The Second Moment of $\mathrm{GL}_3 \times \mathrm{GL}_2$ $L$-functions at Special Points

Let $\phi$ be a fixed Hecke--Maass form for $\mathrm{SL}_3 (\mathbb{Z})$ and $u_j$ traverse an orthonormal basis of Hecke--Maass forms for $\mathrm{SL}_2 (\mathbb{Z})$. Let $1/4+t_j^2$ be the Laplace eigenvalue of $u_j$. In this paper, we prove the mean Lindel\"of hypothesis for the second moment of $L (1/2+it_j, \phi \times u_j)$ on $T < t_j \leqslant T + \sqrt{T}$. Previously, this was proven by Young on $t_j \leqslant T$. Our approach is more direct as we do not apply the Poisson summation formula to detect the `Eisenstein--Kloosterman' cancellation.