Timeline for Unitary representations of lattices
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 24, 2019 at 11:13 | vote | accept | CommunityBot | ||
| Nov 23, 2019 at 13:07 | comment | added | YCor | (I implicitly assumed $F$ of char. zero; indeed the conclusion is different in finite char., as explained in Venkataramana's more complete answer.) | |
| Nov 23, 2019 at 9:25 | answer | added | Venkataramana | timeline score: 3 | |
| Nov 20, 2019 at 9:39 | comment | added | user130903 | Thanks! That helps. Cheers. | |
| Nov 20, 2019 at 9:23 | history | edited | YCor |
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| Nov 20, 2019 at 9:22 | comment | added | YCor | I also think that every representation $\Gamma\to\mathrm{GL}_n(\mathbf{C})$ has an image with compact closure, by superrigidity (while there clearly exists such a faithful representation for some $n$). | |
| Nov 20, 2019 at 9:19 | comment | added | YCor | Probably yes. Let $p$ be the residual characteristic of $F$. By arithmeticity, I think $\Gamma$ is commensurable with $H(\mathbf{Z}[1/p])$ for some semisimple $\mathbf{Q}$-group $H$ (I'm not sure arithmeticity says exactly this), with $G(F)$ isomorphic (up to finite index and finite kernel) to a cocompact direct factor in $H(\mathbf{Q}_p)$. Then $H(\mathbf{R})$ is compact, so inclusion $H(\mathbf{Z}[1/p])\subset H(\mathbf{R})\subset \mathrm{U}_n$ yield a faithful finite-dim unitary rep. | |
| Nov 20, 2019 at 9:05 | history | asked | user130903 | CC BY-SA 4.0 |