I asked this on Math.SE and got no answer, so I'll try my luck here.
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ 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) for $\mu$ a measure on $G$ 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 $SL(2,\mathbb{R})$. Moreover, on compact groups the $q$-harmonic functions (for any $q$) are constants, so they coincide with $\mu$-harmonic functions for $\mu$ the Haar measure (which is equivalent, modulo a constant, to the Riemannian volume). Is there any similar (although probably weaker) relationship beyond the "trivial"/compact case?
