Mathematical 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
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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.
References
- 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.