Timeline for Branching laws for orthogonal groups
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Feb 24, 2017 at 20:47 | comment | added | paul garrett | Since the sum of the dimension of the $H=SO(n-1)$ and the dimension of the minimal parabolic $P$ in the (maximally split) $G=SO(n)$ is greater than or equal to the dimension of $G$, it is likely that the double coset space $P\backslash G/H$ is finite. So a Mackey-Bruhat computation should determine conditions for that space of intertwinings to be non-trivial. | |
| Feb 24, 2017 at 19:24 | comment | added | David Loeffler | That sounds very interesting, but how do I make it more concrete? If I know the Satake parameter of $\pi$, how do I tell if it's a theta lift? | |
| Feb 24, 2017 at 19:22 | history | edited | David Loeffler | CC BY-SA 3.0 |
added 51 characters in body
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| Feb 24, 2017 at 18:47 | comment | added | user104335 | This is a comment, but I don't have enough reputation points to comment. I think the representations of $\mathrm{SO}(n)$ that have nonzero periods over some embedded $\mathrm{SO}(n-1)$ are the ones that are lifts from $\widetilde{SL}_2$ via the theta correspondence. (So to be clear: I think the answer to your question is "Yes, precisely when $\pi$ is a theta lift from $\widetilde{SL}_2(\mathbf{Q}_p)$".) | |
| Feb 24, 2017 at 18:38 | history | asked | David Loeffler | CC BY-SA 3.0 |