Consider the map

${\displaystyle q:G/T\times T\to G,\,(gT,t)\mapsto gtg^{-1}}$ .

The Weyl group W acts on T by conjugation and on ${\displaystyle G/T}$  from the left by: for ${\displaystyle nT\in W}$ ,

${\displaystyle nT(gT)=gn^{-1}T.}$ 

Let ${\displaystyle G/T\times _{W}T}$  be the quotient space by this W-action. Then, since the W-action on ${\displaystyle G/T}$  is free, the quotient map

${\displaystyle p:G/T\times T\to G/T\times _{W}T}$ 

is a smooth covering with fiber W when it is restricted to regular points. Now, ${\displaystyle q}$  is ${\displaystyle p}$  followed by ${\displaystyle G/T\times _{W}T\to G}$  and the latter is a homeomorphism on regular points and so has degree one. Hence, the degree of ${\displaystyle q}$  is ${\displaystyle \#W}$  and, by the change of variable formula, we get:

${\displaystyle \#W\int _{G}f\,dg=\int _{G/T\times T}q^{*}(f\,dg).}$ 

Here, ${\displaystyle q^{*}(f\,dg)|_{(gT,t)}=f(t)q^{*}(dg)|_{(gT,t)}}$  since ${\displaystyle f}$  is a class function. We next compute ${\displaystyle q^{*}(dg)|_{(gT,t)}}$ . We identify a tangent space to ${\displaystyle G/T\times T}$  as ${\displaystyle {\mathfrak {g}}/{\mathfrak {t}}\oplus {\mathfrak {t}}}$  where ${\displaystyle {\mathfrak {g}},{\mathfrak {t}}}$  are the Lie algebras of ${\displaystyle G,T}$ . For each ${\displaystyle v\in T}$ ,

${\displaystyle q(gv,t)=gvtv^{-1}g^{-1}}$ 

and thus, on ${\displaystyle {\mathfrak {g}}/{\mathfrak {t}}}$ , we have:

${\displaystyle d(gT\mapsto q(gT,t))({\dot {v}})=gtg^{-1}(gt^{-1}{\dot {v}}tg^{-1}-g{\dot {v}}g^{-1})=(\operatorname {Ad} (g)\circ (\operatorname {Ad} (t^{-1})-I))({\dot {v}}).}$ 

Similarly we see, on ${\displaystyle {\mathfrak {t}}}$ , ${\displaystyle d(t\mapsto q(gT,t))=\operatorname {Ad} (g)}$ . Now, we can view G as a connected subgroup of an orthogonal group (as it is compact connected) and thus ${\displaystyle \det(\operatorname {Ad} (g))=1}$ . Hence,

${\displaystyle q^{*}(dg)=\det(\operatorname {Ad} _{{\mathfrak {g}}/{\mathfrak {t}}}(t^{-1})-I_{{\mathfrak {g}}/{\mathfrak {t}}})\,dg.}$ 

To compute the determinant, we recall that ${\displaystyle {\mathfrak {g}}_{\mathbb {C} }={\mathfrak {t}}_{\mathbb {C} }\oplus \oplus _{\alpha }{\mathfrak {g}}_{\alpha }}$  where ${\displaystyle {\mathfrak {g}}_{\alpha }=\{x\in {\mathfrak {g}}_{\mathbb {C} }\mid \operatorname {Ad} (t)x=e^{\alpha (t)}x,t\in T\}}$  and each ${\displaystyle {\mathfrak {g}}_{\alpha }}$  has dimension one. Hence, considering the eigenvalues of ${\displaystyle \operatorname {Ad} _{{\mathfrak {g}}/{\mathfrak {t}}}(t^{-1})}$ , we get:

${\displaystyle \det(\operatorname {Ad} _{{\mathfrak {g}}/{\mathfrak {t}}}(t^{-1})-I_{{\mathfrak {g}}/{\mathfrak {t}}})=\prod _{\alpha >0}(e^{-\alpha (t)}-1)(e^{\alpha (t)}-1)=\delta (t){\overline {\delta (t)}},}$ 

as each root ${\displaystyle \alpha }$  has pure imaginary value.

The Weyl character formula is a consequence of the Weyl integral formula as follows. We first note that ${\displaystyle W}$  can be identified with a subgroup of ${\displaystyle \operatorname {GL} ({\mathfrak {t}}_{\mathbb {C} }^{*})}$ ; in particular, it acts on the set of roots, linear functionals on ${\displaystyle {\mathfrak {t}}_{\mathbb {C} }}$ . Let

${\displaystyle A_{\mu }=\sum _{w\in W}(-1)^{l(w)}e^{w(\mu )}}$ 

where ${\displaystyle l(w)}$  is the length of w. Let ${\displaystyle \Lambda }$  be the weight lattice of G relative to T. The Weyl character formula then says that: for each irreducible character ${\displaystyle \chi }$  of ${\displaystyle G}$ , there exists a ${\displaystyle \mu \in \Lambda }$  such that

${\displaystyle \chi |T\cdot \delta =A_{\mu }}$ .

To see this, we first note

1. ${\displaystyle \|\chi \|^{2}=\int _{G}|\chi |^{2}dg=1.}$ 
2. ${\displaystyle \chi |T\cdot \delta \in \mathbb {Z} [\Lambda ].}$ 

The property (1) is precisely (a part of) the orthogonality relations on irreducible characters.

• Adams, J. F. (1982), Lectures on Lie Groups, University of Chicago Press, ISBN 978-0-226-00530-0
• Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics 98, Springer-Verlag, Berlin, 1995.