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Let $\mathcal{A}$ be a generalized Weyl or $q$-Weyl algebra, $M$ be an$\mathcal{A}$-$\overline{\mathcal{A}}$ bimodule. Choosing an automorphism$\rho$ of $\mathcal{A}$ we can define the notion of an invariant Hermitianform: $(au,v)=(u,v\rho(a))$ for all $a\in \mathcal{A}$ and $u,v\in M$. Papers[P. Etingof, D. Klyuev, E. Rains, D. Stryker. Twisted traces and positive formson quantized Kleinian singularities of type A. SIGMA 17 (2021), 029, 31 pages,arXiv:2009.09437] and [D. Klyuev. Twisted traces and positive forms ongeneralized q-Weyl algebras. SIGMA 18 (2022), 009, 28 pages, arXiv:2105.12652]obtained a classification of invariant positive definite Hermitian forms in thecase when $M=\mathcal{A}=\overline{\mathcal{A}}$, the case of the regularbimodule. We obtain a classification of invariant positive definite forms on$M$ in the case when $\mathcal{A}$ has no finite-dimensional representations.

Unitarizability of Harish-Chandra bimodules over generalized Weyl and $q$-Weyl algebras