Weyl's integral formula

In mathematics , Weyl's integral formula or Weyl's integral formula is a formula for calculating the integral of functions on compact Lie groups, with which, in particular, the calculation of the integral of class functions can be reduced to an integration over the maximum torus. It is named after Hermann Weyl .

Let be a compact , connected Lie group , a maximal torus and a continuous function . Then ${\displaystyle G}$ ${\displaystyle T\subset G}$ ${\displaystyle f\colon G\to \mathbb {C} }$

${\displaystyle \int _{G}f(g)\,dg={\frac {1}{\sharp W}}\int _{T}\det(Id-Ad_{G/T}(t^{-1}))\int _{G/T}f(gtg^{-1})\,dg\,dt}$,

where the Weyl group of and means the restriction of the adjoint representation to the first summand of the -invariant decomposition . ${\displaystyle W}$ ${\displaystyle G}$ ${\displaystyle Ad_{G/T}\colon T\to Aut(T_{e}G/T)}$ ${\displaystyle Ad\mid _{T}}$ ${\displaystyle Ad\mid _{T}}$ ${\displaystyle {\mathfrak {g}}=T_{e}(G/T)\oplus {\mathfrak {t}}}$

In particular one obtains for a continuous class function

${\displaystyle \int _{G}f(g)\,dg={\frac {1}{\sharp W}}\int _{T}\det(Id-Ad_{G/T}(t^{-1}))f(t)\,dt}$,

one only needs to integrate over the maximum torus.

The proof follows from the properties of the by

${\displaystyle q(g,t)=gtg^{-1}}$

defined figure

${\displaystyle q\colon G/T\times T\to G}$,

namely

${\displaystyle \deg(q)=\sharp W}$

for the degree of mapping and

${\displaystyle \det(Dq(gT,t))=\det(Ad_{G/T}(t^{-1})-Id)}$

for the determinant of the differential of . ${\displaystyle q}$

literature

• T. Bröcker, T. tom Dieck: Representations of compact Lie groups. Springer Verlag New York 1985.
• M. Sepanski: Compact Lie groups. Springer Verlag New York 2007.