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arXiv · 2604.10818

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

Kin Ming Tsang
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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.

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Kin Ming Tsang. 2026-04-12. Conjectural decomposition of symmetric powers of automorphic representations for $\mathrm{GL}(n)$. https://arxiv.org/abs/2604.10818

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