A note on the Howe Duality conjecture for symplectic-orthogonal and unitary pairs

In this short note we expand on recent results on the degenerate principle series $I(s,\chi )$ of classical groups associated to $s\in \mathbb{C}$ and a quadratic character $\chi$. In particular, we strengthen the result for $s\in {\mathbb{R}}_{\ge 0}$, which allows us to give as a corollary a new proof of the Howe duality conjecture for symplectic-orthogonal and unitary pairs.