Harmonicity on semisimple groups

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?

Answer

Attribution
Source : Link , Question Author : Timurlhan , Answer Author : Community

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