Weyl integration formula

In mathematics, the Weyl integration formula, introduced by Hermann Weyl, is an integration formula for a compact connected Lie group G in terms of a maximal torus T. Precisely, it says there exists a real-valued continuous function u on T such that for every class function f on G (function invariant under conjugation by $G$):

$\int _{G}f(g)\,dg=\int _{T}f(t)u(t)\,dt.$

Moreover, $u$ is explicitly given as: $u=|\delta |^{2}/\#W$ where $W=N_{G}(T)/T$ is the Weyl group determined by T and

$\delta (t)=\prod _{\alpha >0}\left(e^{\alpha (t)/2}-e^{-\alpha (t)/2}\right),$

the product running over the positive roots of G relative to T. More generally, if $f$ is an arbitrary integrable function, then

$\int _{G}f(g)\,dg=\int _{T}\left(\int _{G}f(gtg^{-1})\,dg\right)u(t)\,dt.$

The formula can be used to derive the Weyl character formula. (The theory of Verma modules, on the other hand, gives a purely algebraic derivation of the Weyl character formula.)

Derivation

Consider the map

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

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

$nT(gT)=gn^{-1}T.$

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

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

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

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

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

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

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

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

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

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

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

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

as each root $\alpha$ has pure imaginary value.

Weyl character formula

Main article: Weyl character formula

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

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

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

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

To see this, we first note

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

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

References

• 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.