{Ad}}_{t^{-1}}\left(\right)+Y-X\right).\end{array}$$ So $J_{\left(gT,t\right)}$ is the determinant of the linear transformation of $𝔤$ given by $${\mathrm{Ad}}_{𝔤}\left(g\right)\left(\begin{array}{rc}{\mathrm{Ad}}_{𝔤/𝔥}\left(t^{-1}\right)-{\mathrm{id}}_{𝔤/𝔥}& 0\\ 0& {\mathrm{id}}_{𝔥}\end{array}\right),$$ where the second factor is a block $2\times 2$ 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, $\det \left({\mathrm{Ad}}_g\right)=1,$ and since $𝔤/𝔥=\underset{\alpha \in R}{\oplus }{𝔤}_{\alpha },$

$$J_{\left(gT,t\right)}=\prod _{\alpha \in R}\left(X^{\alpha }\left(t^{-1}\right)-1\right)=\prod _{\alpha \in R}\left(X^{\alpha }\left(t\right)-1\right),$$

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

$$\begin{array}{c}{\int }_G\chi \left(g\right)\end{array}$$ 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.

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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