Timeline for Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$
Current License: CC BY-SA 3.0
13 events
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| May 22, 2020 at 16:49 | comment | added | LSpice | @Kimball, I thought the same as you, but I think here $D$ stands for 'derived'. (I still don't think it's a good notation, though; something like $G_{\text{der}}$ or $DG$ or $\mathcal DG$ is probably better.) | |
| Feb 14, 2018 at 9:50 | comment | added | Kimball | Why the notation $G_D$? For me, $D$ is usually a division algebra. | |
| Feb 13, 2018 at 14:55 | vote | accept | D_S | ||
| Feb 13, 2018 at 13:53 | history | edited | Paul Broussous |
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| Feb 13, 2018 at 13:47 | answer | added | Paul Broussous | timeline score: 5 | |
| Feb 13, 2018 at 13:47 | comment | added | Paul Broussous | The representation $c-{\rm Ind}_{G_D}^G\, \pi$ is smooth but never admissible in the sense that $(c-{\rm Ind}_{G_D}^G\, \pi )^K$ is not finite dimensional for all compact open subgroups $K$. | |
| Feb 10, 2018 at 22:23 | comment | added | paul garrett | It may already be interesting to look at unramified principal series of $GL_2$ restricted to $SL_2$, where one can easily do the calculations directly. So, generically, irreducible u.r.p.s. stay irreducible, except at special parameters where they fall into two pieces. (This is quite parallel to the case of $GL_2$ and $SL_2$ over finite fields, too.) | |
| Feb 10, 2018 at 22:06 | history | edited | D_S | CC BY-SA 3.0 |
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| Feb 10, 2018 at 22:02 | comment | added | Not a grad student | As group schemes, $GL_n=SL_n . G_m$. | |
| Feb 10, 2018 at 21:58 | comment | added | D_S | Yes, you are right. | |
| Feb 10, 2018 at 21:47 | comment | added | Will Sawin | The claim that $GL_n(F)$ is generated by $SL_n(F)$ and the center $F^\times$ of $GL_n(F)$ seems wrong. In fact the quotient by the subgroup generated by those two is $F^\times / (F^\times)^n$. | |
| Feb 10, 2018 at 21:42 | history | edited | D_S | CC BY-SA 3.0 |
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| Feb 10, 2018 at 21:37 | history | asked | D_S | CC BY-SA 3.0 |