Let $G=O(p,q)$ and $M$ the locally symmetric space obtained by taking th symmetric space of $O(p,q)$ and quotienting by an arithmetic group $\Gamma$. In INTERSECTION NUMBERS OF CYCLES ON LOCALLY SYMMETRIC SPACES Kudla-Millson construct a bilinear pairing $$(-,-):H^i_c(M,\mathbb{C})\times H^{\text{dim}(M)-i}_{ct}(G,\mathcal{S}(V^n))^{\mathfrak{q}}_{\chi_m}\to \left(\text{Siegel modular forms of weight } \frac{p+q}{2}\right).$$
If we replace $G=U(p,q)$, then we instead Hermetian modular forms of weight $m$. My question is if we have some idea of what the image of this pairing is? The reason I ask is that I've read/heard people refer to the "Kudla-Millson lift" as a map from Siegel modular forms to $H^i_c(M)$ and that for sufficently large weight the lift is surjective (for example in these notes, at the begining of section 5). Lift (at least naively) sounds to me like the bilinear pairing $(,)$ might surjective (allowing us to give a section "the lift") but I've not seen this actually being claimed anywhere, which is why I ask.
