Weyl integration formula explained

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} \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 x 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 x _{W} T be the quotient space by this W-action. Then, since the W-action on G/T is free, the quotient mapp: G/T x T \to G/T x _{W} T is a smooth covering with fiber W when it is restricted to regular points. Now, q is p followed by G/T x _{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} q ^{*(f} dg). Here, q ^{*(f} dg)| _{(gT,} = 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 x T as ak{g}/ ak{t} ⊕ ak{t} where ak{g}, ak{t} are the Lie algebras of G, T . For each v \in T ,q(gv, t) = gvtv ^{-1} g ^{-1} and thus, on ak{g}/ ak{t} , we have:d(gT \mapsto q(gT, t))( • v) = gtg ^{-1} (gt ^{-1} • v t g ^{-1} - g • v g ^{-1} ) = (\operatorname{Ad}(g) \circ (\operatorname{Ad}(t ^{-1} ) - I))( • v). Similarly we see, on ak{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} _{ak{g} / ak{t}}(t ^{-1} ) - I _{ak{g} / ak{t}}) dg. To compute the determinant, we recall that ak{g} _{C} = ak{t} _{C} ⊕ ⊕ _{\alpha} ak{g} _{\alpha} where ak{g} _{\alpha} = \{ x \in ak{g} _{C} \mid \operatorname{Ad}(t) x = e ^{\alpha(t)} x, t \in T \} and each ak{g} _{\alpha} has dimension one. Hence, considering the eigenvalues of \operatorname{Ad} _{ak{g} / ak{t}}(t ^{-1} ) , we get:\det(\operatorname{Ad} _{ak{g} / ak{t}}(t ^{-1} ) - I _{ak{g} / ak{t}}) = \prod _{\alpha} (e ^{-\alpha(t)} - 1 )(e ^{\alpha(t)} - 1 ) = \delta(t) \overline{\delta(t)}, as each root \alpha has pure imaginary value.

Weyl character formula

See 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 \operatorname{GL}( ak{t} ^{*)} _{C} ; in particular, it acts on the set of roots, linear functionals on ak{t} _{C} . LetA _{\mu} = \sum _{w} ( -1 ) ^{l(w)} e ^{w(\mu)} where l(w) is the length of w. Let Λ 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 Λ such that\chi|T ⋅ \delta = A _{\mu} .To see this, we first note\|\chi \| ^{2} = \int _{G} |\chi| ^{2} dg = 1 . \chi|T ⋅ \delta \in Z [ Λ ]. The property (1) is precisely (a part of) the orthogonality relations on irreducible characters.

References

• • Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics 98, Springer-Verlag, Berlin, 1995.

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