Weyl integration formula

Short description: Mathematical 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:

[math]\displaystyle{ \int_G f(g) \, dg = \int_T f(t) u(t) \, dt. }[/math]

Moreover, [math]\displaystyle{ u }[/math] is explicitly given as: [math]\displaystyle{ u = |\delta |^2 / \# W }[/math] where [math]\displaystyle{ W = N_G(T)/T }[/math] is the Weyl group determined by T and

[math]\displaystyle{ \delta(t) = \prod_{\alpha \gt 0} \left( e^{\alpha(t)/2} - e^{-\alpha(t)/2} \right), }[/math]

the product running over the positive roots of G relative to T. More generally, if [math]\displaystyle{ f }[/math] is only a continuous function, then

[math]\displaystyle{ \int_G f(g) \, dg = \int_T \left( \int_G f(gtg^{-1}) \, dg \right) u(t) \, dt. }[/math]

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

[math]\displaystyle{ q : G/T \times T \to G, \, (gT, t) \mapsto gtg^{-1} }[/math].

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

[math]\displaystyle{ nT(gT) = gn^{-1} T. }[/math]

Let [math]\displaystyle{ G/T \times_W T }[/math] be the quotient space by this W-action. Then, since the W-action on [math]\displaystyle{ G/T }[/math] is free, the quotient map

[math]\displaystyle{ p: G/T \times T \to G/T \times_W T }[/math]

is a smooth covering with fiber W when it is restricted to regular points. Now, [math]\displaystyle{ q }[/math] is [math]\displaystyle{ p }[/math] followed by [math]\displaystyle{ G/T \times_W T \to G }[/math] and the latter is a homeomorphism on regular points and so has degree one. Hence, the degree of [math]\displaystyle{ q }[/math] is [math]\displaystyle{ \# W }[/math] and, by the change of variable formula, we get:

[math]\displaystyle{ \# W \int_G f \, dg = \int_{G/T \times T} q^*(f \, dg). }[/math]

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

[math]\displaystyle{ q(gv, t) = gvtv^{-1}g^{-1} }[/math]

and thus, on [math]\displaystyle{ \mathfrak{g}/\mathfrak{t} }[/math], we have:

[math]\displaystyle{ 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). }[/math]

Similarly we see, on [math]\displaystyle{ \mathfrak{t} }[/math], [math]\displaystyle{ d(t \mapsto q(gT, t)) = \operatorname{Ad}(g) }[/math]. Now, we can view G as a connected subgroup of an orthogonal group (as it is compact connected) and thus [math]\displaystyle{ \det(\operatorname{Ad}(g)) = 1 }[/math]. Hence,

[math]\displaystyle{ q^*(dg) = \det(\operatorname{Ad}_{\mathfrak{g}/\mathfrak{t}}(t^{-1}) - I_{\mathfrak{g}/\mathfrak{t}})\, dg. }[/math]

To compute the determinant, we recall that [math]\displaystyle{ \mathfrak{g}_{\mathbb{C}} = \mathfrak{t}_{\mathbb{C}} \oplus \oplus_\alpha \mathfrak{g}_\alpha }[/math] where [math]\displaystyle{ \mathfrak{g}_{\alpha} = \{ x \in \mathfrak{g}_{\mathbb{C}} \mid \operatorname{Ad}(t) x = e^{\alpha(t)} x, t \in T \} }[/math] and each [math]\displaystyle{ \mathfrak{g}_\alpha }[/math] has dimension one. Hence, considering the eigenvalues of [math]\displaystyle{ \operatorname{Ad}_{\mathfrak{g}/\mathfrak{t}}(t^{-1}) }[/math], we get:

[math]\displaystyle{ \det(\operatorname{Ad}_{\mathfrak{g}/\mathfrak{t}}(t^{-1}) - I_{\mathfrak{g}/\mathfrak{t}}) = \prod_{\alpha \gt 0} (e^{-\alpha(t)} - 1)(e^{\alpha(t)} - 1) = \delta(t) \overline{\delta(t)}, }[/math]

as each root [math]\displaystyle{ \alpha }[/math] 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 [math]\displaystyle{ W }[/math] can be identified with a subgroup of [math]\displaystyle{ \operatorname{GL}(\mathfrak{t}_{\mathbb{C}}^*) }[/math]; in particular, it acts on the set of roots, linear functionals on [math]\displaystyle{ \mathfrak{t}_{\mathbb{C}} }[/math]. Let

[math]\displaystyle{ A_{\mu} = \sum_{w \in W} (-1)^{l(w)} e^{w(\mu)} }[/math]

where [math]\displaystyle{ l(w) }[/math] is the length of w. Let [math]\displaystyle{ \Lambda }[/math] be the weight lattice of G relative to T. The Weyl character formula then says that: for each irreducible character [math]\displaystyle{ \chi }[/math] of [math]\displaystyle{ G }[/math], there exists a [math]\displaystyle{ \mu \in \Lambda }[/math] such that

[math]\displaystyle{ \chi|T \cdot \delta = A_{\mu} }[/math].

To see this, we first note

1. [math]\displaystyle{ \|\chi \|^2 = \int_G |\chi|^2 dg = 1. }[/math]
2. [math]\displaystyle{ \chi|T \cdot \delta \in \mathbb{Z}[\Lambda]. }[/math]

The property (1) is precisely (a part of) the orthogonality relations on irreducible characters.

References

• Adams, J. F. (1969), Lectures on Lie Groups, University of Chicago Press 
• Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics 98, Springer-Verlag, Berlin, 1995.