Ibukiyama correspondences on automorphic forms on ${\mathrm{Mp}}_4({\mathbb{A}}_{\mathbb{Q}})$ and ${\mathrm{SO}}_5({\mathbb{A}}_{\mathbb{Q}})$ generating large discrete series representations at the real place

In our previous paper we gave proofs of Ibukiyama’s correspondences on holomorphic Siegel modular forms of degree 2 of half-integral weight and integral weight. In this paper, we formulate and prove similar correspondences on automorphic forms on ${\mathrm{Mp}}_4({\mathbb{A}}_{\mathbb{Q}})$ or ${\mathrm{SO}}_5({\mathbb{A}}_{\mathbb{Q}})$ generating large discrete series representations at the real components. In addition, we show that the correspondences can be described in terms of local theta correspondences.