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 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.)
Consider the map The Weyl group W acts on T by conjugation and on
be the quotient space by this W-action.
is free, the quotient map is a smooth covering with fiber W when it is restricted to regular points.
and the latter is a homeomorphism on regular points and so has degree one.
and, by the change of variable formula, we get: Here,
{\displaystyle q^{*}(f\,dg)|_{(gT,t)}=f(t)q^{*}(dg)|_{(gT,t)}}
is a class function.
We next compute
{\displaystyle q^{*}(dg)|_{(gT,t)}}
We identify a tangent space to
are the Lie algebras of
, t ) ) = Ad ( g )
{\displaystyle d(t\mapsto q(gT,t))=\operatorname {Ad} (g)}
Now, we can view G as a connected subgroup of an orthogonal group (as it is compact connected) and thus
det ( Ad ( g ) ) = 1
{\displaystyle \det(\operatorname {Ad} (g))=1}
Hence, To compute the determinant, we recall that
α
α
has pure imaginary value.
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
be the weight lattice of G relative to T. The Weyl character formula then says that: for each irreducible character
such that To see this, we first note The property (1) is precisely (a part of) the orthogonality relations on irreducible characters.