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

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 $SL(n)$.