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 :
Moreover, is explicitly given as: where is the Weyl group determined by T and
the product running over the positive roots of G relative to T. More generally, if is an arbitrary integrable function, then
Note that the inner integral is over the manifold, the quotient of the group over the maximal torus, and is some Borel measure on this manifold.
The formula can be used to derive the Weyl character formula.
Derivation
Consider the mapThe Weyl group W acts on T by conjugation and on from the left by: for,
Let be the quotient space by this W-action. Then, since the W-action on is free, the quotient map
is a smooth covering with fiber W when it is restricted to regular points. Now, is followed by and the latter is a homeomorphism on regular points and so has degree one. Hence, the degree of is and, by the change of variable formula, we get:
Here, since is a class function. We next compute. We identify a tangent space to as where are the Lie algebras of. For each,
and thus, on, we have:
Similarly we see, on,. Now, we can view G as a connected subgroup of an orthogonal group and thus. Hence,
To compute the determinant, we recall that where and each has dimension one. Hence, considering the eigenvalues of, we get:
as each root 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 can be identified with a subgroup of ; in particular, it acts on the set of roots, linear functionals on. Letwhere is the length of w. Let be the weight lattice of G relative to T. The Weyl character formula then says that: for each irreducible character of, there exists a such that
To see this, we first note