of the propagator in the space-time picture as done in [@Maldacena:2001km]. We also clarify some aspects regarding the definition of spectral flowed vertex operators in the $x$-basis. In section \[sect5\], we generalize the expression of the reflection symmetry for a sector with $\omega \ne 0$, and use it to explicitly compute the contact term of the propagator in section \[sect6\]. In section \[sect7\], we discuss the impact of the reflection symmetry on the integral definition of a vertex with a single spectral flow charge. Finally, in section \[disc\], we present our conclusions.

The H$_3^+$-WZNW model and the reflection symmetry {#sect2}
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Although the WZNW model describing string propagation in AdS$_3$ is the one based on the universal covering group of SL$(2,\mathbb R)$, some aspects of the theory can be read off more easily from its Euclidean counterpart, namely, the H$_3^+$-WZNW model. In this section, we review some basics of the H$_3^+$-WZNW model, as many of the formulas obtained in this context remain valid for the unflowed sector of the SL$(2,\mathbb R)$-WZNW model once the dependence on the space-time momentum is analytically continued. Most of these formulas will be properly generalized for sectors with non trivial spectral flow number in later 