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

TL;DR

This paper characterizes the unitarity of good parity irreducible representations of split $p$-adic groups $ ext{SO}_{2n+1}$ and $ ext{Sp}_{2n}$, showing they are unitary if and only if of Arthur type, enabling explicit unitarity checks.

Contribution

It establishes a complete criterion linking unitarity and Arthur type for good parity representations of these groups, with an explicit algorithm for verification.

Findings

Unitarity of good parity representations is equivalent to being of Arthur type.

Provides an explicit algorithm for unitarity testing of irreducible representations.

Determines the set of unitary representations as local components of automorphic spectra.

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.