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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:
Moreover, u {\displaystyle u} is explicitly given as: u = | δ | 2 / # W {\displaystyle u=|\delta |^{2}/\#W} where W = N G / T {\displaystyle W=N_{G}/T} is the Weyl group determined by T and
the product running over the positive roots of G relative to T. More generally, if f {\displaystyle f} is only a continuous function, then
The formula can be used to derive the Weyl character formula.