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 only a continuous function, then
The formula can be used to derive the Weyl character formula.

Derivation

Consider the map
The 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. Let
where 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
The property is precisely the orthogonality relations on irreducible characters.