Normalized local theta correspondence and the duality of inner product formulas (Q1760378)

In the paper under review, the author gives a conjectural normalization of local theta correspondence in terms of the leading coefficient of certain weighted local integrals and verifies it in two cases. In more detail, let \(\mathrm{Mp}(W)\) be the 2-fold metaplectic cover of a symplectic group \(\mathrm{Sp}(W)\). Let \((G,H)\) be a reductive pair in \(\mathrm{Sp}(W)\) and \((MG, MH)\) be their preimage in \(\mathrm{MSp}(W)\). The author defines three types of weighted period integrals, which are integrating matrix coefficients of the oscillator representation of \(\mathrm{Mp}(W)\) and genuine representations of \(MG\) and \(MH\) against the weight function with a complex variable \(s\) introduced in the author's paper [Int. Math. Res. Not. 2012, No. 2, 239--298 (2012; Zbl 1238.22010)]. Then the author conjectures that those weighted period integrals have meromorphic continuation in the whole \(s\)-plane. In addition, using the generalized formal degree defined by the author, he gives a conjectural formula for the leading coefficients at \(s=0\) of those integrals. In particular, the author verifies this conjecture for the pairs \((\mathrm{SL}(2), \mathrm{PGL}(2))\) and \((\mathrm{SL}(2), \mathrm{SO}(2,2))\) over \(p\)-adic fields and the real field. As an application, the author gives global inner product formulas of global theta correspondences in these two cases.