Explicit Teichm\"uller curves with complemetary series

We construct an explicit family of arithmetic Teichm\"uller curves $\mathcal{C}_{2k}$, $k\in\mathbb{N}$, supporting $\textrm{SL}(2,\mathbb{R})$-invariant probabilities $\mu_{2k}$ such that the associated $\textrm{SL}(2,\mathbb{R})$-representation on $L^2(\mathcal{C}_{2k}, \mu_{2k})$ has complementary series for every $k\geq 3$. Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichm\"uller geodesic flow restricted to these explicit arithmetic Teichm\"uller curves $\mathcal{C}_{2k}$ has arbitrarily slow rate of exponential mixing.