Question

Let $F$ be a local field. Consider the group extension (split) $$ PSL(n,F) \rightarrow PGL(n,F) \rightarrow F^\times / (F^\times)^n.$$ What knowledge about $PGL(n)$ is necessary in order to understand the representation of $PSL(n)$ from this?

Answer 1

In the case of a non-Archimedean local field $F$, one may reduce the representation theory of $H={\rm SL}(n)$ to that of $G={\rm GL}(n)$. For instance supercuspidal representations of $H$ are obtained as constituents of the restriction to $H$ of the supercuspidal representations of $G$ (these restrictions are semisimple). In fact restriction of representations from $G$ to $H$ is an instance of Langlands functorialities. It corresponds to the natural projection between $L$-groups :

$$ {}^{\rm L}G ={\rm GL}(n,{\mathbb C})\longrightarrow {}^{\rm L}H={\rm PGL}(n,{\mathbb C}) $$

Representations of ${\rm PSL}(n)$ are just the representations of $H$ with trivial central character.

A good reference is :

MR1253507 (94k:22035) Bushnell, Colin J.; Kutzko, Philip C. The admissible dual of ${\rm SL}(N)$. II. Proc. London Math. Soc. (3) 68 (1994), no. 2, 317–379.

MR1209709 (94a:22033) Bushnell, Colin J.; Kutzko, Philip C. The admissible dual of ${\rm SL}(N)$. I. Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 2, 261–280.