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Jan 4, 2022 at 2:19 vote accept Andrew
Jan 2, 2022 at 14:13 answer added David Loeffler timeline score: 4
Jan 2, 2022 at 13:15 comment added Andrew @DavidLoeffler, thank you very much for the helpful comment! The definition of orthogonal (or symplectic) of $\pi_v$ I mean is that if $\phi: WD_F \to \operatorname{GL}(n)$ is the Weil-Deligne representation corresponding to $\pi_v$, then we say $\pi_v$ is orthogonal (or symplectic) if the image of $\phi$ can be replaced to $\operatorname{SO}(n)$ (or $\operatorname{Sp}(n)$.) So if $\pi$ is symplectic (orthogonal), then it seems $\pi_v$’s are symplectic (or orthogonal) for all places $v$.
Jan 2, 2022 at 8:03 comment added David Loeffler It's easy to see that ($\pi$ is either symplectic or orthogonal) $\Leftrightarrow$ ($\pi$ is selfdual) $\Leftrightarrow$ ($\pi_v$ is selfdual for all $v$), but it's not so obvious if the local factors are symplectic / orthogonal if $\pi$ is.
Jan 1, 2022 at 20:12 history edited LSpice CC BY-SA 4.0
`\operatorname`, `\bigotimes`, and backwards apostrophe
Jan 1, 2022 at 18:29 history edited Andrew CC BY-SA 4.0
edited title
Jan 1, 2022 at 18:18 history asked Andrew CC BY-SA 4.0