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:
![{\displaystyle \int _{G}f(g)\,dg=\int _{T}f(t)u(t)\,dt.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/a2269b0533e51448e5c3f50165d211626f3b390c)
Moreover,
is explicitly given as:
where
is the Weyl group determined by T and
![{\displaystyle \delta (t)=\prod _{\alpha >0}\left(e^{\alpha (t)/2}-e^{-\alpha (t)/2}\right),}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/a52666e5b5e8efd7efc8d5e661ba1b86e25cd998)
the product running over the positive roots of G relative to T. More generally, if
is only a continuous function, then
![{\displaystyle \int _{G}f(g)\,dg=\int _{T}\left(\int _{G}f(gtg^{-1})\,dg\right)u(t)\,dt.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/0ede5c7c78f1ba26bca12d244a42158e9b6115f7)
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[edit]
Consider the map
.
The Weyl group W acts on T by conjugation and on
from the left by: for
,
![{\displaystyle nT(gT)=gn^{-1}T.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/98c36346b9f448548d4d087efc77cf117beab56b)
Let
be the quotient space by this W-action. Then, since the W-action on
is free, the quotient map
![{\displaystyle p:G/T\times T\to G/T\times _{W}T}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/24660cb71a4a849138cf82ad114fc60ea3a21311)
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:
![{\displaystyle \#W\int _{G}f\,dg=\int _{G/T\times T}q^{*}(f\,dg).}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/f9056e9933c10a60993ab962a047154bdd89095c)
Here,
since
is a class function. We next compute
. We identify a tangent space to
as
where
are the Lie algebras of
. For each
,
![{\displaystyle q(gv,t)=gvtv^{-1}g^{-1}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/b811746ff8df9de6e152391c1081eeede71d5b10)
and thus, on
, we have:
![{\displaystyle d(gT\mapsto q(gT,t))({\dot {v}})=gtg^{-1}(gt^{-1}{\dot {v}}tg^{-1}-g{\dot {v}}g^{-1})=(\operatorname {Ad} (g)\circ (\operatorname {Ad} (t^{-1})-I))({\dot {v}}).}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/7a4ee462924f11460f4b2411e60a601ac362da4b)
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,
![{\displaystyle q^{*}(dg)=\det(\operatorname {Ad} _{{\mathfrak {g}}/{\mathfrak {t}}}(t^{-1})-I_{{\mathfrak {g}}/{\mathfrak {t}}})\,dg.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/bd3746e5c2376acc3d187ca374489c1e8bab3e47)
To compute the determinant, we recall that
where
and each
has dimension one. Hence, considering the eigenvalues of
, we get:
![{\displaystyle \det(\operatorname {Ad} _{{\mathfrak {g}}/{\mathfrak {t}}}(t^{-1})-I_{{\mathfrak {g}}/{\mathfrak {t}}})=\prod _{\alpha >0}(e^{-\alpha (t)}-1)(e^{\alpha (t)}-1)=\delta (t){\overline {\delta (t)}},}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/424bade7e5d1bd71307e912ec4c89f30c9f9dbbf)
as each root
has pure imaginary value.
Weyl character formula[edit]
![[icon]](//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png) | This section needs expansion. You can help by adding to it. (April 2020) |
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
![{\displaystyle A_{\mu }=\sum _{w\in W}(-1)^{l(w)}e^{w(\mu )}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/eebde5e2dd5d2852223199d021753da644c27178)
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
![{\displaystyle \|\chi \|^{2}=\int _{G}|\chi |^{2}dg=1.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/e32991060311e73e929b487e86e49a987bbb60ad)
![{\displaystyle \chi |T\cdot \delta \in \mathbb {Z} [\Lambda ].}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/f6bcc57170ece1a15533e2e5773a9018cf8e2a92)
The property (1) is precisely (a part of) the orthogonality relations on irreducible characters.
References[edit]
- 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.