On Classical groups detected by the tensor third representation

TL;DR

This paper investigates how the tensor third representation detects classical groups in the context of Langlands' functoriality, focusing on conditions involving partitions and Schur functors for specific group types.

Contribution

It provides new insights into the detection of classical groups by the tensor third representation related to Langlands' $L$-functions and Schur functors for certain group types.

Findings

Conditions on partitions for detection by tensor third representation

Analysis of detection for groups of types B, C, D

Connections to Langlands' functoriality and $L$-functions

Abstract

Motivated by the Langlands' beyond endoscopy proposal for establishing functoriality, we study the representation $\otimes^3$ in a setting related to the Langlands $L$-functions $L(s,\pi,\,\otimes^3),$ where $\pi$ is a cuspidal automorphic representation of $G$ where $G$ is either $\mathrm{SO}(2n+1)$, $\mathrm{Sp}(2n)$ and $\mathrm{SO}(2n)$. In particular, under what conditions on partitions $\lambda$, we examine whether or not $\otimes^3$ detects the subgroups $\mathbb{S}_{[\lambda]}(G)$ for $G$ with type $B_n$ and $D_{2n}$ or $\mathbb{S}_{\langle\lambda\rangle}(G)$ for $G$ with type $C_n$. Here $\mathbb{S}_{[\lambda]}$ and $\mathbb{S}_{\langle\lambda\rangle}$ are the usual Schur functors associated to the partition $\lambda$.