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There is a recent theorem by Kleiner (based on previous work by Colding & Minicozzi), that reads: If $G$ is a finitely generated group of polynomial growth, then for every $d$ the space of harmonic functions with polynomial growth of at most $d$ is finite dimensional. Kleiner later uses this to obtain a non-trivial representation of $G$.

Is an analogue for the setting of Lie groups known? that is, is it known whether for a polynomially growing (equivalently, type (R)) Lie group, the space of polynomially growing harmonic functions of some fixed rate is finite dimensional?

Edit: the measure setting is a probability measure $\mu$ such that:

  1. $\mu$ is adapted, in the sense that there is no proper subgroup $H$ such that $\mu(H)=1$.
  2. $\mu$ is compactly supported.
  3. $\mu$ is absolutely continuous w.r.t. the Haar measure.
  4. $\mu$ is symmetric.
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    $\begingroup$ "Harmonic function" on a group has no meaning; what makes sense is a harmonic function on a group endowed with a probability measure. And Kleiner's theorem is for the case of the uniform probability wrt a finite symmetric generating subset (probably it works on a more general setting, but maybe with some constraints anyway). In a locally compact (Lie or not) group, you need to be more specific about what you require about the probability measure (has density wrt Haar measure? compactly supported?), and also about the functions (continuous? measurable and $L^1$-loc?...). $\endgroup$
    – YCor
    Commented Jul 27, 2015 at 15:24
  • $\begingroup$ Well, I meant it as part of the question, but indeed I had in mind a compactly supported, absolutely continuous w.r.t. Haar, and such that there is no proper subgroup $H$ with $\mu(H)=1$ probability measure $\mu$. I don't know what restriction on the functions are necessary other than of course being measurable. $\endgroup$
    – Snoop Catt
    Commented Jul 27, 2015 at 18:06
  • $\begingroup$ Oh, and symmetric $\endgroup$
    – Snoop Catt
    Commented Jul 27, 2015 at 18:24
  • $\begingroup$ @YCor: I think that Corollary 8.2 of Colding & Minicozzi's "Harmonic functions on manifolds" holds the answer $\endgroup$
    – Snoop Catt
    Commented Jul 28, 2015 at 8:56
  • $\begingroup$ this paper is here math.uchicago.edu/~shmuel/SAMS2.pdf I couldn't find any definition of being harmonic there (one could imagine several choices, maybe they chose the one with the Laplacian, although it should also be specified if one considers smooth functions, distributions, or what - possibly this does not matter); I understand it relies on a choice of Riemannian metric (which in the Lie case is expected to be invariant), so I expect it addresses only the case of connected Lie groups (instead of general Lie groups, which include discrete groups). $\endgroup$
    – YCor
    Commented Jul 28, 2015 at 9:23

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