How many local factors does it take to determine a representation?

Abstract: In 1993, Henniart proved that the (then conjectural) Local Langlands Correspondence for $\operatorname{GL}(n$) is determined by gamma-factors of pairs. More precisely, he proved a local converse theorem: for $F$ a non-archimedean local field, an irreducible (smooth complex) representation $\pi$ of $\operatorname{GL}(n,F)$ is determined by the collection of gamma-factors of the pairs $(\pi,\tau)$ as $\tau$ runs through the irreducible representations of all $\operatorname{GL}(m,F)$ with $m < n$. More recently, Chai (2019) and Jacquet—Liu (2018) showed that one only needs to consider $m\leq n/2$ to determine $\pi$. This bound on $m$ is best possible, at least when $n$ is less that the residual characteristic of $F$, by work of Adrian—Liu—Stevens—Tam (2018) and Adrian (2023).

For groups other than $\operatorname{GL}(n)$ there are additional complications. I’ll explain what I know is already known about this problem and report on some joint work with Moshe Adrian, which gives some answers but also leaves many questions.

number theory

Audience: researchers in the topic

London number theory seminar

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