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 ${\displaystyle G}$):

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

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

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

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

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

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: