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 (function invariant under conjugation by

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.}

Moreover,

u

{\displaystyle u}

is explicitly given as:

u

=

|

δ

|

2

/

#

W

{\displaystyle u=|\delta |^{2}/\#W}

where

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),}

the product running over the positive roots of G relative to T. More generally, if

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.}

Note that the inner integral is over the manifold

G

/

T

{\displaystyle G/T}

, the quotient of the group

G

{\displaystyle G}

over the maximal torus

T

{\displaystyle T}

, and

d

(

g

T

)

{\displaystyle 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

×

T

→

G

,

(

g

T

,

t

)

↦

g

t

g

−

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}

from the left by: for

n

T

∈

W

{\displaystyle nT\in W}

,

n

T

(

g

T

)

=

g

n

−

1

T

.

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

Let

G

/

T

×

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}

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}

is a smooth covering with fiber W when it is restricted to regular points. Now,

q

{\displaystyle q}

is

p

{\displaystyle p}

followed by

G

/

T

×

W

T

→

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}

is

#

W

{\displaystyle \#W}

and, by the change of variable formula, we get:

#

W

∫

G

f

d

g

=

∫

G

/

T

×

T

q

∗

(

f

d

g

)

.