I asked [this][1] on Math.SE and got no answer, so I'll try my luck here.

Let $G$ be a semisimple real Lie group, $\mu$ its Haar measure, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued function on $G$. We then have the following notions

1) we say $f$ is $\mu$-harmonic if $f(g)=\int_Gf(hg)d\mu(h)$

2) we say that $f$ is $\mathfrak{g}$-harmonic if $\Omega f=0$

3) for a left-invariant Riemannian metric $q$ on $G$ we say that $f$ is $q$-harmonic if $\Delta_qf=0$ (where $\Delta_q$ is the Laplace-Beltrami operator associated to the metric $q$)

Questions: what, if any, are the relationships (i.e. logical implications) between these notions?

I do know that when $q$ is actually bi-invariant then $\Omega$ and $\Delta_q$ coincide, but among semisimple Lie groups only the compact ones have such metrics, so this leaves out basic examples like $SL(2,\mathbb{R})$.


  [1]: http://math.stackexchange.com/questions/1752043/harmonicity-on-semisimple-groups