Editing Maximal torus (section)

==Weyl integral formula==
Suppose ''f'' is a continuous function on ''G''. Then the integral over ''G'' of ''f'' with respect to the normalized Haar measure ''dg'' may be computed as follows:
: <math>\displaystyle{\int_G f(g)\, dg = |W|^{-1} \int_T |\Delta(t)|^2\int_{G/T}f\left(yty^{-1}\right)\,d[y]\, dt,}</math>
where <math>d[y]</math> is the normalized volume measure on the quotient manifold <math>G/T</math> and <math>dt</math> is the normalized Haar measure on ''T''.<ref>{{harvnb|Hall|2015}} Theorem 11.30 and Proposition 12.24</ref> Here  Δ is given by the [[Weyl denominator formula]] and <math>|W|</math> is the order of the Weyl group. An important special case of this result occurs when ''f'' is a [[class function]], that is, a function invariant under conjugation. In that case, we have
: <math>\displaystyle{\int_G f(g)\, dg = |W|^{-1} \int_T f(t) |\Delta(t)|^2\, dt.}</math>
Consider as an example the case <math>G=SU(2)</math>, with <math>T</math> being the diagonal subgroup. Then the Weyl integral formula for class functions takes the following explicit form:<ref>{{harvnb|Hall|2015}} Example 11.33</ref>

: <math>\displaystyle{\int_{SU(2)} f(g)\, dg = \frac{1}{2} \int_0^{2\pi} f\left(\mathrm{diag}\left(e^{i\theta}, e^{-i\theta}\right)\right)\, 4\,\mathrm{sin}^2(\theta)  \, \frac{d\theta}{2\pi}.}</math>
Here <math>|W|=2</math>, the normalized Haar measure on <math>T</math> is <math>\frac{d\theta}{2\pi}</math>, and <math>\mathrm{diag}\left(e^{i\theta}, e^{-i\theta}\right)</math> denotes the diagonal matrix with diagonal entries <math>e^{i\theta}</math> and <math>e^{-i\theta}</math>.