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[1] 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. (The theory of Verma modules, on the other hand, gives a purely algebraic derivation of 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 (as it is compact connected) 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 (1) is precisely (a part of) the orthogonality relations on irreducible characters.

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References

  1. Adams, Theorem 6.1.
  • Adams, J. F. (1969), Lectures on Lie Groups, University of Chicago Press
  • Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics 98, Springer-Verlag, Berlin, 1995.
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