sections. We shall follow [@Teschner:1997ft] closely.

The spectrum $\mathcal V_{\mbox{\scriptsize H}_3^+}$ of the H$_3^+$-WZNW model carries a representation of two commuting isomorphic $\mathfrak{sl}(2,\mathbb C)$ current algebras generated by the modes $J^a_n$ and $\bar J^a_n$, with $a=+,0,-$ and $n\in\mathbb Z$. The holomorphic modes satisfy $$\begin{aligned}
&&\left[J^0_n,J^0_m\right]=-\frac{1}{2}k  n \delta_{n+m,0},\nonumber\\
&&\left[J^0_n,J^{\pm}_m\right]=\pm J^{\pm}_{n+m},\nonumber\\
&&\left[J^-_n,J^+_m\right]=2J^{0}_{n+m}+k n \delta_{n+m,0},\nonumber\end{aligned}$$ where $k$ denotes the level of the current algebra. Identical relations hold for the antiholomorphic generators as well.

As usual, there are two commuting Virasoro algebras in the universal enveloping algebra of the current algebra with generators $L_n$ and $\bar L_n$, with $n\in\mathbb Z$, defined according to the Sugawara construction, namely, $$L_n=\frac{1}{2(k-2)}\sum_{k\in\mathbb Z}:J^+_{-k}J^-_{n+k}+J^-_{-k}J^+_{n+k}-2J^0_{-k}J^0_{n+k}:,\nonumber$$ where the normal ordering is defined as $$:J^a_nJ^b_m:=\left\{\begin{array}{ll}
J^a_nJ^b_m&\mbox{if}~n<m,\\
\frac{1}{2}\left(J^a_nJ^b_n+J^b_nJ^a_n\right)&\mbox{if}~n=m,\\
J^b_mJ^a_n&\mbox{if}~n>m,
\end{array}\right.\nonumber$$ and correspondingly for the antiholomorphic sector. The central charge is given by $$c=\frac{3k}{k-2}.\nonumber$$

The space $\mathcal V_{\mbox{\scriptsize H}_3^+}$ decomposes into irreducible representations of the current algebra as $$\label{decomposition}
\mathcal V_{\mbox{\scriptsize H}_3^+}=\int^{\oplus}_{\mathcal C^+}dj\mathcal R_j,$$ where $\mathcal C^+=-1/2+i\mathbb R_+$. The module $\mathcal R_j$ is constructed standardly. As a first step, one considers $\mathcal P_j=P_j\otimes P_j$, where $P_j$ denotes the unitary principal series of the $\mathfrak{sl}(2,\mathbb C)$ algebra generated by the zero-modes. As pointed out in [@Teschner:1997ft], these principal series can be realized on the Schwartz 