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 {\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 .