Weyl integration formula

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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[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 {\displaystyle G} {\displaystyle G}):

∫ G f ( g ) d g = ∫ T f ( t ) u ( t ) d t . {\displaystyle \int _{G}f(g)\,dg=\int _{T}f(t)u(t)\,dt.} {\displaystyle \int _{G}f(g)\,dg=\int _{T}f(t)u(t)\,dt.}

Moreover, u {\displaystyle u} {\displaystyle u} is explicitly given as: u = | δ | 2 / # W {\displaystyle u=|\delta |^{2}/\#W} {\displaystyle u=|\delta |^{2}/\#W} where W = N G ( T ) / T {\displaystyle W=N_{G}(T)/T} {\displaystyle W=N_{G}(T)/T} is the Weyl group determined by T and

δ ( t ) = ∏ α > 0 ( e α ( t ) / 2 − e − α ( t ) / 2 ) , {\displaystyle \delta (t)=\prod _{\alpha >0}\left(e^{\alpha (t)/2}-e^{-\alpha (t)/2}\right),} {\displaystyle \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 {\displaystyle f} {\displaystyle f} is an arbitrary integrable function, then

∫ G f ( g ) d g = ∫ T ( ∫ G / T f ( g t g − 1 ) d ( g T ) ) u ( t ) d t . {\displaystyle \int _{G}f(g)\,dg=\int _{T}\left(\int _{G/T}f(gtg^{-1})\,d(gT)\right)u(t)\,dt.} {\displaystyle \int _{G}f(g)\,dg=\int _{T}\left(\int _{G/T}f(gtg^{-1})\,d(gT)\right)u(t)\,dt.}

Note that the inner integral is over the manifold G / T {\displaystyle G/T} {\displaystyle G/T}, the quotient of the group G {\displaystyle G} {\displaystyle G} over the maximal torus T {\displaystyle T} {\displaystyle T}, and d ( g T ) {\displaystyle d(gT)} {\displaystyle d(gT)} is some Borel measure on this manifold.[2]

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 × T → G , ( g T , t ) ↦ g t g − 1 {\displaystyle q:G/T\times T\to G,\,(gT,t)\mapsto gtg^{-1}} {\displaystyle 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 {\displaystyle G/T} {\displaystyle G/T} from the left by: for n T ∈ W {\displaystyle nT\in W} {\displaystyle nT\in W},

n T ( g T ) = g n − 1 T . {\displaystyle nT(gT)=gn^{-1}T.} {\displaystyle nT(gT)=gn^{-1}T.}

Let G / T × W T {\displaystyle G/T\times _{W}T} {\displaystyle G/T\times _{W}T} be the quotient space by this W-action. Then, since the W-action on G / T {\displaystyle G/T} {\displaystyle G/T} is free, the quotient map

p : G / T × T → G / T × W T {\displaystyle p:G/T\times T\to G/T\times _{W}T} {\displaystyle 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 {\displaystyle q} {\displaystyle q} is p {\displaystyle p} {\displaystyle p} followed by G / T × W T → G {\displaystyle G/T\times _{W}T\to G} {\displaystyle 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 {\displaystyle q} {\displaystyle q} is # W {\displaystyle \#W} {\displaystyle \#W} and, by the change of variable formula, we get:

# W ∫ G f d g = ∫ G / T × T q ∗ ( f d g ) . {\displaystyle \#W\int _{G}f\,dg=\int _{G/T\times T}q^{*}(f\,dg).} {\displaystyle \#W\int _{G}f\,dg=\int _{G/T\times T}q^{*}(f\,dg).}

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

q ( g v , t ) = g v t v − 1 g − 1 {\displaystyle q(gv,t)=gvtv^{-1}g^{-1}} {\displaystyle q(gv,t)=gvtv^{-1}g^{-1}}

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

d ( g T ↦ q ( g T , t ) ) ( v ˙ ) = g t g − 1 ( g t − 1 v ˙ t g − 1 − g v ˙ g − 1 ) = ( Ad ⁡ ( g ) ∘ ( Ad ⁡ ( t − 1 ) − I ) ) ( v ˙ ) . {\displaystyle d(gT\mapsto q(gT,t))({\dot {v}})=gtg^{-1}(gt^{-1}{\dot {v}}tg^{-1}-g{\dot {v}}g^{-1})=(\operatorname {Ad} (g)\circ (\operatorname {Ad} (t^{-1})-I))({\dot {v}}).} {\displaystyle d(gT\mapsto q(gT,t))({\dot {v}})=gtg^{-1}(gt^{-1}{\dot {v}}tg^{-1}-g{\dot {v}}g^{-1})=(\operatorname {Ad} (g)\circ (\operatorname {Ad} (t^{-1})-I))({\dot {v}}).}

