Casselman-Wallach property for homological theta lifting

In this paper, we establish the Casselman-Wallach property for homological theta lifting over archimedean local fields. As a consequence, the Euler-Poincaré characteristic is a well-defined element in the Grothendieck group of Casselman-Wallach representations. Our main tool is a corank-one parabolic stable filtration on the Weil representation.