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

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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

  1.  
  2.  

The property (1) is precisely (a part of) the orthogonality relations on irreducible characters.

References

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  1. ^ Adams 1982, Theorem 6.1.
  • 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.