Degenerate principal series and local theta correspondence III: The case of complex groups

Let (G′,G) be one of the following complex dual pairs: (O(m,C),Sp(2n,C)), (GL(m,C),GL(2n,C)), (Sp(2m,C),O(4n,C)). We determine the relationship between the space of G′-coinvariants and certain degenerate principal series representations of G. We also show that the space of G′-invariant distributions is generated by the Dirac distribution as a G-representation.