# Some results on reducibility of parabolic induction for classical groups

## TL;DR

This paper investigates conditions under which the parabolic induction of a representation of a classical group is reducible, providing criteria based on the supercuspidal support of the inducing representation.

## Contribution

It establishes a reducibility criterion for parabolic induction in classical groups based on the supercuspidal support, extending previous understanding of representation theory.

## Key findings

- Reducibility of $
ho\rtimes\sigma$ implies reducibility of $\pi\rtimes\sigma$ when $\rho$ is in the support of $\pi$.
- Provides irreducibility criteria in special cases where the support condition is not met.
- Enhances understanding of the structure of induced representations in classical groups.

## Abstract

Given a (complex, smooth) irreducible representation $\pi$ of the general linear group over a non-archimedean local field and an irreducible supercuspidal representation $\sigma$ of a classical group, we show that the (normalized) parabolic induction $\pi\rtimes\sigma$ is reducible if there exists $\rho$ in the supercuspidal support of $\pi$ such that $\rho\rtimes\sigma$ is reducible. In special cases we also give irreducibility criteria for $\pi\rtimes\sigma$ when the above condition is not satisfied.