Similarly we see, on t {\displaystyle {\mathfrak {t}}} {\displaystyle {\mathfrak {t}}}, d ( t ↦ q ( g T , t ) ) = Ad ⁡ ( g ) {\displaystyle d(t\mapsto q(gT,t))=\operatorname {Ad} (g)} {\displaystyle 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 ( Ad ⁡ ( g ) ) = 1 {\displaystyle \det(\operatorname {Ad} (g))=1} {\displaystyle \det(\operatorname {Ad} (g))=1}. Hence,

q ∗ ( d g ) = det ( Ad g / t ⁡ ( t − 1 ) − I g / t ) d g . {\displaystyle q^{*}(dg)=\det(\operatorname {Ad} _{{\mathfrak {g}}/{\mathfrak {t}}}(t^{-1})-I_{{\mathfrak {g}}/{\mathfrak {t}}})\,dg.} {\displaystyle q^{*}(dg)=\det(\operatorname {Ad} _{{\mathfrak {g}}/{\mathfrak {t}}}(t^{-1})-I_{{\mathfrak {g}}/{\mathfrak {t}}})\,dg.}

To compute the determinant, we recall that g C = t C ⊕ ⨁ α g α {\displaystyle {\mathfrak {g}}_{\mathbb {C} }={\mathfrak {t}}_{\mathbb {C} }\oplus \bigoplus _{\alpha }{\mathfrak {g}}_{\alpha }} {\displaystyle {\mathfrak {g}}_{\mathbb {C} }={\mathfrak {t}}_{\mathbb {C} }\oplus \bigoplus _{\alpha }{\mathfrak {g}}_{\alpha }} where g α = { x ∈ g C ∣ Ad ⁡ ( t ) x = e α ( t ) x , t ∈ T } {\displaystyle {\mathfrak {g}}_{\alpha }=\{x\in {\mathfrak {g}}_{\mathbb {C} }\mid \operatorname {Ad} (t)x=e^{\alpha (t)}x,t\in T\}} {\displaystyle {\mathfrak {g}}_{\alpha }=\{x\in {\mathfrak {g}}_{\mathbb {C} }\mid \operatorname {Ad} (t)x=e^{\alpha (t)}x,t\in T\}} and each g α {\displaystyle {\mathfrak {g}}_{\alpha }} {\displaystyle {\mathfrak {g}}_{\alpha }} has dimension one. Hence, considering the eigenvalues of Ad g / t ⁡ ( t − 1 ) {\displaystyle \operatorname {Ad} _{{\mathfrak {g}}/{\mathfrak {t}}}(t^{-1})} {\displaystyle \operatorname {Ad} _{{\mathfrak {g}}/{\mathfrak {t}}}(t^{-1})}, we get:

det ( Ad g / t ⁡ ( t − 1 ) − I g / t ) = ∏ α > 0 ( e − α ( t ) − 1 ) ( e α ( t ) − 1 ) = δ ( t ) δ ( t ) ¯ , {\displaystyle \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)}},} {\displaystyle \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 α {\displaystyle \alpha } {\displaystyle \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 {\displaystyle W} {\displaystyle W} can be identified with a subgroup of GL ⁡ ( t C ∗ ) {\displaystyle \operatorname {GL} ({\mathfrak {t}}_{\mathbb {C} }^{*})} {\displaystyle \operatorname {GL} ({\mathfrak {t}}_{\mathbb {C} }^{*})}; in particular, it acts on the set of roots, linear functionals on t C {\displaystyle {\mathfrak {t}}_{\mathbb {C} }} {\displaystyle {\mathfrak {t}}_{\mathbb {C} }}. Let

A μ = ∑ w ∈ W ( − 1 ) l ( w ) e w ( μ ) {\displaystyle A_{\mu }=\sum _{w\in W}(-1)^{l(w)}e^{w(\mu )}} {\displaystyle A_{\mu }=\sum _{w\in W}(-1)^{l(w)}e^{w(\mu )}}

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

χ | T ⋅ δ = A μ {\displaystyle \chi |T\cdot \delta =A_{\mu }} {\displaystyle \chi |T\cdot \delta =A_{\mu }}.

To see this, we first note

  1. ‖ χ ‖ 2 = ∫ G | χ | 2 d g = 1. {\displaystyle \|\chi \|^{2}=\int _{G}|\chi |^{2}dg=1.} {\displaystyle \|\chi \|^{2}=\int _{G}|\chi |^{2}dg=1.}
  2. χ | T ⋅ δ ∈ Z [ Λ ] . {\displaystyle \chi |T\cdot \delta \in \mathbb {Z} [\Lambda ].} {\displaystyle \chi |T\cdot \delta \in \mathbb {Z} [\Lambda ].}

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

References

  1. Adams 1982, Theorem 6.1.
  2. Zhang 2014
  • 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
  • Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics 98, Springer-Verlag, Berlin, 1995.