On the conjectural decomposition of symmetric powers of automorphic representations for GL(3) and GL(4)

TL;DR

This paper explores the structure of symmetric power lifts of automorphic representations for GL(3) and GL(4), providing bounds on the number of cuspidal summands under certain automorphy assumptions.

Contribution

It establishes bounds on the number of cuspidal components in symmetric power lifts of automorphic representations for GL(3) and investigates similar problems for GL(4).

Findings

Bound of 3 cuspidal summands for k ≥ 7 in GL(3) case

Bound of 2 cuspidal summands for k ≥ 19 with k ≡ 1 mod 3 in GL(3) case

Extension of the problem to GL(4) representations

Abstract

Given a cuspidal automorphic representation $\pi$ for GL(3) over a number field and a positive integer $k$, assume that the symmetric $m$th power lifts of $\pi$ are isobaric automorphic for $m \leq k$, cuspidal for $m \leq k-1$, and that certain associated Rankin-Selberg products are isobaric automorphic. Then the number of cuspidal isobaric summands in the $k$th symmetric power lift is bounded above by 3 when $k \geq 7$, and bounded above by 2 when $k \geq 19$ with $k \equiv 1 \pmod 3$. We then investigate the analogous problem for GL(4).