Timeline for Plancherel measure and dimension
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 18, 2018 at 8:35 | comment | added | LSpice | The compact-group Plancherel formula is a general thing; it doesn't matter whether the group comes from an algebraic or Lie group, or in any other way, but is purely topological. I think you mean $\mu(\pi) = \dim(\pi)$ (no inverse). | |
| Dec 18, 2018 at 8:10 | comment | added | TheStudent | @LSpice Sorry for the lack of details, I modified the question to match this requirement, and these are exactly the representations I am interested in! | |
| Dec 18, 2018 at 8:07 | history | edited | TheStudent | CC BY-SA 4.0 |
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| Dec 18, 2018 at 7:48 | comment | added | LSpice | @TheStudent, quaternion algebras (assuming you mean their multiplicative groups) aren't semisimple; but you are right, that even their unit groups (or adjoint quotients) have non-trivial, finite-dimensional representations. I should have specified 'non-compact' for my $p$-adic claim. | |
| Dec 18, 2018 at 7:30 | history | edited | TheStudent | CC BY-SA 4.0 |
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| Dec 18, 2018 at 7:27 | comment | added | TheStudent | @LSpice What about representations of quaternion algebras at ramified places? | |
| Dec 18, 2018 at 6:13 | comment | added | LSpice | @Zero, it depends on the ground field! For the reals, there are probably lots. For the $p$-adics, a torus can have some, but a semisimple group has none. | |
| Dec 18, 2018 at 5:48 | comment | added | TheStudent | @Zero I modified the question to talk about compact groups, I am interested in this case and want to know if the same relation than for finite groups hold or not. | |
| Dec 18, 2018 at 5:47 | history | edited | TheStudent | CC BY-SA 4.0 |
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| Dec 18, 2018 at 3:31 | comment | added | user130903 | I don't think there are finite-dimensional unitary representations for a connected reductive group other than the trivial one. | |
| Dec 18, 2018 at 0:25 | history | asked | TheStudent | CC BY-SA 4.0 |