On unitarity of some representations of classical p-adic groups II

TL;DR

This paper investigates whether a certain correspondence in classical p-adic group representations preserves unitarizability, focusing on cases where the associated representation has the same infinitesimal character as a generalized Steinberg representation.

Contribution

It completes the proof that unitarizability is preserved under this correspondence when the associated representation shares the infinitesimal character with a generalized Steinberg representation.

Findings

Confirmed unitarizability preservation in specific cases

Extended previous partial results on the correspondence

Clarified conditions under which unitarizability is maintained

Abstract

C. Jantzen has defined a correspondence which attaches to an irreducible representation of a classical $p$-adic group, a finite set of irreducible representations of classical $p$-adic groups supported in a single or in two cuspidal lines (the case of the single cuspidal lines is interesting for the unitarizability). It would be important to know if this correspondence preserves the unitarizability (in both directions). The main aim of this paper is to complete the proof started in the previous paper of the fact that if we have an irreducible unitarizable representation $\pi$ of a classical $p$-adic group whose one attached representation $X_\rho(\pi)$ supported by a cuspidal line, has the same infinitesimal character as the generalized Steinberg representation supported in that line, then $X_\rho(\pi)$ is unitarizable.