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 says¹ there exists a real-valued continuous function u on T such that for every class function f on G (function invariant under conjugation by ${\displaystyle G}$):

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

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

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

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

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

Note that the inner integral is over the manifold ${\displaystyle G/T}$, the quotient of the group ${\displaystyle G}$ over the maximal torus ${\displaystyle T}$, and ${\displaystyle d(gT)}$ is some Borel measure on this manifold.²

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.)

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 ${\displaystyle {\mathfrak {g}}_{\mathbb {C} }={\mathfrak {t}}_{\mathbb {C} }\oplus \bigoplus _{\alpha }{\mathfrak {g}}_{\alpha }}$ 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.