Conjectural decomposition of symmetric powers of automorphic representations for $\mathrm{GL}(n)$

TL;DR

This paper provides a conditional upper bound on the number of cuspidal summands in symmetric power lifts of automorphic representations for GL(n), assuming certain Langlands functoriality conjectures.

Contribution

It establishes a new conditional bound on symmetric power decompositions of automorphic representations, extending to cases with relaxed cuspidality assumptions.

Findings

Bound is independent of k for large k

Conditional on automorphy and cuspidality of symmetric powers

Extends analysis to non-cuspidal symmetric power lifts

Abstract

Let $\pi$ be a cuspidal automorphic representation for $\mathrm{GL}(n)$ over a number field. We establish a conditional upper bound on the number of cuspidal isobaric summands in the symmetric $k$-th power lift of $\pi$, assuming that the symmetric $m$-th power lift of $\pi$ is automorphic and cuspidal for all $m \leq k-1$, along with other specified Langlands functoriality conjectures. For sufficiently large $k$, the resulting bound is independent of the specific value of $k$. We further extend our study to cases in which the cuspidality assumptions on the symmetric power lifts are relaxed.