Pullback formulae for nearly holomorphic Saito-Kurokawa lifts

We give explicit pullback formulae for nearly holomorphic Saito-Kurokawa lifts restrict to product of upper half-plane against with product of elliptic modular forms. We generalize the formula of Ichino to modular forms of higher level and free the restriction on weights. The explicit formulae provide non-trivial examples for the refined Gan-Gross-Prasad conjecture for $({\rm SO}_5,{\rm SO}_4)$ in the non-tempered cases. As an application, we obtain Deligne’s conjecture for critical values of certain automorphic $L$-functions for ${\mathrm{G}\mathrm{L}}_3\times {\mathrm{G}\mathrm{L}}_2$. We also expect to apply our pullback formulae to construct two-variables $p$-adic $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$ in the future.