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{{harvnb|Adams|1982|loc=Theorem 6.1.}} there exists a real-valued continuous function u on T such that for every class function f on G (function invariant under conjugation by $G$ ):

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

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 an arbitrary integrable 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 $\mathfrak{g}_{\mathbb{C}} = \mathfrak{t}_{\mathbb{C}} \oplus \bigoplus_\alpha \mathfrak{g}_\alpha$ 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

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

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

References

{{citation |url={{Google books|TC7d3ZcqjfsC|page=142|plainurl=yes}}| title=Lectures on Lie Groups | isbn=978-0-226-00530-0 | last1=Adams | first1=J. F. | date=1982 | publisher=University of Chicago Press }}
Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics 98, Springer-Verlag, Berlin, 1995.

Category:Differential geometry