Timeline for Some basic question on the parabolic induction
Current License: CC BY-SA 4.0
8 events
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| Jul 15, 2018 at 5:13 | comment | added | Monty | @Tsai, Thank you very much! Your answer helped me a lot! If you don’t mind, would you please see my earlier question? I think you are the very person who can answer my question. My earlier question is below: mathoverflow.net/questions/304707/… | |
| Jul 15, 2018 at 0:07 | comment | added | Cheng-Chiang Tsai | @Monty Pardon that I probably shouldn't stress the adjective; it's nothing serious. When $F$ is non-archimedean (and our representation is over a field in which $p$ is invertible), by "nice" representation we definitely like a smooth representation. When $F=\mathbb{R}$ many know which category to work on much better than me. | |
| Jul 14, 2018 at 16:44 | comment | added | paul garrett | I think the canonical isomorphism is best understood via the associativity of tensor products (albeit over rings with many idempotents, but not units), for the non-archimedean case. This was P. Cartier's approach in his Corvallis notes. | |
| Jul 14, 2018 at 12:02 | comment | added | Monty | @Tsai, Thank you very much! What is the ‘nice’ do you mean? | |
| Jul 14, 2018 at 7:55 | comment | added | Cheng-Chiang Tsai | The two representations are canonically isomorphic. The general statement is the following: let $P=MN$ be a parabolic of $G$ and $P'=M'N'$ be a parabolic of $M$. Then the preimage of $P^*:=P'N=M'(N'N)$ is another parabolic of $G$ with $M'$ again as its Levi, and we'd like to say $\mathrm{Ind}_{P^*}^G\sigma=\mathrm{Ind}_P^G\mathrm{Ind}_{P'}^M\sigma$ for any (nice) representation of $M'$. This can be checked from definition. To get your situation we have $P=P_{n_1,n_2}$ and $P'=P_{a,b}\times P_{c,d}$. The same statement is true for normalized induction, as the modular characters will match up. | |
| Jul 14, 2018 at 7:54 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
typo in the title
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| Jul 14, 2018 at 7:21 | history | edited | Monty | CC BY-SA 4.0 |
edited title
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| Jul 14, 2018 at 6:59 | history | asked | Monty | CC BY-SA 4.0 |