Unitary dual of $p$-adic split $\mathrm{SO}_{2n+1}$ and $\mathrm{Sp}_{2n}$: The good parity case (and slightly beyond)

Abstract

Let $F$ be a $p$-adic field, and let $G$ be either the split special orthogonal group $\mathrm{SO}_{2n+1}(F)$ or the symplectic group $\mathrm{Sp}_{2n}(F)$, with $n \geq 0$. We prove that a smooth irreducible representation of good parity of $G$ is unitary if and only if it is of Arthur type. Combined with the algorithms of the first author or Hazeltine-Liu-Lo for detecting Arthur type representations, our result leads to an explicit algorithm for checking the unitarity of any given irreducible representation of good parity. Finally, we determine the set of unitary representations that may appear as local components of the discrete automorphic spectrum.