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:
 * $$\int_G f(g) \, dg = \int_T f(t) u(t) \, dt.$$

Moreover, $$u$$ is explicitly given as: $$u = |\delta |^2 / \# W$$ where $$W = N_G(T)/T$$ is the Weyl group determined by T and
 * $$\delta(t) = \prod_{\alpha > 0} \left( e^{\alpha(t)/2} - e^{-\alpha(t)/2} \right),$$

the product running over the positive roots of G relative to T. More generally, if $$f$$ is only a continuous function, then
 * $$\int_G f(g) \, dg = \int_T \left( \int_G f(gtg^{-1}) \, dg \right) u(t) \, dt.$$

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
 * $$q : G/T \times T \to G, \, (gT, t) \mapsto gtg^{-1}$$.

The Weyl group W acts on T by conjugation and on $$G/T$$ from the left by: for $$nT \in W$$,
 * $$nT(gT) = gn^{-1} T.$$

Let $$G/T \times_W T$$ be the quotient space by this W-action. Then, since the W-action on $$G/T$$ is free, the quotient map
 * $$p: G/T \times T \to G/T \times_W T$$

is a smooth covering with fiber W when it is restricted to regular points. Now, $$q$$ is $$p$$ followed by $$G/T \times_W T \to G$$ and the latter is a homeomorphism on regular points and so has degree one. Hence, the degree of $$q$$ is $$\# W$$ and, by the change of variable formula, we get:
 * $$\# W \int_G f \, dg = \int_{G/T \times T} q^*(f \, dg).$$

Here, $$q^*(f \, dg)|_{(gT, t)} = f(t) q^*(dg)|_{(gT, t)}$$ since $$f$$ is a class function. We next compute $$q^*(dg)|_{(gT, t)}$$. We identify a tangent space to $$G/T \times T$$ as $$\mathfrak{g}/\mathfrak{t} \oplus \mathfrak{t}$$ where $$\mathfrak{g}, \mathfrak{t}$$ are the Lie algebras of $$G, T$$. For each $$v \in T$$,
 * $$q(gv, t) = gvtv^{-1}g^{-1}$$

and thus, on $$\mathfrak{g}/\mathfrak{t}$$, we have:
 * $$d(gT \mapsto q(gT, t))(\dot v) = gtg^{-1}(gt^{-1} \dot v t g^{-1} - g \dot v g^{-1}) = (\operatorname{Ad}(g) \circ (\operatorname{Ad}(t^{-1}) - I))(\dot v).$$

Similarly we see, on $$\mathfrak{t}$$, $$d(t \mapsto q(gT, t)) = \operatorname{Ad}(g)$$. Now, we can view G as a connected subgroup of an orthogonal group (as it is compact connected) and thus $$\det(\operatorname{Ad}(g)) = 1$$. Hence,
 * $$q^*(dg) = \det(\operatorname{Ad}_{\mathfrak{g}/\mathfrak{t}}(t^{-1}) - I_{\mathfrak{g}/\mathfrak{t}})\, dg.$$

To compute the determinant, we recall that $$\mathfrak{g}_{\mathbb{C}} = \mathfrak{t}_{\mathbb{C}} \oplus \oplus_\alpha \mathfrak{g}_\alpha$$ where $$\mathfrak{g}_{\alpha} = \{ x \in \mathfrak{g}_{\mathbb{C}} \mid \operatorname{Ad}(t) x = e^{\alpha(t)} x, t \in T \}$$ and each $$\mathfrak{g}_\alpha$$ has dimension one. Hence, considering the eigenvalues of $$\operatorname{Ad}_{\mathfrak{g}/\mathfrak{t}}(t^{-1})$$, we get:
 * $$\det(\operatorname{Ad}_{\mathfrak{g}/\mathfrak{t}}(t^{-1}) - I_{\mathfrak{g}/\mathfrak{t}}) = \prod_{\alpha > 0} (e^{-\alpha(t)} - 1)(e^{\alpha(t)} - 1) = \delta(t) \overline{\delta(t)},$$

as each root $$\alpha$$ has pure imaginary value.

Weyl character formula
The Weyl character formula is a consequence of the Weyl integral formula as follows. We first note that $$W$$ can be identified with a subgroup of $$\operatorname{GL}(\mathfrak{t}_{\mathbb{C}}^*)$$; in particular, it acts on the set of roots, linear functionals on $$\mathfrak{t}_{\mathbb{C}}$$. Let
 * $$A_{\mu} = \sum_{w \in W} (-1)^{l(w)} e^{w(\mu)}$$

where $$l(w)$$ is the length of w. Let $$\Lambda$$ be the weight lattice of G relative to T. The Weyl character formula then says that: for each irreducible character $$\chi$$ of $$G$$, there exists a $$\mu \in \Lambda$$ such that
 * $$\chi|T \cdot \delta = A_{\mu}$$.

To see this, we first note The property (1) is precisely (a part of) the orthogonality relations on irreducible characters.
 * 1) $$\|\chi \|^2 = \int_G |\chi|^2 dg = 1.$$
 * 2) $$\chi|T \cdot \delta \in \mathbb{Z}[\Lambda].$$