Let $G$ be a semisimple Lie group with Iwasawa decomposition $G=KAN$ and consider the determinant of the adjoint representation $\operatorname{Ad}$ of $AN$.
I want to determine what the derived representation looks like on $\mathfrak{a}$ (on $\mathfrak{n}$ it is obviously zero). I suspect that one can calculate this values using the root space decomposition w.r.t the root system $(\mathfrak{g},\mathfrak{a})$.
I know that the derivative of the determinant is the trace. Does this help? Thanks for any hints.
My question is answered in Knapps book Lie groups beyond an introduction on page 472 (Integration, Application to Reductive Lie Groups). The answer is $2\rho\log(a)$, where $\rho$ is the half sum of positive roots (counted with multiplicities).