It is known that no nontrivial connected cover of $\operatorname{SL}(2,\mathbb R)$ admits a faithful finite dimensional linear representation (see, for example, page 143 in Fulton-Harris and Exercise 11.9 therein). I am looking for a reference with a proof of this fact and an information who observed it first.


**EDIT:** Textbook reference: Proposition 16.1.7 and Example 16.1.8 in Hilgert and Neeb *Structure and Geometry of Lie Groups*.



Reference for nonlinearity of covers of $\operatorname{SL}(2,\mathbb R)$
See two papers by Kubota: *Ein arithmetischer Satz über eine Matrizengruppe* (1966, MR0188194) and *Topological Covering of SL(2) Over a Local Field* (1967, MR0204422). Maybe these are the first references, although in isome sense it goes back (at least) to Weil's famous Acta paper *Sur certains groupes d'opérateurs unitaires* (1964, MR0165033).


