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 (function invariant under conjugation by $G$):

$\underset{G}{\int }f(g)dg=\underset{T}{\int }f(t)u(t)dt.$

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

$\delta (t)=\prod _{\alpha >0}(e^{\alpha (t)/2}-e^{-\alpha (t)/2}),$

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

$\underset{G}{\int }f(g)dg=\underset{T}{\int }(\underset{G/T}{\int }f(gt{g}^{-1})d(gT))u(t)dt.$

Note that the inner integral is over the manifold $G/T$, the quotient of the group $G$ over the maximal torus $T$, and $d(gT)$ is some Borel measure on this manifold.

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

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

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

$nT(gT)=g{n}^{-1}T.$

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 $\mathrm{\#}W$ and, by the change of variable formula, we get:

$\mathrm{\#}W\underset{G}{\int }fdg=\underset{G/T\times T}{\int }{q}^{*}(fdg).$

Here, $q^{*}(fdg){|}_{(gT,t)}=f(t)q^{*}(dg){|}_{(gT,t)}$ 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 $v\in T$,

$q(gv,t)=gvt{v}^{-1}{g}^{-1}$

and thus, on $\mathfrak{𝔤}/\mathfrak{𝔱}$, we have:

$d(gT\mapsto q(gT,t))(\dot{v})=gt{g}^{-1}(g{t}^{-1}\dot{v}t{g}^{-1}-g\dot{v}{g}^{-1})=(\mathrm{Ad}(g)\circ (\mathrm{Ad}(t^{-1})-I))(\dot{v}).$

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

$q^{*}(dg)=\det ({\mathrm{Ad}}_{\mathfrak{𝔤}/\mathfrak{𝔱}}(t^{-1})-I_{\mathfrak{𝔤}/\mathfrak{𝔱}})dg.$

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

$\det ({\mathrm{Ad}}_{\mathfrak{𝔤}/\mathfrak{𝔱}}(t^{-1})-I_{\mathfrak{𝔤}/\mathfrak{𝔱}})=\prod _{\alpha >0}(e^{-\alpha (t)}-1)(e^{\alpha (t)}-1)=\delta (t)\overline{\delta (t)},$

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

Weyl character formula

Main article: 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 $\mathrm{GL}({\mathfrak{𝔱}}_{\mathbb{ℂ}}^{*})$; 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 $\mathrm{\Lambda }$ 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 $\mu \in \mathrm{\Lambda }$ such that

$\chi |T\cdot \delta =A_{\mu }$.

To see this, we first note

1. $\Vert \chi {\Vert }^2=\underset{G}{\int }|\chi {|}^{2}dg=1.$
2. $\chi |T\cdot \delta \in \mathbb{\mathbb{Z}}[\mathrm{\Lambda }].$

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

References

• Zhang, L. (2014). Matrix integrals over unitary groups: An application of Schur-Weyl duality. arXiv preprint arXiv:1408.3782.
• Adams, J. F. (1982), Lectures on Lie Groups, University of Chicago Press, ISBN 978-0-226-00530-0, https://books.google.com/books?id=TC7d3ZcqjfsC&pg=PA142 
• Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics 98, Springer-Verlag, Berlin, 1995.