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

TL;DR

This paper strengthens recent results on classical groups' degenerate principal series, providing a new proof of the Howe duality conjecture for symplectic-orthogonal and unitary pairs by expanding on previous findings.

Contribution

It offers an improved result for real parameters in the degenerate principal series, leading to a novel proof of the Howe duality conjecture for specific classical group pairs.

Findings

Strengthened the result for $s \\in \\mathbb{R}_{\\ge 0}$ in the degenerate principal series.

Provided a new proof of the Howe duality conjecture for symplectic-orthogonal and unitary pairs.

Expanded the understanding of the degenerate principle series in classical groups.

Abstract

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.