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(g) its universal enveloping algebra, let Ω be the Casimir element in U(g) and let f be a smooth (or analytic) real-valued function on G. We then have the following notions
1) for μ a measure on G we say f is μ-harmonic if f(g)=∫Gf(hg)dμ(h)
2) we say that f is g-harmonic if Ωf=0
3) for a left-invariant Riemannian metric q on G we say that f is q-harmonic if Δqf=0 (where Δq 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 Ω and Δq coincide, but among semisimple Lie groups only the compact ones have such metrics, so this leaves out basic examples like SL(2,R). Moreover, on compact groups the q-harmonic functions (for any q) are constants, so they coincide with μ-harmonic functions for μ 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?