symmetries, the vertex operators in the space-time picture are not even Virasoro quasi-primary fields. Finally, the form of the overall constant in the propagator could be relevant for describing transport properties of conformal field theories associated by duality with string physics in AdS$_3$, a subject that has recently aroused interest [@Polchinski:2012nh; @Hartnoll:2016apf]. Indeed, this constant admits a factorized form that is well suited for a proper normalization of spectral flowed vertex operators in the space-time scenario. This normalization become singular for some configurations suggesting that it could have an impact while studying singularities of the propagator and their interpretation in dual models.

The paper is organized as follows. In sections \[sect2\] and \[sect3\] we review some generalities of the H$_3^+$-WZNW model and its Lorentzian counterpart. We stress that by the latter we understand the WZNW model whose target space is the universal cover of the SL$(2,\mathbb R)$ group manifold. After introducing the spectra of both theories, we discuss the reflection symmetry in the Euclidean model and its emergence in the SL$(2,\mathbb R)$-WZNW model case, including the winding sector. In section \[sect4\], we introduce the vertex operators associated with spectrally flowed states and we review the computation of the regular term 