a microscopic state if $\Phi_j(m|z)$ is associated with a state in a discrete series. Notice that (\[mellin\]) has a meaning as long as $m-\bar m \in \mathbb Z$, a fact that we shall always assume.

After applying this formula to (\[2pt\]) we get the following expression for the two-point function involving only unflowed states, $$\label{2ptsl2}
\left\langle\Phi_{j_1}(m_1|z_1)\Phi_{j_2}(m_2|z_2)\right\rangle=\left[\delta\left(j_{12}^+\right)+Y^{-1-j_1}_{m_1}\delta\left(j_{12}\right)\right]\delta^2(m_1+m_2)\left|z_{12}\right|^{-4\Delta_{01}},$$ where $$Y^j_{m}=\frac{\pi B_j}{\gamma(-2j)}\frac{\gamma(-j-m)}{\gamma(1+j-m)},
  \qquad \qquad \gamma(x)=\frac{\Gamma(x)}{\Gamma(1-\bar x)},\nonumber$$ and $$\delta^2(m)=\int_{\mathbb C} d^2x \, \left|x^{m-1}\right|^2= 4\pi^2 \delta(m+\bar m)\delta_{m,\bar m}.\nonumber$$ In order to obtain Eq. (\[2ptsl2\]) we have repeatedly used the following complex extension of Euler integral: $$\label{euler}
\int_{\mathbb C} d^2x \left|x^{a-1}(1-x)^{b-1}\right|^2=\pi\frac{\gamma(a)\gamma(b)}{\gamma(a+b)},$$ and the identity $$\gamma(x)\gamma(1-\bar x)=1.\nonumber$$

The reflection symmetry also has a counterpart in the Lorentzian model. Indeed, Eq. (\[mellin\]) when applied to (\[reflexion\])-(\[intertwining\]) gives $$\label{reflem}
\Phi_j(m|z)=Y^{-1-j}_{m}\Phi_{-1-j}(m|z).$$ For $j\in\mathcal C$, this formula defines the intertwining between $\mathcal C_{j\alpha}$ and $\mathcal C_{-1-j,\alpha}$. For a real value of $j$, this expression lacks this interpretation and Eq. (\[reflem\]) is just a functional relation between the analytic continuations of vertex operators, [*i.e.*]{} microscopic states.

Correlation functions in the SL$(2,\mathbb R)$-WZNW model can violate spectral flow number conservation according to certain selection rules (see [@Maldacena:2001km] for more details), their computation being more involved than those with trivial total spectral flow charge. 