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I believe that the Gelfand-Kirillov (GK) dimension for a generalized Weyl algebra $D(\sigma,a)$ is just the GKdim$(D) + 1$.

Does anyone have a reference for this?

I can find partial results, and I am sure this is implied by more general results regarding GK dimension, but I would like to have a precise reference for generalized Weyl algebras.

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This is addressed in the section "GK Dimension" of Ebrahim - The prime spectrum and representation theory of the 2×2 reflection equation algebra.

If the automorphism $\sigma$ is locally algebraic, then indeed $\operatorname{GKdim}(D(\sigma,a))=\operatorname{GKdim}(D)+1$.

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You can find your answer in Zhao, Mo, and Zhang - Gelfand–Kirillov dimension of generalized Weyl algebras, Lemma 3.2.

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    $\begingroup$ As the editor tells you, you should enter a link description when you include a link. I have edited in this case. $\endgroup$
    – LSpice
    Commented Jun 18, 2020 at 16:37

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