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

TL;DR

This paper classifies invariant positive definite Hermitian forms on Harish-Chandra bimodules over generalized Weyl and $q$-Weyl algebras, extending understanding of unitarizability in these quantum algebraic structures.

Contribution

It provides a classification of invariant positive definite forms on Harish-Chandra bimodules over generalized Weyl and $q$-Weyl algebras for generic parameters.

Findings

Complete classification of invariant positive definite forms

Extension of unitarizability theory to generalized Weyl algebras

Results applicable to Coulomb branch quantizations

Abstract

Let $\mathcal{A}$ be a quantized ($K$-theoretic) BFN Coulomb branch with $G=\mathbb{C}^*$ and any $N$, that is, $\mathcal{A}$ is a generalized Weyl or $q$-Weyl algebra. Let $M$ be an $\mathcal{A}$-$\overline{\mathcal{A}}$ bimodule. Choosing an automorphism $\rho$ of $\mathcal{A}$ we can define the notion of an invariant Hermitian form: $(au,v)=(u,v\rho(a))$ for all $a\in \mathcal{A}$ and $u,v\in M$. We obtain a classification of invariant positive definite forms on $M$ in the case when $M$ is Harish-Chandra in the sense of Losev and quantization parameter is generic.