In this talk we prove a conjecture of Kudla and Rallis on degenerate principal series, namely that all of its irreducible quotients appear with multiplicity one. In particular, we obtain a new proof of the conservation relation of the local Theta correspondence in Type I. The main idea is to consider a filtration of the parabolic reduction of a degenerate principal series, which is induced from a suitable stratification of the underlying Lagrangian Grassmanian.
This allows us to reduce the conjecture to the Howe correspondence in Type II. All of the presented results are work in progress.