Let $G=KAN$ be the Iwasawa decomposition of $G$ and $k(g):=k, a(g):=a$ be the corresponding projections onto $K$ resp. $A,g=kan$. Then I want to proof for any continuous $f:K\to\Bbb C$ that $$\int_K f(k)dk=\int_K f(k(gk))e^{-2\rho(\log (a(gk)))}dk.$$ I know that the Haar measure $dg$ on $G$ can be expressed as $$dg=e^{2\rho(\log(a))}dkdadn.$$ Here $2\rho$ denotes the sum of all positive $\mathfrak{a}$ roots.
Sadly I don't know how to conclude the result with this formula. Any hints will be gratefully appreciated.