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Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-projectivized to give a representation of $\Gamma$? That is, do cocycles arising from projective representations of $G$ untwist when restricted to $\Gamma$?

For $G=PSL(2,\mathbb{R})$ and $\Gamma \cong F_m$, the free group on $m$ generators, this is true, by virtue of the fact that free groups have trivial Schur multiplier. The case that mainly interests me is $G=PGL(n,F)$, $n \geq 2$, $F$ a non-archimedean local field of characteristic $0$ and residue field of order relatively prime to $n$. For lattices in higher-rank simple Lie groups, we can always find some cocycle that does not untwist -- but perhaps those cocycles that come from projective representations of the ambient group $G$ do untwist in $\Gamma$?

(This question arose in the course of studying certain representations of group von Neumann algebras. I'm just getting started understanding cocycles and projective representations by reading Karpilovsky's "The Schur Multiplier" and Section 4.2 of Zimmmer's "Ergodic Theory and Semisimple Groups," so general tips on how to think about these things, or corrections to my use of the terminology, would be welcome!)

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  • $\begingroup$ are you considering finite dimensional projective representations? The answers may be different for finite or infinite dimensional representations. $\endgroup$ Jul 22, 2018 at 4:05
  • $\begingroup$ @Venkataramana, thank you, I should have specified: infinite-dimensional. I have edited the question. $\endgroup$
    – L.C. Ruth
    Jul 22, 2018 at 12:47

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