Does the Haar measure change when switching representations on a Lie group?

[Luck0]

I'm not sure if this question belongs to here, but here it goes

Suppose you have to integrate over a lie group in the fundamental representation. If you pass to the adjoint representation of that group, does the Haar measure have to change? I think that it should not change because it is invariant under left and right translations, is it correct?

 

[dextercioby]

Integrate what?

 

[Luck0]

Integrate what?

Some function f: G -> ℝ, where G is the Lie group. For example, the Itzykson-Zuber integral

∫dUexp(-tr(XUYU^†)), where X, Y are n x n hermitean matrices and U ∈ U(n)

 

[fresh_42]

Maybe this can help you (section 4, 34 and a few more):
https://www.ams.org/open-math-notes/omn-view-listing?listingId=110677

 

[Luck0]

Maybe this can help you (section 4, 34 and a few more):
https://www.ams.org/open-math-notes/omn-view-listing?listingId=110677

This is very helpful, thanks!