Discrete Series of Representations for the Modular Double of the Quantum Group $U_q(\operatorname{sl}(2,\mathbb{R}))$

The modular double of the quantum group $U_q(\operatorname{sl}(2))$ with deformation parameter $q=e^{i\pi\tau}$ is a natural object explicitly taking into account the duality $\tau\mapsto 1/\tau$. The use of the modular double in conformal field theory allows one to consider the region $1<c<25$ for the central charge of the Virasoro algebra when $|\tau|=1$. In this paper, a new discrete series of representations for the modular double of $U_q(\operatorname{sl}(2,\mathbb{R}))$ is found for such $\tau$.