A note on the multiplicity of $SL(n)$ over function fields

TL;DR

This paper explores the relationship between Lafforgue's decomposition of automorphic representations for $SL(n)$ over function fields and classical $L$-packet decompositions, providing bounds on multiplicities of extensions of unramified Hecke characters.

Contribution

It demonstrates that Lafforgue's decomposition for $SL(n)$ aligns with classical $L$-packet decomposition and establishes an upper bound on multiplicities based on extensions of unramified Hecke characters.

Findings

Lafforgue's decomposition coincides with classical $L$-packet decomposition for $SL(n)$.

Number of extensions bounds the multiplicity of $SL(n)$ representations.

Provides a natural upper bound on the multiplicity of $SL(n)$ in automorphic forms.

Abstract

In \cite{lafforgue2012chtoucas}, Vicent Lafforgue attaches a semisimple Langlands parameter (or, what amounts to the same thing, a $\hat{G}$-pseudocharacter) to every cuspidal automorphic representation of a reductive group $G$ over the field of functions of a smooth projective algebraic curve $X$ over a finite field. Hence, gets a decomposition of the space of cusp forms. In this note, we show that in the case of $G = SL(n)$, Lafforgue's decomposition coincides with the classical decomposition using $L$-packets, and moreover, the number of ($G$-equivalence classes of) extensions of an unramified Hecke character of $G$ to $\hat{G}$-pseudocharacters serves as a natural upper bound on the multiplicity of $SL(n)$.