On the socles of certain parabolically induced representations of $p$-adic classical groups

TL;DR

This paper analyzes the socles of certain parabolically induced representations of p-adic classical groups, providing methods to compute their maximal semisimple subrepresentations and identify new unitary representations.

Contribution

It introduces a new approach to compute socles of induced representations from Speh and Arthur-type representations, aiding reducibility and unitarity analysis.

Findings

Computed socles for a class of induced representations

Identified many new unitary (complementary series) representations

Provided criteria for reducibility of these representations

Abstract

In this paper, we consider representations of $p$-adic classical groups parabolically induced from the products of shifted Speh representations and unitary representations of Arthur type of good parity. We describe how to compute the socles (the maximal semisimple subrepresentations) of these representations. As a consequence, we can determine whether these representations are reducible or not. In particular, our results produce many unitary representations, which are called complementary series.