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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.

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    $\begingroup$ It is easier to give a proof; every linear representation from the universal cover of $SL_2(\mathbb {R})$ into $GL_n(\mathbb {R})$ yields a complex representation of Lie algebras $\mathfrak {sl}_2(\mathbb {C})$ into $\mathfrak{gl}_n(\mathbb {C})$; since the group $SL_2(\mathbb {C})$ is simply connected, this extends to a holomorphic (algebraic) representation $SL_2(\mathbb {C}) \rightarrow GL_n(\mathbb {C}$; restricting , we get a representation $SL_2(\mathbb{R}\rightarrow GL_n(\mathbb {R})$. Hence the representation can never be faithful on any finite cover of $SL_2(\mathbb{R})$. $\endgroup$
    – Venkataramana
    Commented Jun 29, 2016 at 12:08
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    $\begingroup$ Also see the link: mathoverflow.net/questions/110208/… $\endgroup$
    – Venkataramana
    Commented Jun 29, 2016 at 12:11
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    $\begingroup$ This is addressed here and here. $\endgroup$
    – Sean Lawton
    Commented Jun 29, 2016 at 12:12
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    $\begingroup$ Thank you. Well, of course, it is not so difficult to give a proof. I thought that such a well known (and not obvious) fact should be proven in a textbook. $\endgroup$
    – Jarek Kędra
    Commented Jul 1, 2016 at 7:17
  • $\begingroup$ It's unclear that this result has a precise author, because I guess that the finite-dimensional representations of the corresponding Lie algebra were classified (middle 19th? Clebsch?) much before the universal covering of $\mathrm{SL}_2(\mathbf{R})$ was defined (no idea where it appeared first!), and the only ingredient of the proof beyond basic Lie theory is this classification of representations of the Lie algebra. $\endgroup$
    – YCor
    Commented Jul 12, 2016 at 12:43

2 Answers 2

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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).

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Within the same textbook (Hilgert and Neeb) look at Example 9.5.18 which deals with the universal cover $\widetilde{SL_2(\mathbb{R})}$ of $SL_2(\mathbb{R})$ the proof follows similarly for an arbitrary cover with non-trivial kernel.

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