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
:$\delta(t) = \prod_ \left( e^ - e^ \right),$
the product running over the positive roots of ''G'' relative to ''T''. More generally, if $f$ is only a continuous function, then
:$\int_G f(g) \, dg = \int_T \left( \int_G f(gtg^) \, dg \right) u(t) \, dt.$

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
:$q : G/T \times T \to G, \, (gT, t) \mapsto gtg^$.
The Weyl group ''W'' acts on ''T'' by conjugation and on $G/T$ from the left by: for $nT \in W$,
:$nT(gT) = gn^ T.$
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:
:$\# W \int_G f \, dg = \int_ q^*(f \, dg).$
Here, $q^*(f \, dg), _ = f(t) q^*(dg), _$ since $f$ is a class function. We next compute $q^*(dg), _$. We identify a tangent space to $G/T \times T$ as $\mathfrak/\mathfrak \oplus \mathfrak$ where $\mathfrak, \mathfrak$ are the Lie algebras of $G, T$. For each $v \in T$,
:$q(gv, t) = gvtv^g^$
and thus, on $\mathfrak/\mathfrak$, we have:
:$d(gT \mapsto q(gT, t))(\dot v) = gtg^(gt^ \dot v t g^ - g \dot v g^) = (\operatorname(g) \circ (\operatorname(t^) - I))(\dot v).$
Similarly we see, on $\mathfrak$, $d(t \mapsto q(gT, t)) = \operatorname(g)$. Now, we can view ''G'' as a connected subgroup of an orthogonal group (as it is compact connected) and thus $\det(\operatorname(g)) = 1$. Hence,
:$q^*(dg) = \det(\operatorname_(t^) - I_)\, dg.$
To compute the determinant, we recall that $\mathfrak_ = \mathfrak_ \oplus \oplus_\alpha \mathfrak_\alpha$ where $\mathfrak_ = \$ and each $\mathfrak_\alpha$ has dimension one. Hence, considering the eigenvalues of $\operatorname_(t^)$, we get:
:$\det(\operatorname_(t^) - I_) = \prod_ (e^ - 1)(e^ - 1) = \delta(t) \overline,$
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 $\operatorname(\mathfrak_^*)$; in particular, it acts on the set of roots, linear functionals on $\mathfrak_$. Let
:$A_ = \sum_ (-1)^ e^$
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
:$\chi, T \cdot \delta = A_$.
To see this, we first note
# $\, \chi \, ^2 = \int_G , \chi, ^2 dg = 1.$
# \chi, T \cdot \delta \in \mathbb Lambda
The property (1) is precisely (a part of) the orthogonality relations on irreducible characters.

References

*{{citation, last=Adams, first= J. F., title= Lectures on Lie Groups, publisher=University of Chicago Press, year= 1969
*Theodor Bröcker and Tammo tom Dieck, ''Representations of compact Lie groups'', Graduate Texts in Mathematics 98, Springer-Verlag, Berlin, 1995.

Differential geometry