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 {\displaystyle G} ):
∫ G f ( g ) d g = ∫ T f ( t ) u ( t ) d t . {\displaystyle \int _{G}f(g)\,dg=\int _{T}f(t)u(t)\,dt.}
Moreover, u {\displaystyle u} is explicitly given as: u = | δ | 2 / # W {\displaystyle u=|\delta |^{2}/\#W} where W = N G ( T ) / T {\displaystyle W=N_{G}(T)/T} is the Weyl group determined by T and
δ ( t ) = ∏ α > 0 ( e α ( t ) / 2 − e − α ( t ) / 2 ) , {\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 f {\displaystyle f} is an arbitrary integrable function, then
∫ G f ( g ) d g = ∫ T ( ∫ G / T f ( g t g − 1 ) d ( g T ) ) u ( t ) d t . {\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 G / T {\displaystyle G/T} , the quotient of the group G {\displaystyle G} over the maximal torus T {\displaystyle T} , and d ( g T ) {\displaystyle d(gT)} is some Borel measure on this manifold. 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 × T → G , ( g T , t ) ↦ g t g − 1 {\displaystyle 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 {\displaystyle G/T} from the left by: for n T ∈ W {\displaystyle nT\in W} ,
n T ( g T ) = g n − 1 T . {\displaystyle nT(gT)=gn^{-1}T.}
Let G / T × W T {\displaystyle G/T\times _{W}T} be the quotient space by this W-action. Then, since the W-action on G / T {\displaystyle G/T} is free, the quotient map
p : G / T × T → G / T × W T {\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, q {\displaystyle q} is p {\displaystyle p} followed by G / T × W T → G {\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 q {\displaystyle q} is # W {\displaystyle \#W} and, by the change of variable formula, we get:
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